US 20050250651 A1 Abstract A computer program product with computer program mechanism embedded therein is provided. The mechanism has instructions for initializing a quantum system, which includes a plurality of qubits, to an initialization Hamiltonian H
_{O}. The system is capable of being in one of at least two configurations at any give time including H_{O }and a problem Hamiltonian H_{P}. Each respective first qubit in the plurality of qubits is arranged with respect to a respective second qubit in the plurality of qubits such that the first respective qubit and the second respective qubit define a predetermined coupling strength. The predetermined coupling strengths between the qubits in the plurality of qubit collectively define a computational problem to be solved. The mechanism further comprises instructions for adiabatically changing the system until it is described by the ground state of the problem Hamiltonian H_{P }and instructions for reading out the state of the system. Claims(48) 1. A computer program product for use in conjunction with a computer system, the computer program product comprising a computer readable storage medium and a computer program mechanism embedded therein, the computer program mechanism comprising:
instructions for initializing a quantum system comprising a plurality of superconducting qubits to an initialization Hamiltonian H _{O}, wherein
the quantum system is capable of being in one of at least two configurations at any give time, the at least two configurations including:
a first configuration characterized by the initialization Hamiltonian H
_{O}, and a second configuration characterized by a problem Hamiltonian H
_{P}, and wherein each respective first superconducting qubit in said plurality of superconducting qubits is arranged with respect to a respective second superconducting qubit in the plurality of superconducting qubits such that the first respective superconducting qubit and the second respective superconducting qubit define a predetermined coupling strength and wherein the predetermined coupling strengths between each said first respective superconducting qubit and second respective superconducting qubit collectively define a computational problem to be solved; instructions for adiabatically changing the quantum system until it is described by the ground state of the problem Hamiltonian H _{P}; and instructions for reading out the state of the quantum system. 2. The computer program product of 3. A computer program product for use in conjunction with a computer system, the computer program product comprising a computer readable storage medium and a computer program mechanism embedded therein, the computer program mechanism for determining a quantum state of a first target superconducting qubit in a plurality of superconducting qubits,.the computer program mechanism comprising:
instructions for initializing a plurality of superconducting qubits so that they are described by a problem Hamiltonian, wherein the problem Hamiltonian describes (i) the quantum state of the plurality of superconducting qubits and (ii) each coupling energy between qubits in the plurality of qubits, and wherein the problem Hamiltonian is at or near a ground state; instructions for adding an rf-flux to the first target superconducting qubit, wherein the rf-flux has an amplitude that is less than one flux quantum; and instructions for adiabatically varying an amount of an additional flux in the first target superconducting qubit and observing a presence or an absence of a dip in a voltage response of a tank circuit that is inductively coupled with the first target superconducting qubit during a time when said instructions for adiabatically varying are executing 4. The computer program product of 5. The computer program product of 6. The computer program product of 7. The computer program product of 8. The computer program product of 9. The computer program product of 10. The computer program product of 11. The computer program product of 12. The computer program product of 13. The computer program product of 14. The computer program product of 15. The computer program product of 16. The computer program product of 17. The computer program product of 18. A computer system for determining a quantum state of a first target superconducting qubit in a plurality of superconducting qubits, the computer system comprising:
a central processing unit; a memory, coupled to the central processing unit, the memory storing instructions for biasing all or a portion of the qubits in the plurality of superconducting qubits other than the first target superconducting qubit, wherein a problem Hamiltonian describes (i) the biasing on the qubits in the plurality of superconducting qubits and (ii) each coupling energy between respective superconducting qubit pairs in the plurality of superconducting qubits, and wherein the problem Hamiltonian is at or near a ground state; instructions for adding an rf-flux to the first target superconducting qubit, wherein the rf-flux has an amplitude that is less than one flux quantum; and instructions for adiabatically varying an amount of an additional flux in the first target superconducting qubit and observing a presence or an absence of a dip in a voltage response of a tank circuit that is inductively coupled with the first target superconducting qubit during a time when said instructions for adiabatically varying are executed. 19. A structure for adiabatic quantum computing comprising:
a plurality of superconducting qubits, wherein said plurality of superconducting qubits are capable of being in any one of at least two configurations at any give time, the at least two configurations including: a first configuration characterized by an initialization Hamiltonian H _{0}, and a second Hamiltonian characterized by a problem Hamiltonian H_{P}, the problem Hamiltonian having a ground state, wherein each respective first superconducting qubit in said plurality of superconducting qubits is coupled with a respective second superconducting qubit in the plurality of superconducting qubits such that the first respective superconducting qubit and the corresponding second respective superconducting qubit define a predetermined coupling strength and wherein the predetermined coupling strength between each said first respective superconducting qubit and corresponding second respective superconducting qubit collectively defines a computational problem to be solved; and a tank circuit inductively coupled to all or a portion of said plurality of superconducting qubits. 20. The structure of 21. The structure of 22. The structure of 23. The structure of the superconducting material of the pancake coil is niobium, and there is a spacing of 1 about micrometer between the first turn and the second turn of the pancake coil. 24. The structure of 25. The structure of 26. The structure of 27. The structure of 28. The structure of _{T }that is determined by the equality: f _{T}=ω_{T}/2π=1/{square root}{square root over (L_{T}C_{T})} wherein
L
_{T }is an inductance of the tank circuit; and C
_{T }is a capacitance of the tank circuit. 29. The structure of 30. The structure of 31. The structure of 32. The structure of 33. The structure of 34. The structure of 35. The structure of 36. The structure of _{DC}. 37. The structure of _{RF }of a frequency co close to the resonance frequency ω_{0 }of the tank circuit. 38. The structure of _{E}, is Φ _{E}=Φ_{DC}+Φ_{RF } wherein
Φ
_{RF }is an amount of applied magnetic flux contributed to the superconducting qubit by the alternating current I_{RF}; and Φ
_{DC }is an amount of applied magnetic flux that is determined by the direct bias current I_{DC}. 39. The structure of Φ _{DC}=Φ_{A} +f(t)Φ_{0}, wherein,
Φ
_{0 }is one flux quantum; f(t)Φ
_{0 }is constant or is slowly varying and is generated by the direct bias current I_{DC}; and Φ _{A} =B _{A} ×L _{Q}, wherein,
B
_{A }is a magnitude of the magnetic field applied on the superconducting qubit by the means for applying the magnetic field; and L
_{Q }is an inductance of the superconducting qubit. 40. The structure of 41. The structure of 42. The structure of 43. The structure of _{RF }has a magnitude between about 10^{−5}Φ_{0 }and about 10^{−1}Φ_{0}. 44. The structure of _{A}, and/or Φ_{RF}. 45. The structure of _{RF }in accordance with a small amplitude fast function. 46. The structure of _{RF }in accordance with a small amplitude fast function is a microwave generator that is in electrical communication with said tank circuit. 47. The structure of an amplifier connected across the tank circuit; and means for measuring a total impedance of the tank circuit, expressed through the phase angle χ between driving current I _{RF }and the tank voltage. 48. The structure of Description This application claims benefit, under 35 U.S.C. § 119(e), of U.S. Provisional Patent Application No. 60/557,748, filed on Mar. 29, 2004, which is hereby incorporated by reference in its entirety. This application also claims benefit, under 35 U.S.C. § 119(e), of U.S. Provisional Patent Application No. 60/588,002, filed on Jul. 13, 2004, which is hereby incorporated by reference in its entirety. This application is further related to concurrently filed application Ser. No. ______, Attorney Docket No. 706700-999193, entitled “Adiabatic Quantum Computation with Superconducting Qubits,” and application Ser. No. ______, Attorney Docket No. 706700-999207, entitled “Adiabatic Quantum Computation with Superconducting Qubits,” each of which is hereby incorporated by reference in its entirety. This invention relates to superconducting circuitry. More specifically, this invention relates to devices for quantum computation. Research on what is now called quantum computing may have begun with a paper published by Richard Feynman. See Feynman, 1982, A major activity in the quantum computing art is the identification of physical systems that can support quantum computation. This activity includes finding suitable qubits as well as developing systems and methods for controlling such qubits. As detailed in the following sections, a qubit serves as the basis for performing quantum computation. The physical systems that are used in quantum computing are quantum computers. A quantum bit or “qubit” is the building block of a quantum computer in the same way that a conventional binary bit is a building block of a classical computer. A qubit is a quantum bit, the counterpart in quantum computing to the binary digit or bit of classical computing. Just as a bit is the basic unit of information in a classical computer, a qubit is the basic unit of information in a quantum computer. A qubit is conventionally a system having two or more discrete energy states. The energy states of a qubit are generally referred to as the basis states of the qubit. The basis states of a qubit are termed the |0> and |1> basis states. In the mathematical modeling of these basis states, each state is associated with an eigenstate of the sigma-z (σ The state of a qubit can be in any superposition of two basis states, making it fundamentally different from a bit in an ordinary digital computer. A superposition of basis states arises in a qubit when there is a non-zero probability that the system occupies more than one of the basis states at a given time. Qualitatively, a superposition of basis states means that the qubit can be in both basis states |0> and |1> at the same time. Mathematically, a superposition of basis states means that the wave function that characterizes the overall state of the qubit, denoted |Ψ>, has the form
To complete a quantum computation using a qubit, the state of the qubit is typically measured (e.g., read out). When the state of the qubit is measured the quantum nature of the qubit is temporarily lost and the superposition of basis states collapses to either the |0> basis state or the |1> basis state, thus regaining its similarity to a conventional bit. The actual state of the qubit after it has collapsed depends on the amplitudes a and b immediately prior to the readout operation. A survey of exemplary physical systems from which qubits can be formed is found in Braunstein and Lo (eds.), Superconducting qubits generally fall into two categories; phase qubits and charge qubits. Phase qubits store and manipulate information in the phase states of the device. Charge qubits store and manipulate information in the elementary charge states of the device. In superconducting materials, phase is a property of the material whereas elementary charges are represented by pairs of electrons called Cooper pairs. The division of such devices into two classes is outlined in Makhlin et al., 2001, “Quantum-State Engineering with Josephson-Junction Devices,” Phase and charge are related values in superconductors and, at energy scales where quantum effects dominate, the Heisenberg uncertainty principle causes certainty in phase to lead to uncertainty in charge and, conversely, causes certainty in charge to lead to uncertainty in the phase of the system. Superconducting phase qubits are devices formed out of superconducting materials having a small number of distinct phase states and many charge states, such that when the charge of the device is certain, information stored in the phase states becomes delocalized and evolves quantum mechanically. Therefore, fixing the charge of a phase qubit leads to delocalization of the phase states of the qubit and subsequent useful quantum behavior in accordance with well-known principles of quantum mechanics. Experimental realization of superconducting devices as qubits was made by Nakamura et al., 1999, Superconducting qubits have two modes of operation related to localization of the states in which information is stored. When the qubit is initialized or measured, the information is classical, 0 or 1, and the states representing that classical information are also classical in order to provide reliable state preparation. Thus, a first mode of operation of a qubit is to permit state preparation and measurement of classical information. A second mode of operation occurs during quantum computation, where the information states of the device become dominated by quantum effects such that the qubit can evolve controllably as a coherent superposition of those states and, in some instances, even become entangled with other qubits in the quantum computer. Thus, qubit devices provide a mechanism to localize the information states for initialization and readout operations, and de-localize the information states during computation. Efficient functionality of both of these modes and, in particular, the transition between them in superconducting qubits is a challenge that has not been satisfactorily resolved in the prior art. A proposal to build a quantum computer from superconducting qubits was published in 1997. See Bocko et al., 1997, The superconducting phase qubit is well known and has demonstrated long coherence times. See, for example, Orlando et al., 1999, The difference between any two Josephson junctions in the persistent current qubit is characterized by a coefficient, termed α, which typically ranges from about 0.5 to about 1.3. In some instances, the term α for a pair of Josephson junctions in the persistent current qubit is the ratio of the critical current between the two Josephson junctions in the pair. The critical current of a Josephson junction is the minimum current through the junction at which the junction is no longer superconducting. That is, below the critical current, the junction is superconducting whereas above the critical current, the junction is not superconducting. Thus, for example, the term α for junctions Referring to Persistent current qubit When wells There are broad classes of condensed matter systems that have states defined by the presence and absence of extra charge, or the excess charge exists in either a ground or an excited state. Such systems are diverse and have long held theoretical and experimental interest, e.g. Millikan, 1911, Phys. Rev. 32, pp. 349-397, which is hereby incorporated by reference in its entirety. There has been attention directed to semiconductor systems such as quantum dots. Research has been conducted using single particle electronics because they hold promise for conventional computers. Subsequently, proposals were made for these systems as quantum computers. While these systems collectively could be called charge qubits, this term is reserved herein for superconducting qubits. Specifically, a superconducting charge qubit has as basis states the presence (charge=2e, or some multiple thereof of 2e) or absence (charge=0) of charge on a small superconducting island. For charge qubits, the Coulomb energy E A charge qubit is a small (mesoscopic) island of superconductor separated by a Josephson junction from a large superconductor (reservoir), see Computer scientists concerned with complexity routinely use the definitions of different complexity classes. The number of complexity classes is ever changing, as new ones are defined and existing ones merge through advancements made in computer science. The complexity classes known as non-deterministic polynomial-time (NP), NP-complete (NPC), and NP-hard (NPH) are all classes of decision problems. Decision problems have binary outcomes. Problems in NP are computational problems for which there exists polynomial time verification. That is, it takes no more than polynomial time (class P) in the size of the problem to verify a potential solution. It may take more than polynomial time to create a potential solution. NP-hard problems take longer to verify a potential solution. For each NP-hard problem, there is an NP-complete problem that can be reduced to the NP-hard problem. However, NP-complete problems that can be reduced to a NP-hard problem do not enjoy polynomial time verification. Problems in NPC can be defined as problems in NP that have been shown to be equivalent to, or harder to solve, than a known problem in NPC. Equivalently, the problems in NPC are problems in NP that are also in NPH. This can be expressed as NPC=NP ∩ NPH. A problem is equivalent, or harder to solve, than a known problem in NPC if there exists a polynomial time reduction to the instant problem from the known problem in NPC. Reduction can be regarded as a generalization of mapping. The mappings can be one to one functions, many to one functions, or make use of oracles, etc. The concepts of complexity classes and how they define the intractability of certain computational problems is found in, for example, Garey and Johnson, 1979, Analogous to the way a classical computer is built using wires and logic gates, a quantum computer can be built using quantum circuits comprised of “wires” and “unitary gates.” Here, the wire is not a physical entity. Rather, it represents the state of the qubit in time. The “unitary gates” are applied at precise times to specific qubits to effect evolution of the qubit in accordance with the circuit model for quantum computing. The circuit model of quantum computing is a standard and universal model used by many practitioners in the art. The circuit model is universal in the sense that it is able to convert any input state into any output state. The elements of the circuit model are that a small set of one- and two-qubit unitary gates are applied to the qubits with precise timing. The circuit model of quantum computing can implement algorithms such as Shor's algorithm for factoring numbers or Grover's algorithm for searching databases. Shor's algorithm provides an exponential speedup relative to classical (non-quantum) computers for factoring numbers. Grover's application provides a polynomial speed up relative to classical computers for searching databases. See, for example, Nielsen and Chuang, 2000, An example of the circuit model is shown in The following subsections discuss the adiabatic theorem of quantum mechanics and introduce adiabatic quantum computing. One definition of an adiabatic process is a process that occurs in a system without heat entering or leaving the system. There exists a theorem in quantum mechanics that provides a suitable framework for such processes. The adiabatic theorem of quantum mechanics has several versions but a notable element of many such versions is as follows. A quantum system prepared in its ground state will remain in the ground state of the various instantaneous Hamiltonians through which it passes, provided the changes are made sufficiently slowly. This form of change is termed adiabatic change. Such a system is adiabatic because the population of the various states of the quantum system has not been altered as a result of the change. Hence, if the populations have not changed, the temperature of the system has not changed, and therefore no heat has entered or left the system. In 2000, a form of quantum computing, termed adiabatic quantum computing, was proposed. See, for example, Farhi et al., 2001, Science 292, pp. 472-475, which is hereby incorporated by reference in its entirety. In adiabatic quantum computing (AQC), the problem to be solved is encoded into a physical system such that departures from the solution to the problem incur a net energy cost to the system. AQC is universal in that it is able to convert any input state into any output state. However, unlike the circuit model of quantum computing, there is no application of a predetermined set of one- and two-qubit unitary gates at precise times. It is believed that AQC can be used to find solutions to some problems with greater efficiency than the circuit model. Such problems include problems contained in, and related to, the NP, NP-hard, and NP-complete classes. As shown in One computational problem that can be solved with adiabatic quantum computing is the The -
- INSTANCE: Graph G=(V, E), weight w(e) ε Z
^{+}for each e ε E, for positive integer K - QUESTION: Is there a partition of V into disjoint sets V
_{1 }and V_{2 }such that the sum of the weights of the edges form E that have one endpoint in V_{1 }and one endpoint in V_{2 }is at least K?
- INSTANCE: Graph G=(V, E), weight w(e) ε Z
Consider an instance of a positive number K and a graph G=(V, E), having a set of vertices V={v Mathematically, solving One computational problem that can be solved with adiabatic quantum computing is the -
- INSTANCE: Graph G=(V, E), positive integer K≦|V|.
- QUESTION: Does G contain an independent set of size K or more, i.e., as subset of V′
__⊂__V with |V′|≧K such that no two vertices in V′ are joined by an edge in E? where emphasis is added to show differences between theINDEPENDENT SET problem and another problem, known asCLIQUE , that is described below. Expanding upon this definition, consider an undirected edge-weighted graph having a set of vertices and a set of edges, and a positive integer K that is less than or equal to the number of vertices of the graph. TheINDEPENDENT SET problem, expressed as a decision problem, asks whether there is a subset of vertices of size K, such that no two vertices in the subset are connected by an edge of the graph. Many other permutations of the problem exist and include optimization problems based on this decision problem. An example of an optimization problem is the identification of the independent set of the graph that yields the maximum K. This is calledMAX INDEPENDENT SET .
Mathematically, solving Mathematically, solving -
- INSTANCE: Graph G=(V, E), positive integer K≦|V|.
- QUESTION: Does G contain a clique of size K or more, i.e., as subset of V′
__⊂__V with |V′|≧K such that every two vertices in V′ are joined by an edge in E? Here, emphasis has been added to show differences betweenCLIQUE andINDEPENDENT SET . It can also be shown howCLIQUE is related to the problemVERTEX COVER . Again, all problems in NP-complete are reducible to each other within polynomial time, making devices that solve one NP-complete problem efficiently, useful for other NP-complete problems.
The question of whether superconducting qubits can be used to implement adiabatic quantum computing (AQC) has been posed in the art. However, such proposals are unsatisfactory because they either lack enabling details on the physical systems on which AQC would be implemented or they rely on qubits that have not been shown to successfully perform an n-qubit quantum computation, where n is greater than 1 and the quantum computation requires entanglement of qubits. For example, Kaminsky and Lloyd, 2002, “Scalable Architecture for Adiabatic Quantum Computing of NP-Hard Problems,” in Accordingly, given the above background, there is a need in the art for improved systems and methods for adiabatic quantum computing. Discussion or citation of a reference herein shall not be construed as an admission that such reference is prior art to the present invention. The present invention addresses the need in the art for improved systems and methods for adiabatic quantum computing. In some embodiments of the present invention, a graph based computing problem, such as In one aspect of the present invention, the plurality of qubits that represents the graph is initialized to a first state that does not permit the qubits to quantum tunnel. Then, the plurality of qubits is set to an intermediate state in which quantum tunneling between individual basis states within each qubit in the plurality of qubits can occur. In preferred embodiments, the change to the intermediate state occurs adiabatically. In other words, for any given instant t that occurs during the change to the intermediate state or while the qubits are in the intermediate state, the plurality of qubits are in the ground state of an instantaneous Hamiltonian that describes the plurality of qubits at the instant t. The qubits remain in the intermediate state that permits quantum tunneling between basis states for a period of time that is sufficiently long enough to allow the plurality of qubits to reach a solution for the computation problem represented by the plurality of qubits. Once the qubits have been permitted to quantum tunnel for a sufficient period of time, the state of the qubits is adjusted such that they reach some final state that either does not permit quantum tunneling or, at least, does not permit rapid quantum tunneling. In preferred embodiments, the change to the final state occurs adiabatically. In other words, for any given instant t that occurs during the change to the final state, the plurality of qubits are in the ground state of an instantaneous Hamiltonian that describes the plurality of qubits at the instant t. In other examples of the systems and methods of the present invention, the plurality of qubits that represents the graph is initialized to a first state that does permit the qubits to quantum tunnel. The state of the quantum system is changed once the qubits have been permitted to quantum tunnel for a sufficient period of time. The state of the qubits is adjusted such that they reach some final state that either does not permit quantum tunneling or, at least, does not permit rapid quantum tunneling. In preferred embodiments, the change to the final state occurs adiabatically. Some embodiments of the present invention are universal quantum computers in the adiabatic quantum computing model. Some embodiments of the present invention include qubits with single-qubit Hamiltonian terms and at least one two-qubit Hamiltonian term. A first aspect of the invention provides a method for quantum computing using a quantum system comprising a plurality of superconducting qubits. The quantum system is characterized by an impedance. Also, the quantum system is capable of being in any one of at least two configurations at any given time. These at least two configurations include a first configuration characterized by an initialization Hamiltonian H In some embodiments in accordance with the first aspect of the invention, the reading step comprises measuring an impedance of the quantum system. In some embodiments the reading step comprises determining a state of a superconducting qubit in the plurality of superconducting qubits. In some embodiments, the reading step differentiates a ground state of the superconducting qubit from an excited state of the superconducting qubit. In some embodiments, a superconducting qubit in the plurality of superconducting qubits is a persistent current qubit. In some embodiments, the reading step measures a quantum state of the superconducting qubit as a presence or an absence of a voltage. In some embodiments, a superconducting qubit in the plurality of superconducting qubits is capable of tunneling between a first stable state and a second stable state when the quantum system is in the first configuration. In some embodiments, a superconducting qubit in the plurality of superconducting qubits is capable of tunneling between a first stable state and a second stable state during the adiabatic changing step. In some embodiments, the adiabatic changing step occurs during a time period that is between 1 nanosecond and 100 microseconds. In some embodiments, the initializing step includes applying a magnetic field to the plurality of superconducting qubits in the direction of a vector that is perpendicular to a plane defined by the plurality of superconducting qubits. In some embodiments, a superconducting qubit in the plurality of superconducting qubits is a persistent current qubit. A second aspect of the invention provides a method for quantum computing using a quantum system that comprises a plurality of superconducting qubits. The quantum system is coupled to an impedance readout device. The quantum system is capable of being in any one of at least two configurations at any given time. The at least two configurations include a first configuration characterized by an initialization Hamiltonian H In some embodiments in accordance with this second aspect of the invention, the reading step measures a quantum state of a superconducting qubit in the plurality of superconducting qubits as a presence or an absence of a voltage. In some embodiments, the reading step differentiates a ground state of the superconducting qubit from an excited state of the superconducting qubit. In some embodiments, a superconducting qubit in the plurality of superconducting qubits is (i) a phase qubit in the charge regime or (ii) a persistent current qubit. In some embodiments, a superconducting qubit in the plurality of superconducting qubits is capable of tunneling between a first stable state and a second stable state when the quantum system is in the first configuration. In some embodiments, a superconducting qubit in the plurality of superconducting qubits is capable of tunneling between a first stable state and a second stable state during the adiabatic changing step. In some embodiments, the adiabatic changing step occurs during a time period that is greater than 1 nanosecond and less than 100 microseconds. In some embodiments, the initializing step includes applying a magnetic field to the plurality of superconducting qubits in the direction of a vector that is perpendicular to a plane defined by the plurality of superconducting qubits. In some embodiments, a superconducting qubit in the plurality of superconducting qubits is a persistent current qubit. A third aspect of the invention provides a method of determining a quantum state of a first target superconducting qubit. The method comprises presenting a plurality of superconducting qubits including a first target superconducting qubit in the plurality of superconducting qubits. A problem Hamiltonian describes (i) the quantum state of the plurality of superconducting qubits and (ii) each coupling energy between qubits in the plurality of qubits. The problem Hamiltonian is at or near a ground state. An rf-flux is added to the first target superconducting qubit. The rf-flux has an amplitude that is less than one flux quantum. An amount of an additional flux in the first target superconducting qubit is adiabatically varied. A presence or an absence of a dip in a voltage response of a tank circuit that is inductively coupled with the first target superconducting qubit during the adiabatically varying step is observed thereby determining the quantum state of the first target superconducting qubit. In some embodiments in accordance with this third aspect of the invention, each superconducting qubit in the plurality of superconducting qubits is in a quantum ground state during all or a portion of the adding step, the adiabatically varying step, and the observing step. In some embodiments, the problem Hamiltonian corresponds to a terminus of an adiabatic evolution of the plurality of superconducting qubits. In some embodiments, the method further comprises biasing all or a portion of the superconducting qubits in the plurality of superconducting qubits. The problem Hamiltonian further describes a biasing on the first target superconducting qubit. In some embodiments, an energy of the biasing step exceeds the tunneling energy of a tunneling element of the Hamiltonian of the first target superconducting qubit, thereby causing tunneling to be suppressed in the first target superconducting qubit during an instance of the biasing step, adding step and the adiabatically varying step. In some embodiments in accordance with this third aspect of the invention, the method further comprises adiabatically removing additional flux that was added to the first target superconducting qubit during the adiabatically varying step. In some embodiments, the adiabatically varying step comprises adiabatically varying the additional flux in accordance with a waveform selected from the group consisting of periodic, sinusoidal, triangular, and trapezoidal. In some embodiments, the adiabatically varying step comprises adiabatically varying the additional flux in accordance with a low harmonic Fourier approximation of a waveform selected from the group consisting of periodic, sinusoidal, triangular, and trapezoidal. In some embodiments, the additional flux has a direction that is deemed positive or negative. In some embodiments, the adiabatically varying step is characterized by a waveform that has an amplitude that grows with time. The amplitude of the waveform corresponds to an amount of additional flux that is added to the first target superconducting qubit during the adiabatically varying step. In some embodiments, the additional flux has an equilibrium point that varies with time. In some embodiments, the additional flux is either unidirectional or bidirectional. In some embodiments, the additional flux has a frequency of oscillation between about 1 cycle per second and about 100 kilocycles per second. In some embodiments in accordance with the third aspect of the invention, the adding step comprises adding the rf-flux using (i) an excitation device that is inductively coupled to the first target superconducting qubit or (ii) a the tank circuit. In some embodiments, the method further comprises repeating the adding step and the adiabatically varying step between 1 time and 100 times. In such embodiments, the presence or absence of the dip in the voltage response of the tank circuit is observed as an average of the voltage response of the tank circuit across each instance of the adiabatically varying step. In some embodiments in accordance with the third aspect of the invention, the first target superconducting qubit is flipped from an original basis state to an alternate basis state during the adiabatically varying step. The method further comprises returning the first target superconducting qubit to its original basis state by adiabatically removing additional flux in the qubit after the adiabatically varying step. In some embodiments, the adiabatically varying step does not alter the quantum state of each of superconducting qubits in the plurality of superconducting qubits other than the first target superconducting qubit. In some embodiments, the method further comprises recording a presence or an absence of the dip in the voltage response of the tank circuit. In some embodiments in accordance with the third aspect of the invention, the method further comprises adding a second rf-flux to a second target superconducting qubit in the plurality of superconducting qubits. The second rf-flux has an amplitude that is less than one flux quantum. Then an amount of a second additional flux in the second target superconducting qubit is adiabatically varied. A presence or an absence of a second dip in a voltage response of a tank circuit that is inductively coupled with the second target superconducting qubit during said adiabatically varying is observed, thereby determining the quantum state of the second target superconducting qubit. In some embodiments in accordance with the third aspect of the invention, the method further comprises designating a different superconducting qubit in the plurality of superconducting qubits as the first target superconducting qubit. The adding step and the adiabatically varying step are then reperformed with the different superconducting qubit as the first target superconducting qubit. The designating and reperforming are repeated until all or a portion (e.g., most, almost all, at least eighty percent) of the superconducting qubits in the plurality of superconducting qubits has been designated as the first target superconducting qubit. In some embodiments in accordance with the third aspect of the invention, a tank circuit is inductively coupled with the first target superconducting qubit. The method further comprises performing an adiabatic quantum computation step for an amount of time with the plurality of superconducting qubits prior to the adding step. The amount of time is determined by a factor the magnitude of which is a function of a number of qubits in the plurality of superconducting qubits. An amount of an additional flux in the first target superconducting qubit is adiabatically varied. Then, a presence or an absence of a dip in the voltage response of a tank circuit during the adiabatically varying step is observed, thereby determining the quantum state of the first target superconducting qubit. In some embodiments, the presence of a dip in the voltage response of the tank circuit corresponds to the first target superconducting qubit being in a first basis state. The absence of a dip in the voltage response of the tank circuit corresponds to the target superconducting qubit being in a second basis state. In some embodiments in accordance with the third aspect of the invention, the adiabatically varying step further comprises identifying an equilibrium point for the additional flux using an approximate evaluation method. In some embodiments, the method further comprises classifying the state of the first target qubit as being in the first basis state when the dip in the voltage across the tank circuit occurs to the left of the equilibrium point and classifying the state of the first target qubit as being in the second basis state when the dip in the voltage across the tank circuit occurs to the right of the equilibrium point. A fourth aspect of the present invention comprises a method for adiabatic quantum computing using a quantum system comprising a plurality of superconducting qubits. The quantum system is capable of being in any one of at least two quantum configurations at any give time. The at least two quantum configurations include a first configuration described by an initialization Hamiltonian H In some embodiments in accordance with the fourth aspect of the invention, each respective first superconducting qubit in the plurality of superconducting qubits is arranged with respect to a respective second superconducting qubit in the plurality of superconducting qubits such that the first respective superconducting qubit and the corresponding second respective superconducting qubit define a predetermined coupling strength. The predetermined coupling strength between each of the first respective superconducting qubits and corresponding second respective superconducting qubits in the plurality of superconducting qubits collectively define a computational problem to be solved. In some instances, the problem Hamiltonian H In some embodiments in accordance with the fourth aspect of the invention, the reading out step comprises probing an observable of the σ A fifth aspect of the present invention provides a structure for adiabatic quantum computing comprising a plurality of superconducting qubits. The plurality of superconducting qubits are capable of being in any one of at least two configurations at any give time. The at least two configurations include a first configuration characterized by an initialization Hamiltonian H In some embodiments in accordance with the fifth aspect of the invention, a superconducting qubit in the plurality of superconducting qubits is a persistent current qubit. In some embodiments, the tank circuit has a quality factor that is greater than 1000. In some embodiments, the tank circuit comprises an inductive element. The inductive element comprises a pancake coil of superconducting material. In some embodiments, the pancake coil of a superconducting material comprising a first turn and a second turn. The superconducting material of the pancake coil is niobium. Furthermore, there is a spacing of 1 about micrometer between the first turn and the second turn of the pancake coil. In some embodiments in accordance with the fifth aspect of the invention, the tank circuit comprises an inductive element and a capacitive element that are arranged in parallel or in series with respect to each other. In some embodiments, the tank circuit comprises an inductive element and a capacitive element that are arranged in parallel with respect to each other and the tank circuit has an inductance between about 50 nanohenries and about 250 nanohenries. In some embodiments, the tank circuit comprises an inductive element and a capacitive element that are arranged in parallel with respect to each other and the tank circuit has a capacitance between about 50 picofarads and about 2000 picofarads. In some embodiments, the tank circuit comprises an inductive element and a capacitive element that are arranged in parallel with respect to each other and the tank circuit has a resonance frequency between about 10 megahertz and about 20 megahertz. In some embodiments, the tank circuit has a resonance frequency f -
- L
_{T }is an inductance of the tank circuit; and - C
_{T }is a capacitance of the tank circuit.
- L
In some embodiments in accordance with the fifth aspect of the invention, the tank circuit comprises one or more Josephson junctions. In some embodiments, the structure further comprises means for biasing the one or more Josephson junctions of the tank circuit. In some embodiments, the structure further comprises an amplifier connected across the tank circuit in such a manner that the amplifier can detect a change in voltage across the tank circuit. In some embodiments, the amplifier comprises a high electron mobility field-effect transistor (HEMT) or a pseudomorphic high electron mobility field-effect transistor (PHEMT). In some embodiments, the amplifier comprises a multi-stage amplifier. In some embodiments, the multi-stage amplifier comprises two, three, or four transistors In some embodiments, structure further comprises a helium-3 pot of a dilution refrigerator that is thermally coupled to all or a portion of the plurality of superconducting qubits. to. In some embodiments in accordance with the fifth aspect of the invention, the structure further comprising means for driving the tank circuit by a direct bias current I -
- Φ
_{RF }is an amount of applied magnetic flux contributed to the superconducting qubit by the alternating current I_{RF}; and - Φ
_{DC }is an amount of applied magnetic flux that is determined by the direct bias current I_{DC}. In some embodiments, the structure further comprises means for applying a magnetic field on the superconducting qubit, and wherein Φ_{DC}=Φ_{A}*+f*(*t*)Φ_{0}, where, - Φ
_{0 }is one flux quantum; - f(t)Φ
_{0 }is constant or is slowly varying and is generated by the direct bias current I_{DC}; and Φ_{A}*=B*_{A}*×L*_{Q}, such that - B
_{A }is a magnitude of the magnetic field applied on the superconducting qubit by the means for applying the magnetic field; and - L
_{Q }is an inductance of the superconducting qubit. In some embodiments f(t) has a value between 0 and. In some embodiments, the means for applying a magnetic field on the superconducting qubit comprises a bias line that is magnetically coupled to the superconducting qubit. In some embodiments, the means for applying a magnetic field on the superconducting qubit is an excitation device. In some embodiments, Φ_{RF }has a magnitude between about 10^{−5}Φ_{0 }and about 10^{−1}Φ_{0}. In some embodiments, the structure further comprises means for varying f(t), Φ_{A}, and/or Φ_{RF}. In some embodiments, the structure further comprises means for varying Φ_{RF }in accordance with a small amplitude fast function. In some embodiments, the means for varying Φ_{RF }in accordance with a small amplitude fast function is a microwave generator that is in electrical communication with the tank circuit.
- Φ
In some embodiments in accordance with the fifth aspect of the invention, the structure further comprises an amplifier connected across the tank circuit and means for measuring a total impedance of the tank circuit, expressed through the phase angle χ between driving current I A sixth aspect of the invention provides a computer program product for use in conjunction with a computer system. The computer program product comprises a computer readable storage medium and a computer program mechanism embedded therein. The computer program mechanism comprises instructions for initializing a quantum system comprising a plurality of superconducting qubits to an initialization Hamiltonian H In some embodiments in accordance with this sixth aspect of the invention, the computer program mechanism further comprises instructions for repeating the instructions for biasing, instructions for adding, and instructions for adiabatically varying between 1 time and 100 times inclusive. The presence or absence of the voltage response of the tank circuit is observed as an average of the voltage response of the tank circuit to each instance of the instructions for adiabatically changing that are executed by the instructions for repeating. A seventh aspect of the invention comprises a computer program product for use in conjunction with a computer system. The computer program product comprises a computer readable storage medium and a computer program mechanism embedded therein. The computer program mechanism determines a quantum state of a first target superconducting qubit in a plurality of superconducting qubits. The computer program mechanism comprises instructions for initializing a plurality of superconducting qubits so that they are described by a problem Hamiltonian. The problem Hamiltonian describes (i) the quantum state of the plurality of superconducting qubits and (ii) each coupling energy between qubits in the plurality of qubits. The problem Hamiltonian is at or near a ground state. The computer program mechanism further comprises instructions for adding an rf-flux to the first target superconducting qubit. The rf-flux has an amplitude that is less than one flux quantum. The computer program mechanism further comprises instructions for adiabatically varying an amount of an additional flux in the first target superconducting qubit and observing a presence or an absence of a dip in a voltage response of a tank circuit that is inductively coupled with the first target superconducting qubit during the adiabatically varying step. In some embodiments in accordance with this seventh aspect of the invention, each superconducting qubit in the plurality of superconducting qubits is in a quantum ground state during all or a portion of the instructions for initializing, instructions for adding, and the instructions for adiabatically varying. In some embodiments, the problem Hamiltonian corresponds to a terminus of an adiabatic evolution of the plurality of superconducting qubits. In some embodiments, the computer program product further comprises instructions for biasing all or a portion of the superconducting qubits in the plurality of superconducting qubits. In such embodiments, the problem Hamiltonian additionally describes the biasing on the qubits in the plurality of superconducting qubits. In some embodiments, an energy of the biasing exceeds the tunneling energy of a tunneling element of the Hamiltonian of a superconducting qubit in the plurality of superconducting qubits thereby causing tunneling to be suppressed in the superconducting qubit during an instance of the instructions for biasing, instructions for adding and the instructions for adiabatically varying. In some embodiments in accordance with the seventh aspect of the invention, the computer program mechanism further comprises instructions for adiabatically removing additional flux that was added to the first target superconducting qubit during the instructions for adiabatically varying. In some embodiments, the instructions for adiabatically varying comprise instructions for adiabatically varying the additional flux in accordance with a waveform selected from the group consisting of periodic, sinusoidal, triangular, and trapezoidal. In some embodiments, the instructions for adiabatically varying comprise instructions for adiabatically varying the additional flux in accordance with a low harmonic Fourier approximation of a waveform selected from the group consisting of periodic, sinusoidal, triangular, and trapezoidal. In some embodiments, the additional flux has a direction that is deemed positive or negative. In some embodiments, the instructions for adiabatically varying are characterized by a waveform that has an amplitude that grows with time and such that the amplitude of the waveform corresponds to an amount of additional flux that is added to the first target superconducting qubit during an instance of the instructions for adiabatically varying. In some embodiments in accordance with the seventh aspect of the invention, the additional flux has an equilibrium point that varies with time. In some embodiments, the additional flux is either unidirectional or bidirectional. In some embodiments, the additional flux has a frequency of oscillation between about 1 cycle per second and about 100 kilocycles per second. In some embodiments, the instructions for adding comprise instructions for adding the rf-flux using (i) an excitation device that is inductively coupled to the first target superconducting qubit or (ii) the tank circuit. In some embodiments, the computer program mechanism further comprises instructions for repeating the instructions for adding and the instructions for adiabatically varying between 1 time and 100 times. In such embodiments, the presence or absence of the voltage response of the tank circuit is observed as an average of the voltage response of the tank circuit across each instance of the instructions for adiabatically varying that is executed by the instructions for repeating. An eight aspect of the invention comprises a computer system for determining a quantum state of a first target superconducting qubit in a plurality of superconducting qubits. The computer system comprises a central processing unit and a memory, coupled to the central processing unit. The memory stores instructions for biasing all or a portion of the qubits in the plurality of superconducting qubits other than the first target superconducting qubit. A problem Hamiltonian describes (i) the biasing on the qubits in the plurality of superconducting qubits and (ii) each coupling energy between respective superconducting qubit pairs in the plurality of superconducting qubits. The problem Hamiltonian is at or near a ground state. The memory further stores instructions for adding an rf-flux to the first target superconducting qubit. The rf-flux has an amplitude that is less than one flux quantum. The memory further stores instructions for adiabatically varying an amount of an additional flux in the first target superconducting qubit and observing a presence or an absence of a dip in a voltage response of a tank circuit that is inductively coupled with the first target superconducting qubit during a time when the instructions for adiabatically varying are executed. A ninth aspect of the present invention provides a computation device for adiabatic quantum computing comprising a plurality of superconducting qubits. Each superconducting qubit in the plurality of superconducting qubits comprises two basis states associated with the eigenstates of a σ A tenth aspect of the invention comprises an apparatus comprising a plurality of superconducting charge qubits. Each respective first superconducting charge qubit in the plurality of superconducting charge qubits is coupled with a respective second superconducting charge qubit in the plurality of superconducting charge qubits such that the first respective superconducting charge qubit and the second respective superconducting charge qubit define a predetermined coupling strength. The predetermined coupling strength between each of the first respective superconducting charge qubits and each of the second respective superconducting charge qubits in the plurality of superconducting charge qubits collectively define a computational problem to be solved. Each superconducting charge qubit in the plurality of superconducting charge qubits is capable of being in one of at least two configurations. These at least two configurations include a first configuration in accordance with an initialization Hamiltonian H In some embodiments in accordance with this tenth aspect of the invention, a superconducting charge qubit in the plurality of superconducting charge qubits comprises (i) a mesoscopic island made of superconducting material, (ii) superconducting reservoir, and (iii) a Josephson junction connecting the mesoscopic island to the superconducting reservoir. In some embodiments, the Josephson junction is a split Josephson junction. In some embodiments, the superconducting charge qubit further comprises a flux source configured to apply flux to the split Josephson junction. In some embodiments in accordance with the tenth aspect of the invention, the apparatus further comprises a generator capacitively coupled to a superconducting charge qubit in the plurality of superconducting charge qubits by a capacitor. In some embodiments, the generator is configured to apply a plurality of electrostatic pulses to the superconducting charge qubit. The plurality of electrostatic pulses additionally define the computational problem. In some embodiments in accordance with the tenth aspect of the invention, the apparatus further comprises a variable electrostatic transformer disposed between a first superconducting charge qubit and a second superconducting charge qubit in the plurality of superconducting charge qubits such that the predetermined coupling strength between the first superconducting charge qubit and the second superconducting charge qubit is tunable. In some embodiments, each respective first superconducting charge qubit in the plurality of superconducting charge qubits is arranged with respect to a respective second superconducting charge qubit in the plurality of superconducting charge qubits such that the plurality of superconducting charge qubits collectively form a non-planar graph. An eleventh aspect of the invention provides a method for computing using a quantum system comprising a plurality of superconducting charge qubits. The quantum system is coupled to an electrometer and the quantum system is capable of being in any one of at least two configurations. The at least two configurations includes a first configuration characterized by an initialization Hamiltonian H In some embodiments in accordance with the eleventh aspect of the invention, a first superconducting charge qubit in the plurality of superconducting charge qubits is coupled to a second superconducting charge qubit in the plurality of superconducting charge qubits by a capacitor such that the predetermined coupling strength between the first superconducting charge qubit and the second superconducting charge qubit is predetermined and is a function of the physical properties of the capacitor. In some embodiments in accordance with the eleventh aspect of the invention, a first superconducting charge qubit in the plurality of superconducting charge qubits is coupled to a generator by a device configured to provide a tunable effective charging energy. The device comprises a capacitor and the method further comprises: tuning the value of the effective charging energy of the first superconducting charge qubit by varying the gate voltage on the capacitor of said device. In some embodiments, a superconducting charge qubit in the plurality of superconducting charge qubits comprises a split Josephson junction having a variable effective Josephson energy. In such embodiments, the method further comprises tuning the value of the effective Josephson energy of the superconducting charge qubit by varying a flux applied to the split Josephson junction. In some embodiments, the first configuration is reached by setting the effective Josephson energy of the superconducting charge qubit to a maximum value. In some embodiments in accordance with the eleventh aspect of the invention, the adiabatically changing step comprises changing the configuration of the system from the first configuration characterized by the initialization Hamiltonian H In some embodiments in accordance with an eleventh aspect of the invention, a first superconducting charge qubit in the plurality of superconducting charge qubits is characterized by (i) an effective Josephson energy that is tunable and (ii) an effective charging energy that is tunable. A minimum value of the effective Josephson energy is less than the effective charging energy of the first superconducting charge qubit. A minimum value of the effective Josephson energy is less than a strength of a coupling between the first superconducting charge qubit and a second superconducting charge qubit in the plurality of superconducting charge qubits. The effective charging energy is, at most, equal to a maximum value of the effective Josephson energy of the first superconducting charge qubit. Furthermore, a strength of a coupling between the first superconducting charge qubit and a second superconducting charge qubit in the plurality of superconducting charge qubits is, at most, equal to a maximum value of the effective Josephson energy of the first superconducting charge qubit. In still another embodiment in accordance with the eleventh aspect of the invention, a first superconducting charge qubit in the plurality of superconducting charge qubits is characterized by (i) an effective Josephson energy that is tunable and (ii) an effective charging energy that is tunable. In such embodiments, the adiabatically changing step comprises adiabatically tuning the effective Josephson energy of the first superconducting charge qubit such that the effective Josephson energy of the first superconducting charge qubit reaches a minimum value when the quantum system is described by the ground state of the problem Hamiltonian H In some embodiments in accordance with the eleventh aspect of the invention, a first superconducting charge qubit in the plurality of superconducting charge qubits has a first basis state and a second basis state and, when the quantum system is described by the ground state of the problem Hamiltonian H In some embodiments in accordance with the eleventh aspect of the invention, a first superconducting charge qubit in the plurality of superconducting charge qubits has a first basis state and a second basis state and, when the quantum system is described by the ground state of the problem Hamiltonian H In some embodiments in accordance with the eleventh aspect of the invention, a first superconducting charge qubit in the plurality of superconducting charge qubits is characterized by (i) an effective Josephson energy that is tunable and (ii) an effective charging energy that is tunable. In such embodiments, a minimum value of the effective Josephson energy is less than the effective charging energy of the first superconducting charge qubit; a minimum value of effective Josephson energy is less than a strength of a coupling between the first superconducting charge qubit and a second superconducting charge qubit in the plurality of superconducting charge qubits; the effective charging energy is greater than a maximum value of the effective Josephson energy of the first superconducting charge qubit; and a strength of a coupling between the first superconducting charge qubit and a second superconducting charge qubit in the plurality of superconducting charge qubits is, at most, equal to the maximum effective Josephson energy of the first superconducting charge qubit. In some such embodiments, the initializing step comprises setting the effective charging energy of the first superconducting charge qubit to a minimum value. In some such embodiments, the adiabatically changing step comprises adiabatically tuning the effective Josephson energy of the first superconducting charge qubit such that the effective Josephson energy is at a minimum value when the quantum system is described by the ground state of the problem Hamiltonian H In some embodiments in accordance with the eleventh aspect of the invention, a first superconducting charge qubit in the plurality of superconducting charge qubits is characterized by an effective Josephson energy that is tunable. The initializing step comprises setting the effective Josephson energy of the first superconducting charge qubit to a minimum value, and the adiabatically changing step comprises (i) adiabatically tuning the effective Josephson energy of the first superconducting charge qubit such that the effective Josephson energy is greater than a minimum value for a period of time before the quantum system is described by the ground state of the problem Hamiltonian H Like reference numerals refer to corresponding parts throughout the several views of the drawings. The present invention comprises systems and methods for adiabatic quantum computing using superconducting qubits. In various embodiments of the present invention, adiabatic quantum computing is performed on registers of superconducting qubits that have demonstrated quantum computing functionality. Adiabatic quantum computing is a model of quantum computing that can be used to attempt to find solutions for computationally difficult problems. When choosing a candidate system for adiabatic quantum computing there are a few criteria that can be observed. These criteria can be drawn from those described herein below. However, some embodiments of the present invention may not adhere to all of these criteria. One criterion is that the readout device should a Stem-Gerlach σ Some embodiments of the present invention adhere to the above criteria. Other embodiments of the present invention do not. For instance, in the case of the phase qubit, it is possible to have the tunneling term in the problem Hamiltonian be, not zero, but weak, e.g., for H In accordance with embodiments of the present invention, the general procedure of adiabatic quantum computing is shown in In step In transition In step In step In one embodiment of the present invention, the natural quantum mechanical evolution of the quantum system under the slowly changing Hamiltonian H(t) carries the initial state H A quantum system evolves under the Schrödinger equation:
A general example of the adiabatic evolution of a quantum system is as follows. The states |0;t> and |1;t) are respectively the ground and first excited states of Hamiltonian H(t), with energies E The minimum energy gap between the ground state E In an embodiment of the present invention, T is the time taken to vary a control parameter of a charge qubit, for example induced gate charge or flux for a charge qubit with split Josephson junction. See Section 5.4. In an embodiment of the present invention, time T is a value between about 0.1 nanosecond and about 500 microseconds. In other words, the amount of time between when the quantum system is allowed to begin adiabatically changing from the initial state H In an embodiment of the present invention, T is the time taken to vary a control parameter of a phase qubit, for example flux in a persistent current qubit. See Section 5.3. In an embodiment of the present invention, time T is a value between about 1 nanosecond and about 100 microseconds. In other words, the amount of time between when the quantum system is allowed to begin adiabatically changing from the initial state H Most analyses of adiabatic algorithms emphasize how the gap, g(t), and the ratio of the matrix element <dH/dt> Other embodiments of the present invention can be constructed and operated with a different estimate for the probability of a diabatic transition at step In many embodiments of the present invention, quantum systems for adiabatic quantum computation are designed such that the minimum of the energy gap, the difference in the asymptotic slopes, and the rate of change of the adiabatic evolution parameter ensure that the probability of diabatic transition at step In some embodiments of the present invention, the amount of time required to perform the readout process should be engineered so that the probability that quantum system In an embodiment of the present invention this requisite small probability for transition P Section 5.3 describes quantum computing systems of the present invention that make use of phase qubits. Section 5.4, below, describes quantum computing systems of the present invention that make use of charge qubits. Referring to The two stable states of a qubit In preferred embodiments, the three persistent current qubits, In some embodiments of the present invention, qubit parameters are chosen to satisfy the requirements of the problem to be solved by adiabatic quantum computation and the restrictions of the qubits. For instance, in the case of In one particular embodiment of a qubit Various embodiments of the present invention provide different values for the persistent current that circulates in persistent current qubits Step In accordance with the present invention, the problem of determining the ground state of the three node frustrated ring is encoded into system As shown in All three qubits Embodiments of the present invention, such as those that include a system like In some embodiments of the present invention, the coupling strength between any two abutting qubits is a product of the mutual inductance and the currents in the coupled qubits. The mutual inductance is a function of common surface area and distance. The greater the common surface area the stronger the mutual inductance. The greater the distance the less the mutual inductance. The current in each qubit is a function of the Josephson energy of the qubit, E Step Once system Step Step In an embodiment of the present invention, the result of the adiabatic quantum computation is determined using individual qubit magnetometers. Referring to In an embodiment of the present invention, the result of the adiabatic quantum computation is determined using individual qubit magnetometers laid adjacent to respective qubits in the quantum system. Referring to In an embodiment of the present invention, the qubits In an embodiment of the present invention, qubits In an embodiment of the present invention the tunneling of qubits Referring to Referring to In other embodiments of the present invention, the state of each qubit As described above, for many embodiments of the present invention, the energy levels of the system for adiabatic quantum computing H Considering In an embodiment of the present invention, the system is read out by differentiating various energy levels (solutions) to identify the ground state. In an example of the persistent current qubit, the dephasing rate is currently recorded as being 2.5 microseconds or less. Once the system is in a state of H The states of energy level diagram like that of Knowledge of the initial energy level and the corresponding voltage response can allow the ground state to be determined provided that a sampling of some of the lowest energy levels voltage response has been made. Accordingly, in an embodiment of the present invention, the system is not initialized in the ground state of the initial Hamiltonian. Rather the system is initialized in an excited state of the system such as In operation, when an external signal, such as a magnetic flux in an inductively coupled device, is applied through tank circuit Some embodiments of the present invention make use of the high electron mobility field-effect transistors (HEMT) and their improved pseudomorphic variants (PHEMT) to measure the signal from tank circuit In one embodiment of the present invention, the response voltage of the tank circuit is amplified by a cold amplifier thermally coupled to the helium-3 pot of the dilution refrigerator. The response signal is once more amplified by a room temperature amplifier and detected by an rf lock-in voltmeter. The rf lock-in voltmeter can be used to average the response of the tank circuit over many cycles of the response of the tank. For more information on such an amplifier arrangement, see Oukhanski et al., 2003, Some embodiments of the present invention make use of tank circuit Tank circuit In one embodiment of the present invention, the direct bias current I The terms Φ If the amplitude of Φ The imaginary part of the total impedance of tank circuit This section describes techniques for reading out the state of a quantum system In accordance with an embodiment of the present invention, a method for reading out the state of a superconducting qubit within quantum system The readout of a superconducting qubit involves sweeping the flux applied to the superconducting qubit by an amount and in a direction designed to detect the anticrossing In some embodiments, when the superconducting qubit is in the |0> ground state, e.g. in region This is in contrast to At termini The readout Referring to Now consider the case in which the superconducting qubit is in the |0> state, e.g. at energy level Further consider the case in which a qubit within quantum system In some embodiments, there is an additional magnetic field B The objective of the qubit measurements described in the preceding paragraphs is not to establish the state (ground or excited) that a qubit in quantum system The addition of an external magnetic field that is applied against the superconducting qubit induces a third additional flux Φ In some embodiments, of the present invention, the total flux in the qubit is varied in a triangular pattern, with the magnitude and sign of the flux chosen such that the states of the qubit can be differentiated and the distance to anticrossings or crossings can be found. Details on how such waveforms can accomplish this task are described in Section 5.3.2.2, below. In an embodiment of the present invention, the flux is applied to a superconducting qubit through individual bias wires. In such an embodiment each qubit, or all qubits except for one qubit, in quantum system There are independent parameters that can be altered for each waveform. These parameters include, for example, waveform shape, direction, period, amplitude, amplitude growth function, and equilibrium point. The waveforms can oscillate in both directions about an equilibrium point with a mean of zero (bidirectional), or oscillate away from the equilibrium point in one direction with a non-zero mean (unidirectional). The amplitude for the additional flux Φ In an embodiment of the present invention, the maximum amplitude of the additional flux Φ The frequency of oscillation, f In some embodiments, rather than sweeping through a range of applied fluxes in accordance with a single waveform (oscillation, cycle), the waveform is repeated a plurality of time (plurality of oscillations, plurality of cycles) and voltage response across each of these oscillations (cycles) is averaged. The number of oscillations needed to perform readout, in such embodiments, depends on the architecture of quantum system As shown in In accordance with some embodiments of the present invention, Voltage dips In an embodiment of the present invention, the voltage dips are in fact peaks because the polarity of the leads to tank circuit The fidelity of the readout (step In general, and especially during adiabatic evolution for adiabatic quantum computation (transition One embodiment of the present invention makes use of a negative feedback loop technique to ensure that Landau-Zener transitions do not occur during adiabatic evolution In some embodiments of the present invention, the change in magnitude of the response of tank circuit, χ, ranges from 0.01 radians to about 6 radians for the phase signal. In some embodiments of the present invention, the change in magnitude of the response of the tank circuit, tan(χ), ranges from 0.02 microvolt (μV) to about 1 μV for the amplitude signal. Embodiments of the present invention can make use of an adiabatic process to readout the state of the superconducting qubit during measurement step In contrast to the example provided in Section 5.3.2.4, embodiments of the present invention in accordance with this section make use of a diabatic process to readout the state of a superconducting qubit after the adiabatic quantum computation has been completed. In typical embodiments, this is the only part of the process illustrated in Embodiments of the present invention can make use of repeated adiabatic processes to readout the states of a plurality of superconducting qubits in quantum system In preferred embodiments, each qubit in quantum system As part of measurement step Readout using tank circuit The detailed exact calculation of energy spectra of an instantaneous Hamiltonian H(t) can be intractable due to exponential growth of the problems size as a function of the number of qubits used in an adiabatic computation. Therefore, approximate evaluation techniques are useful as a best guess of the location of the crossings and anticrossings. Accordingly, some embodiments of the present invention make use of an approximate evaluation method to locate anticrossing of the energies, or energy spectra, of the qubits in quantum system In an embodiment of the present invention, techniques collectively known as random matrix theory (RMT) are applied to analyze the quantum adiabatic algorithm during readout. See Bougerol and Lacroix, 1985, In another embodiment of the present invention, spin density functional theory (SDFT) is used as an approximate evaluation method to locate anticrossing of the energies of the system. The probing of the energy spectra by the additional flux can then be used to locate the crossings and anticrossing and perform a readout of the state of superconducting qubits. In another embodiment of the present invention, the approximate evaluation technique comprises a classical approximation algorithm in order to solve NP-Hard problems. When there is a specific instance of a problem to be solved, the problem is mapped to a description of the qubits being used to solve the NP-Hard problem. This process involves finding an approximation algorithm to the problem being solved by the quantum computer and running the approximation algorithm. The approximate solution is then mapped to the quantum computer's state using the mapping that was used to encode the instance of the NP-Hard problem. This provides a good estimate for the state of the superconducting qubits and lessens the requirements on probing the energy levels for crossings and anticrossings. Such a mapping typically involves setting the coupling energies between the qubits being used to solve the NP-Hard problem so that the qubits approximately represent the problem to be solved. For examples of approximation algorithms useful for the present invention see Goemans and Williamson, “0.878-approximation algorithms for MAX CUT and MAX 2SAT,” In In another specific embodiment of the present invention, separations In an embodiment of the present invention, each qubit The adiabatic quantum computing method of the present invention progresses using the system described in A specific adiabatic quantum computation that can be performed with system Step In an embodiment of the present invention, an instance of Step Step Step Measurement of the state of system In some embodiments, steps In accordance with an aspect of the present invention, superconducting charge qubits can be used in adiabatic quantum computation devices (e.g., in quantum systems A superconducting charge qubit suitable for use in embodiments of the present invention is shown in In some embodiments, qubit In an embodiment of the present invention, island The energy states of the qubit are controlled by electrostatic control pulses. The control pulses can be direct current or alternating current The control pulses can be used to induce a gate charge on superconducting island As shown in The Josephson energies of the two Josephson junctions in DC-SQUID As illustrated in At readout, resonator The rate of coherent exchange of a single excitation between the superconducting charge qubit and resonator A single shot readout device is shown in The island of trap The use of trap During operation of the qubit, e.g., the steps of adiabatic quantum computing, the island of trap Embodiments of the present invention can be operated without trap A radio frequency single electron transistor coupled to a superconducting charge qubit is shown in SET island In operation, for superconducting charge qubit read out at the end of the adiabatic quantum computation using system The readout scheme specified above for system The values of capacitance, inductance, and Josephson energy given in relation to superconducting charge qubits above, is provided in order to describe operable embodiments, but these are not the only values that lead to operable embodiments. The values can be changed with and without the need to make corresponding changes to the cooperative components. There are freely available and commercially available software that can aid the designer of superconducting charge qubits. Such software includes, but is not limited to, FASTCAP a tool designed to calculate capacitances of a given layout of superconducting devices. FASTCAP is freely distributed by the Research Laboratory of Electronics of the Massachusetts Institute of Technology, Cambridge, Mass. See also Nabors et al., 1992, IEEE Some embodiments of the present invention comprise a lattice of interconnected superconducting charge qubits that are capacitively coupled to each other. This is a difference between these charge qubits and the systems of As shown in In the embodiment illustrated in The Hamiltonian for lattices of superconducting charge qubits The biasing term proportional to the σ The tunneling term proportional to the σ In embodiments of the present invention, the coupling term proportional to the tensor product of σ In accordance with the general procedure of adiabatic quantum computing as shown in In step In transition In accordance with the adiabatic theorem of quantum mechanics, the plurality of superconducting charge qubits will remain in the ground state of H at every instance the qubits are changed and after the change is complete, provided the change is adiabatic. In some embodiments of the present invention, the plurality of superconducting charge qubits start in an initial state H In step An aspect of the problem Hamiltonian H In step This example describes an embodiment of the invention where the superconducting charge qubits have fixed couplings. To perform adiabatic quantum computation with superconducting charge qubits, in step This example describes an embodiment of the invention where the superconducting charge qubits have fixed couplings. In an embodiment of the present invention where E′ Further embodiments for adiabatic quantum computing with superconducting charge qubits can be constructed from the following protocols for the biasing, tunneling, and coupling of qubits. For the tunneling terms of a plurality of superconducting charge qubits the following protocols can be used. In an embodiment where the tunneling term is fixed, the bias and coupling energies for the initial Hamiltonian are set to be less that the tunneling energies. In the same examples, the bias and coupling energies for the problem Hamiltonian are set to be greater that the tunneling energies. In such a case, the tunneling is weak compared with the final Hamiltonian. In embodiments where the tunneling term is tunable, the tunneling energies for the initial Hamiltonian are set to values approaching and including E For the coupling terms of a plurality of superconducting charge qubits the following protocols can be used. In an embodiment where the coupling terms are fixed at fabrication time, the couplings are set such that they describe the couplings for the problem Hamiltonian. In an embodiment where the coupling terms are tunable, the coupling energies for the initial Hamiltonian are set to zero. During the adiabatic evolution the coupling energies are increased such that they describe the couplings for the problem Hamiltonian. If the tunneling energies are at zero during readout, the coupling energies can be set to any value without the final state of the superconducting charge qubits charging. System System In some embodiments, memory In some embodiments, memory The functionality of controller When introducing elements of the present invention or the embodiment(s) thereof, the articles “a,” “an,” “the,” and “said” are intended to mean that there are one or more of the elements. The terms “comprising,” “including,” and “having” are intended to be inclusive and to mean that there may be additional elements other than the listed elements. Moreover, the term “about” has been used to describe specific parameters. In many instances, specific ranges for the term “about” have been provided. However, when no specific range has been provided for a particular usage of the term “about” herein, than either of two definitions can be used. In the first definition, the term “about” is the typical range of values about the stated value that one of skill in the art would expect for the physical parameter represented by the stated value. For example, a typical range of values about a specified value can be defined as the typical error that would be expected in measuring or observing the physical parameter that the specified value represents. In the second definition of about, the term “about” means the stated value ±0.10 of the stated value. As used herein, the term “instance” means the execution of a step. For example, in a multistep method, a particular step may be repeated. Each repetition of this step is referred to herein as an “instance” of the step. All references cited herein are incorporated herein by reference in their entirety and for all purposes to the same extent as if each individual publication or patent or patent application was specifically and individually indicated to be incorporated by reference in its entirety for all purposes. The present invention can be implemented as a computer program product that comprises a computer program mechanism embedded in a computer readable storage medium. For instance, the computer program product could contain program modules, such as those illustrated in Many modifications and variations of this invention can be made without departing from its spirit and scope, as will be apparent to those skilled in the art. The specific embodiments described herein are offered by way of example only, and the invention is to be limited only by the terms of the appended claims, along with the full scope of equivalents to which such claims are entitled. Referenced by
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