US20160343932A1 - Quantum hardware characterized by programmable bose-hubbard hamiltonians - Google Patents

Quantum hardware characterized by programmable bose-hubbard hamiltonians Download PDF

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US20160343932A1
US20160343932A1 US15/112,642 US201515112642A US2016343932A1 US 20160343932 A1 US20160343932 A1 US 20160343932A1 US 201515112642 A US201515112642 A US 201515112642A US 2016343932 A1 US2016343932 A1 US 2016343932A1
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Masoud MOHSENI
Hartmut Neven
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    • H—ELECTRICITY
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    • H10N60/00—Superconducting devices
    • H10N60/80—Constructional details
    • H10N60/805—Constructional details for Josephson-effect devices
    • H01L39/025—
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N10/00—Quantum computing, i.e. information processing based on quantum-mechanical phenomena
    • G06N99/002—
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N7/00—Computing arrangements based on specific mathematical models
    • G06N7/01—Probabilistic graphical models, e.g. probabilistic networks

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  • the present specification relates to quantum hardware characterized by programmable Bose-Hubbard Hamiltonians.
  • quantum information is represented by multimode quantum hardware, the dynamics of which can be characterized and controlled by a programmable many-body quantum Hamiltonian.
  • the multimode quantum hardware can be programmed as, for example, a quantum processor for certain machine learning problems.
  • Examples of the quantum hardware include neutral atoms on optical lattices, photonic integrated circuits, or superconducting cavity quantum electrodynamics (QED) circuits, and the Hamiltonians characterizing such quantum hardware include dissipative or non-dissipative Bose-Hubbard Hamiltonians.
  • the solution to a machine optimization problem can be encoded into an energy spectrum of a Bose-Hubbard quantum Hamiltonian.
  • the solution is encoded in the ground state of the Hamiltonian.
  • the annealing process may not require tensor product structure of conventional qubits or rotations and measurements of conventional local single qubits.
  • quantum noise or dechoerence can act as a recourse to drive the non-equilibrium quantum dynamics into a non-trivial steady state.
  • the quantum hardware can be used to solve a richer set of problems as compared to quantum hardware represented by an Ising Hamiltonian. Furthermore, instead of the binary representations provided by the Ising Hamiltonians, constraint functions of problems to be solved can have a digital representation according to the density of states in Cavity QED modes.
  • the subject matter of the present disclosure can be embodied in apparatuses that include: a first group of superconducting cavities each configured to receive multiple photons; a second group of superconducting cavities each configured to receive multiple photons; and multiple couplers, in which each coupler couples one superconducting cavity from the first group of superconducting cavities with one superconducting cavity from the second group of superconducting cavities such that the photons in the coupled superconducting cavities interact, and in which a first superconducting cavity of the first group of superconducting cavities is connected to a second superconducting cavity of the second group of superconducting cavities, such that photons of the first and second superconducting cavities are shared by each of the first and second superconducting cavities, the first superconducting cavity is coupled to one or more of the other superconducting cavities of the first group of superconducting cavities to which the second superconducting cavities are coupled, and the second superconducting cavity is coupled to one or more
  • each coupler is configured to annihilate a photon in one superconducting cavity and create a photon in a different superconducting cavity.
  • At least one of the couplers includes a Josephson junction.
  • a Hamiltonian characterizing the apparatus is: ⁇ i h i n i + ⁇ i,j t ij ( ⁇ i ⁇ ⁇ j +h.c.)+ ⁇ i U i n i (n i ⁇ 1), in which n i is a particle number operator and denotes occupation number of a cavity mode i, ⁇ i ⁇ is a creation operator that creates a photon in cavity mode i, ⁇ i is an annihilation operator that annihilates a photon in cavity mode j, h i corresponds to a site disorder, U i corresponds to an on-site interaction, t i,j are the hopping matrix elements, and h.c. is hermitian conjugate.
  • the multiple couplers are trained to produce an output desired probability density function at a subsystem of interest at an equilibrium state of the apparatus.
  • the apparatuses are trained as Quantum Boltz
  • a Hamiltonian characterizing the apparatus is: ⁇ i h i n i + ⁇ i,j t ij ( ⁇ i ⁇ ⁇ j +h.c.)+ ⁇ i U i n i (n i ⁇ 1)+ ⁇ i,j U ij n i n j , in which n i is a particle number operator and denotes occupation number of a cavity mode i, ⁇ i ⁇ is a creation operator that creates a photon in cavity mode i, ⁇ i is an annihilation operator that annihilates a photon in cavity mode j, h i corresponds to a site disorder, U i corresponds to an on-site interaction, t i,j are the hopping matrix elements, and h.c.
  • At least one cavity is a 2D cavity.
  • each cavity can be a 2D cavity.
  • At least one cavity is a 3D cavity.
  • each cavity can be a 3D cavity.
  • each superconducting cavity in the first group of superconducting cavities is connected to a superconducting cavity in the second group of superconducting cavities.
  • the subject matter of the present disclosure can be embodied in methods that include providing an apparatus having: a first group of superconducting cavities each configured to receive multiple photons; a second group of superconducting cavities each configured to receive multiple photons; and multiple couplers, in which each coupler couples one superconducting cavity from the first group of superconducting cavities with one superconducting cavity from the second group of superconducting cavities such that the photons in the coupled superconducting cavities interact, and in which a first superconducting cavity of the first group of superconducting cavities is connected to a second superconducting cavity of the second group of superconducting cavities, such that photons of the first and second superconducting cavities are shared by each of the first and second superconducting cavities, the first superconducting cavity is coupled to one or more of the other superconducting cavities of the first group of superconducting cavities to which the second superconducting cavities are coupled, and the second superconducting cavity is coupled to one
  • the apparatus can be provided in an initial Mott-insulated state.
  • the methods can further include causing a quantum phase transition of the apparatus from the initial Mott-insulator state to a superfluid sate; and adiabatically guiding the apparatus to a problem Hamiltonian.
  • the methods can further include causing a quantum phase transition of the apparatus from the superfluid state to a final Mott-insulator state and reading the state of each superconducting cavity in the apparatus.
  • FIG. 1 is a schematic of an example structure of quantum hardware.
  • FIG. 2A is an example of a selected connection in quantum hardware.
  • FIG. 2B is an example of a full connection in quantum hardware.
  • FIG. 3 is a flow diagram of an example process for encoding a problem in a Hamiltonian containing density-density interactions and programming quantum hardware.
  • FIG. 4 is a flow diagram of an example process for encoding a problem in a dissipative-driven Hamiltonian and programming quantum hardware.
  • FIG. 1 is a schematic of an example structure of quantum hardware 100 that can be characterized by programmable Bose-Hubbard Hamiltonians.
  • the quantum hardware 100 includes QED cavities 104 arranged in columns 110 , and lines 112 . At least some pairs of the QED cavities, such as cavities 110 and 112 , are coupled to each other through coupler 106 .
  • the QED cavities can be superconducting waveguide cavities restricted in dimensionality, e.g., to 1D, 2D or 3D.
  • the couplers 106 can be inductive couplers, and the hardware can be configured with resistors and inductors.
  • the couplers 106 can be Josephson couplers, and in an example, a Josephson coupler is constructed by connecting two superconducting elements separated by an insulator and a capacitance in parallel.
  • the cavities contain photons in optical modes 102 .
  • the cavities can receive a variable amount of photons when the quantum hardware is initialized, or during the use of the quantum hardware.
  • a coupler 106 between two cavities allows the photons of the two cavities to interact with each other. For example, the coupler can create or annihilate photons in a cavity, or move photons between cavities.
  • Each cavity in the hardware 100 can be used as a logical computation unit. The number of photons in a cavity mode of the cavity can be read using photon detectors 108 .
  • the quantum hardware 100 includes a fully connected network of superconducting cavities 104 .
  • each cavity is coupled with all other cavities through couplers 106 .
  • selected pairs of cavities are coupled with each other. The selection can be made based on the need for the quantum computation and the physical confinement of the hardware.
  • FIG. 2A is an example of a selected connection in quantum hardware.
  • the hardware includes cavities A, B, C, D, E, F, G, and H, and pairs of cavities are coupled through couplers 208 .
  • Cavities “A” 202 and “E” 204 are selected to be connected through a connection 206 , so that effectively, they become the same cavity. That is, the connection can be considered an extension of the cavity QED mode. Without the connection 206 , the cavity “A” is coupled to cavities “E”, “F”, “G”, and “H,” but not to cavities B, C, and D.
  • cavity “A” is coupled to all other cavities of the hardware.
  • cavities “B”-“H” are only coupled to selected cavities of the hardware.
  • additional connections similar to the connection 206 can be added. The total amount of interaction between cavities in the hardware can be increased.
  • FIG. 2B An example of a fully connected network is shown in FIG. 2B .
  • FIG. 2B includes connections between “E” and “A” 212 , “F” and “B” 214 , “G” and “C” 216 , and “H” and “D” 218 .
  • the network of FIG. 2B therefore allows for each cavity to interact with all other cavities.
  • FIGS. 1, 2A, and 2B can be characterized by a Bose-Hubbard Hamiltonian:
  • n i is the particle number operator and denotes the occupation number of a cavity mode i
  • ⁇ i ⁇ is a creation operator that creates a photon in cavity mode i
  • ⁇ j is an annihilation operator that annihilates a photon in cavity mode j
  • h i corresponds to a site disorder
  • U i corresponds to an on-site interaction
  • h.c. is the hermitian conjugate.
  • the hardware of FIGS. 1, 2A, and 2B can be used to determine solutions to problems by training the hardware as a Quantum Boltzmann Machine for probabilistic inference on Markov Random Fields.
  • a problem can be defined by a set of observables y i , e.g., photon occupation number at a cavity of the hardware, and a goal is to infer underlying correlations among a set of hidden variables x i . Assuming statistical independence among various pairs of y i and x i , the joint probability distribution would be
  • p ⁇ ( ⁇ x i ⁇ , ⁇ y i ⁇ ) 1 / Z ⁇ ⁇ i , j ⁇ ⁇ ⁇ i , j ⁇ ( x i , x j ) ⁇ ⁇ i ⁇ ⁇ ⁇ i ⁇ ( x i , y i ) ,
  • Z is the partition function (which, for a given system with a fixed energy function or a given Hamiltonian, is constant)
  • ⁇ i,j (x i , x j ) is a pairwise correlation
  • ⁇ i (x i , y i ) is the statistical dependency between a given pair of y i and x i .
  • certain cavity modes can act as the visible/observable input nodes of the Markov Random Field and can be used to train one or more of the Josephson couplers, which connect the hidden nodes x i , to reproduce certain probability distribution of outcomes at the output visible nodes y i .
  • the training can be such that the delocalized energy ground state of the Bose-Hubbard model for each input state can have a probability distribution over the computational, i.e., localized, basis that resembles the output probability distribution function (PDF) of the training example, i.e., p( ⁇ x j ⁇ , ⁇ y i ⁇ ).
  • PDF output probability distribution function
  • the thermalized state of the hardware trained as a Quantum Boltzmann Machine can be sampled to provide a probabilistic inference on the test data according to the Boltzmann distribution function.
  • an energy function can be defined:
  • the quantum hardware of FIGS. 1, 2A, and 2B can be engineered or controlled to allow an additional type of coupling between the coupled cavities characterized by density-density interactions.
  • density-density interactions an additional term can be added to the Bose-Hubbard Hamiltonian:
  • the addition of the density-density interaction term to the Bose-Hubbard Hamiltonian can allow construction of a problem Hamiltonian in which the solution of a wide variety of problems can be encoded.
  • constraint functions of problems can have a digital representation according to the density of states in cavity QED modes with density-density interactions.
  • the hardware is driven with additional fields to compensate for the loss.
  • the quantum hardware can be engineered (an example process of using the additionally engineered hardware is shown in FIG. 3 ) or by controlling the hardware using a dissipative-driven method (an example process of using the controlled hardware is shown in FIG. 4 ).
  • the quantum hardware can additionally be engineered to include, e.g., Kerr non-linearity with Josephson Junction couplers, the Stark effect, or continuous-time C-phase gates between cavities.
  • H total ⁇ i ⁇ ⁇ h i ⁇ n i + ⁇ i , j ⁇ ⁇ t ij ⁇ ( a i ⁇ ⁇ a j + h . c . ) + ⁇ i ⁇ ⁇ U i ⁇ n i ⁇ ( n i - 1 ) + ⁇ i , j ⁇ ⁇ U ij ⁇ n i ⁇ n j ,
  • a time dependent Hamiltonian In use for adiabatic computation, a time dependent Hamiltonian can be represented as:
  • H total (1 ⁇ s ) H i +sH p ,
  • H i is the initial Hamiltonian
  • H i ⁇ i , j ⁇ ⁇ t ij ⁇ ( a i ⁇ ⁇ a j + h . c . ) ,
  • H p is the problem Hamiltonian into which the selected problem is encoded:
  • H p ⁇ i h i n i + ⁇ i U i n i ( n i ⁇ 1)+ ⁇ i,j U ij n i n j .
  • FIG. 3 is a flow diagram of an example process 300 for encoding a problem in a Hamiltonian containing density-density interactions and programming quantum hardware.
  • the hardware undergoes a quantum phase transition from a Mott-insulator state to a superfluid state (step 302 ).
  • the hardware is initially in an insulated state with no phase coherence, and with localized wavefunctions only.
  • the many-body state is therefore a product of local Fock states for each cavity in the hardware:
  • N is the number of photons
  • i is the cavity mode
  • the hardware undergoes a quantum phase transition to a superfluid state so that the wavefunctions are spread out over the entire hardware:
  • ⁇ ⁇ SF ⁇ ⁇ i ⁇ ⁇ a i N ⁇ ⁇ 0 ⁇
  • the hardware transitions from a superfluid state to a non-trivial Mott-insulator state that can capture the solution to the problem (step 306 ).
  • the quantum state of the entire hardware is read out (step 308 ) and can be processed by a classical computer to provide solutions to the given problem.
  • the state of each cavity is determined by the photon occupation number of each cavity mode.
  • the process 300 can be repeated multiple times for the given problem to provide solutions with a statistical distribution.
  • the dynamical effects of density-density interactions can be achieved by an interplay of the Bose-Hubbard Hamiltonian with cavity photon number fluctuations induced by an auxiliary external field.
  • the combination of the hardware and the auxiliary external field is called a dissipative-driven hardware, and the Hamiltonian describing the dissipative-driven hardware is:
  • H BH ⁇ i ⁇ ⁇ h i ⁇ n i + ⁇ i , j ⁇ ⁇ t ij ⁇ ( a i ⁇ ⁇ a j + h . c . ) + ⁇ i ⁇ ⁇ U i ⁇ n i ⁇ ( n i - 1 ) + ⁇ i ⁇ ⁇ [ ⁇ ⁇ ( t ) ⁇ a i ⁇ + ⁇ ⁇ ( t ) * ⁇ a i ] + H SB
  • ⁇ (t) is a slowly-varying envelope of an externally applied field to compensate for photon loss
  • H SB is the Hamiltonian of the interaction between the hardware and the background bath in which the hardware is located:
  • H SB ⁇ i ⁇ ⁇ v ⁇ [ ⁇ i , v ⁇ ( a i ⁇ b v ⁇ + a i ⁇ ⁇ b v ) + ⁇ i , v ⁇ a i ⁇ ⁇ a i ⁇ ( b v + b v ⁇ ) ] ,
  • ⁇ i, ⁇ is the strength of hardware-bath interactions corresponding to the exchange of energy
  • h i corresponds to a site disorder
  • U i corresponds to an on-site interaction
  • t i,j are the hopping matrix elements
  • ⁇ i, ⁇ corresponds to the strength of local photon occupation fluctuations due to exchange of phase with the bath.
  • a solution to a problem can be determined without adiabatically guiding the hardware to the ground state of a problem Hamiltonian as in the process 300 of FIG. 3 .
  • the dissipative-driven hardware is eventually dominated by dissipative dynamics, defining a non-trivial steady state in which the solution to a problem is encoded.
  • FIG. 4 is a flow diagram of an example process 400 for encoding a problem in a dissipative-driven Hamiltonian and programming quantum hardware.
  • the hardware is programmed for a problem to be solved (step 402 ).
  • the problem is an optimization problem or an inference task and is mapped to a Markov Random Field.
  • a problem can be defined by a set of observables y i , e.g., photon occupation number at a cavity of the hardware, and the goal is to infer underlying correlations among a set of hidden variables x i . Assuming statistical independence among pair y i and x i , the joint probability distribution would be:
  • p ⁇ ( ⁇ x i ⁇ , ⁇ y i ⁇ ) 1 / Z ⁇ ⁇ i , j ⁇ ⁇ ⁇ i , j ⁇ ( x i , x j ) ⁇ ⁇ i ⁇ ⁇ ⁇ i ⁇ ( x i , y i ) ,
  • Z is a normalization constant
  • ⁇ i,j (x i , x j ) is a pairwise correlation
  • ⁇ i (x i , y i ) is the statistical dependency between a given pair of y i and x i .
  • p ⁇ ( x N ) ⁇ x ⁇ ⁇ 1 ⁇ ⁇ ⁇ x ⁇ ⁇ 2 ⁇ ⁇ ... ⁇ ⁇ ⁇ x N - 1 ⁇ ⁇ p ⁇ ( ⁇ x i ⁇ , ⁇ y i ⁇ )
  • the marginal probabilities can be computed from the dynamics of the dissipative-driven hardware in a quantum trajectory picture:
  • ⁇ ⁇ ⁇ t - i ⁇ [ H BH + H LS + H decoh , ⁇ ] + ⁇ i , i ′ , j , j ′ ⁇ ⁇ ⁇ i , i ′ , j , j ′ ⁇ a i ⁇ ⁇ a i ′ ⁇ ⁇ ⁇ a j + ⁇ i ⁇ ⁇ ⁇ i ⁇ a i ⁇ ⁇ ⁇ ⁇ ⁇ a i ⁇ ,
  • [ ] is the commutator
  • ⁇ is the density matrix
  • H BH is the Hamiltonian describing the dissipative-driven hardware
  • H LS is the Lamb shift
  • H decoh is an anti-Hermitian term proportional to the dechoerence rate of the hardware that leads to relaxation in the fixed excitation manifold and can be the Fourier transform of the bath correlation functions
  • ⁇ i,i′,j,j′ is a tensor describing the quantum jump rate among fixed-excitation manifolds
  • ⁇ i is a tensor describing quantum jump rates between fixed-excitation manifolds.
  • the dechoerence of the hardware is gradually increased to drive the dynamics of the hardware to a classical regime steady state of dissipative dynamics that encodes the solution to the computational problem (step 404 ).
  • the dynamics of the dissipative-driven hardware can be simplified to:
  • ⁇ ⁇ ⁇ t - 2 ⁇ H decoh ⁇ ⁇ + ⁇ i , i ′ , j , j ′ ⁇ ⁇ ⁇ i , i ′ , j , j ′ ⁇ a i ⁇ ⁇ a i ′ ⁇ ⁇ ⁇ ⁇ a j ′ ⁇ ⁇ a j
  • Local marginal probabilities can then be determined by the hardware and in some implementations a classical computer (step 406 ):
  • tr[ ] is the trace operation which in a density-matrix formulation is used to determine the expectation value of an operator
  • P m is a projector operator corresponding to the occupation density of a local cavity mode m.
  • the second term above retains density-density interactions between photons in a cavity mode i and in a cavity mode j that contribute to the number of photons in the visible cavity mode m.
  • the second term further retains the ⁇ i,i′,j,j′ tensor which can be related to a Markov transition matrix, which is a matrix used in the problem if the problem can be described as a Markov Random Field.
  • the problem e.g., a probabilistic inference
  • the problem can be encoded in a quantum probability distribution of the dissipative Bose-Hubbard Hamiltonian or its extended engineered version; that is using the concept of quantum graphical models.

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