Systems and methods for quantum sampling from probability distributions - Patents.com

Abstract

The system includes a quantum computer and a computing node configured to receive a description of the probability distribution, determine a first Hamiltonian having a ground state that encodes the probability distribution, determine a second Hamiltonian, where the second Hamiltonian is continuously transformable into the first Hamiltonian via a path through at least one quantum phase transition, and provide instructions to the quantum computer to: initialize a quantum system according to the ground state of the second Hamiltonian, and evolve the quantum system from the ground state of the second Hamiltonian to the ground state of the first Hamiltonian according to a path through at least one quantum phase transition. The computing node is further configured to receive measurements on the quantum system from the quantum computer to thereby obtain samples from the probability distribution.

Description

[Technical field]

[0001] REFERENCE TO RELATED APPLICATIONS This application claims the benefit of U.S. Provisional Application No. 62 / 957,400, filed January 6, 2020, which is incorporated by reference in its entirety.

[0002] STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT This invention was made with government support under grants N00014-15-1-2846 awarded by the Department of Naval Research, Office of Defense; and 1125846 and 1506284 awarded by the National Science Foundation. The Government has certain rights in this invention. [Background technology]

[0003] background Efficient algorithms for sampling from probability distributions are of wide practical importance in areas including statistical physics, optimization and machine learning. Quantum systems are naturally adapted to encode sample problems: following the Born rule, projecting measurements of a quantum state |Ψ> in an orthonormal basis [|s>] yields a probability distribution p(s) = |<s|Ψ> | 2 This observation supports recent work aiming to demonstrate the advantages of quantum over classical computers by sampling from probability distributions defined over quantum gate sequences or optical networks. These efforts have yielded impressive experimental proofs, but so far they have limited relevance for problems of practical relevance.

[0004] Thus, there is a continuing need for systems and methods for building and executing quantum algorithms useful in quantum information science. Summary of the Invention

[0005] overview In an exemplary aspect, the disclosure provides a method of sampling from a probability distribution, the method including receiving a description of the probability distribution, determining a first Hamiltonian having a ground state that encodes the probability distribution, determining a second Hamiltonian, where the second Hamiltonian is continuously transformable into the first Hamiltonian via a path through at least one quantum phase transition, initializing a quantum system according to a ground state of the second Hamiltonian, evolving the quantum system from a ground state of the second Hamiltonian to a ground state of the first Hamiltonian according to a path through at least one quantum phase transition, and performing a measurement on the quantum system, thereby obtaining a sample from the probability distribution.

[0006] In another exemplary embodiment, the present disclosure provides a method of configuring a quantum computer to sample from a probability distribution, the method including receiving a description of the probability distribution, determining a first Hamiltonian having a ground state that encodes the probability distribution, determining a second Hamiltonian, where the second Hamiltonian is continuously convertible to the first Hamiltonian via a path through at least one quantum phase transition, and providing instructions to the quantum computer to: initialize a quantum system according to the ground state of the second Hamiltonian, and evolve the quantum system from the ground state of the second Hamiltonian to the ground state of the first Hamiltonian according to a path through at least one quantum phase transition. The method further includes receiving a measurement on the quantum system from the quantum computer, thereby obtaining a sample from the probability distribution.

[0007] In yet another exemplary aspect, the present disclosure provides a computer program for configuring a quantum computer to sample from a probability distribution. product (product), and productincludes a computer-readable storage medium having program instructions embodied therein, the program instructions being executable by a processor to cause the processor to perform a method, the method including: receiving a description of a probability distribution, determining a first Hamiltonian having a ground state that encodes the probability distribution, determining a second Hamiltonian, where the second Hamiltonian is continuously convertible to the first Hamiltonian via a path through at least one quantum phase transition, and providing instructions to a quantum computer to: initialize a quantum system according to a ground state of the second Hamiltonian, and evolve the quantum system from a ground state of the second Hamiltonian to a ground state of the first Hamiltonian according to a path through the at least one quantum phase transition. The method further includes receiving, from the quantum computer, a measurement on the quantum system, thereby obtaining a sample from the probability distribution.

[0008] In yet another exemplary aspect, the present disclosure provides a system including a quantum computer and a computing node configured to receive a description of a probability distribution, to determine a first Hamiltonian having a ground state encoding the probability distribution, to determine a second Hamiltonian, where the second Hamiltonian is continuously convertible to the first Hamiltonian via a path through at least one quantum phase transition, and to provide instructions to the quantum computer to: initialize a quantum system according to the ground state of the second Hamiltonian, and to evolve the quantum system from the ground state of the second Hamiltonian to the ground state of the first Hamiltonian according to a path through at least one quantum phase transition. The computing node is further configured to receive measurements on the quantum system from the quantum computer, thereby obtaining samples from the probability distribution.

[0009] In various embodiments, determining the first Hamiltonian includes deriving the first Hamiltonian from a projected entangled pair state (PEPS) representation of a ground state of the first Hamiltonian.

[0010] In various embodiments, the description of the probability distribution includes a description of a Markov chain whose stationary distribution is the probability distribution. In various embodiments, the Markov chain satisfies detailed balance. In various embodiments, determining the first Hamiltonian includes constructing the first Hamiltonian from a Markov chain. In various embodiments, the description of the Markov chain includes a generator matrix. In various embodiments, the Markov chain includes a single-site update.

[0011] In various embodiments, the ground state of the second Hamiltonian is a product state.

[0012] In various embodiments, the progression is adiabatic.

[0013] In various embodiments, the quantum system includes a plurality of confined neutral atoms. In various embodiments, each of the plurality of confined neutral atoms is configured to sequester at least one other of the plurality of confined neutral atoms when excited to a Rydberg state. In various embodiments, initializing the quantum system includes exciting each of a subset of the plurality of confined neutral atoms according to a ground state of a second Hamiltonian. In various embodiments, evolving includes directing a time-varying beam of coherent electromagnetic radiation to each of the plurality of confined neutral atoms. In various embodiments, the plurality of confined neutral atoms are confined by optical tweezers.

[0014] In various embodiments, the probability distribution comprises a classical Gibbs distribution. In various embodiments, the paths are distinct from paths along a first set of Hamiltonians associated with the Gibbs distribution at different temperatures. In various embodiments, the Gibbs distribution is a Gibbs distribution of weighted independent sets of the graph. In various embodiments, the graph is a unit circle graph. In various embodiments, the graph is a chain graph. In various embodiments, the graph is a star graph with two vertices per branch.

[0015] In various embodiments, the Gibbs distribution is a Gibbs distribution of an Ising model. In various embodiments, the Gibbs distribution is a zero-temperature Gibbs distribution and the Ising model is a ferromagnetic 1D Ising model. In various embodiments, the Gibbs distribution is a Gibbs distribution of a classical Hamiltonian that encodes an unstructured search problem. In various embodiments, the Gibbs distribution is a zero-temperature Gibbs distribution and the unstructured search problem has a single solution.

[0016] In various embodiments, performing the measurement includes imaging the plurality of confined neutral atoms. In various embodiments, imaging the plurality of confined neutral atoms includes quantum gas microscopy.

[0017] The above-described systems and methods have many advantages, including providing quantum speedup for sampling from a probability distribution. [Brief description of the drawings]

[0018] BRIEF DESCRIPTION OF THE DRAWINGS Various objects, features and advantages of the disclosed subject matter may be more fully appreciated with respect to the following detailed description of the disclosed subject matter when considered in conjunction with the following drawings, in which like reference numerals identify like elements and in which: [Figure 1A] FIG. 1A is a flow chart of a method for sampling from a probability distribution used in an exemplary embodiment of the system described herein. [Figure 1B] FIG. 1B is a schematic illustration of a diffusive random walk in a Markov chain as a function of time. [Figure 1C] FIG. 1C is a schematic illustration of ballistic propagation of a domain wall in a quantum system, as used in an exemplary embodiment of the systems described herein. [Figure 2A] FIG. 2A is a schematic phase diagram illustrating the parent Hamiltonian corresponding to the Ising chain used in an exemplary embodiment of the system described herein. [Figure 2B] FIG. 2B is a schematic diagram illustrating the time dependence of J1 / h for an n=100 spin chain used in an exemplary embodiment of the system described herein. [Figure 2C] FIG. 2C is a plot of fidelity as a function of sweep time along the trajectory shown in FIG. 2A, used in an exemplary embodiment of the system described herein. [Figure 2D] FIG. 2D is a plot of the time t a required to achieve a fidelity of greater than 0.999 as a function of the number of spins n used in an exemplary embodiment of the system described herein. [Figure 3A] FIG. 3A is a schematic diagram of a representation of independent sets on a unit circle graph, as used in an exemplary embodiment of the system described herein. [Figure 3B] FIG. 3B is a schematic diagram of a physical realization of the parent Hamiltonian using Rydberg blockade, as used in an exemplary embodiment of the system described herein. [Figure 3C] FIG. 3C is a schematic diagram representing the parameter space and order parameters of the parent Hamiltonian for a chain of length n=30 for the unit circle graph shown in FIG. 3A, used in an exemplary embodiment of the system described herein. [Figure 3D]FIG. 3D is a plot of adiabatic conditioning time t a as a function of n for the two paths shown in FIG. 3C used in an exemplary embodiment of the system described herein. [Figure 4A] FIG. 4A is a schematic diagram illustrating sampling from a star graph used in an exemplary embodiment of the system described herein. [Figure 4B] FIG. 4B is a plot of the entropy per branch S / b of the Gibbs distribution as a function of the inverse temperature β used in an exemplary embodiment of the system described herein. [Figure 5A] FIG. 5A is a plot of a two-dimensional parameter space corresponding to a Hamiltonian associated with an unstructured search problem used in an exemplary embodiment of the system described herein. [Figure 5B] FIG. 5B is a plot of the Hamiltonian gap as a function of V0 used in an exemplary embodiment of the system described herein. [Figure 6] FIG. 6 is a schematic diagram illustrating the phase diagram of the parent Hamiltonian associated with sampling from the 1D Ising model used in an exemplary embodiment of the system described herein. [Figure 7A-D] 7A-7D show the infidelity as a function of ε for the four paths shown in FIG. 2A for chains of different lengths, from n=10 to n=1000, used in an exemplary embodiment of the system described herein.

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[0019] Detailed Description Constructing and executing useful quantum algorithms is one of the challenges in quantum information science. Efficient sampling from a probability distribution is an important computational problem with applications ranging from statistical physics to Monte Carlo and optimal subalgorithms to machine learning. A family of quantum algorithms is introduced herein that provides unbiased samples drawn from a probability distribution by preparing a quantum state that encodes the entire probability distribution. This approach is illustrated by several specific examples, including sampling from the Gibbs distribution of a one-dimensional Ising model and sampling from the Gibbs distribution of weighted independent sets. Since approximating the size of the maximum independent set on a random graph is NP (nondeterministic polynomial) hard, the latter case encompasses computationally challenging problems that may be relevant for applications such as computer vision, biochemistry, and social networks. In some embodiments, this approach is shown to provide speedups over classical Markov chain algorithms for several examples, such as sampling from the Gibbs distribution associated with a one-dimensional Ising model and sampling from the Gibbs distribution of weighted independent sets of two different graphs. In some embodiments, the practical implementation of sampling from independent ensembles based on Rydberg atomic arrays is also described herein. Without being bound by theory, this approach couples computational complexity with phase transitions, provides a physical interpretation of quantum speedups, and opens the door to exploring potentially useful sampling algorithms using near-term quantum devices.

[0020] In some embodiments, the key steps for a method 100 of sampling from a probability distribution are shown in FIG. 1A. Upon receiving 110 a description of a probability distribution p(s) defined over an additive finite event space [s], the quantum system is identified by a complete orthonormal basis [|s>]. In some embodiments, sampling from p(s) is performed by sampling the quantum state

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[0021] Without being bound by theory, the requirement that the path crosses a phase transition arises from the observation that this can result in a speedup of the quantum algorithm described herein when compared to a classical algorithm based on a Markov chain detailed below. Without being bound by theory, in two of the examples shown below, the origin of the speedup can be understood to result from the ballistic (i.e., linear in time) propagation of the domain walls in FIG. 1C enabled by quantum coherent motion at the quantum phase transition, as opposed to the diffusion caused by classical thermal fluctuations shown in FIG. 1B. Since the width of the region explored by diffusion is proportional to the square root of time (dashed curve in FIG. 1B), a quadratic speedup is generally expected. Further speedups are possible if diffusion in the Markov chain is inhibited, for example, by a thermal barrier. Without being bound by theory, an alternative speedup mechanism related to quantum tunneling is not covered in the independent set problem on star graphs, as described further below.

[0022] Parent Hamiltonian In some embodiments, two different constructions of the parent Hamiltonian are described in what follows. The Hamiltonian of a quantum system may represent one or more interactions for the quantum system. The Hamiltonian is an operator that operates on the quantum system. The eigenvalues ​​of the Hamiltonian correspond to the energy spectrum of the quantum system. The ground state of the Hamiltonian is the quantum state of the quantum system that has the minimum energy. The ground state of the Hamiltonian may be the quantum state at zero temperature.

[0023] The first construction of the parent Hamiltonian used to determine the first Hamiltonian follows the provisions of Verstraete et al. for the construction of a first Hamiltonian from a Markov chain. See Verstraete, F., Wolf, MM, Perez-Garcia, D., and Cirac, JI Criticality, the Area Law, and the Computational Power of Projected Entangled Pair States, Phys. Rev. Lett. 96, 220601 (2006), which is incorporated by reference in its entirety. A Markov chain is first defined that samples from a desired probability distribution p(s) where the ratio p(s) / p(s') is known for all pairs of elements s and s' in the event space. The Markov chain may be specified (i.e., described) by a generator matrix M, where the probability distribution q at time t is t (s) is

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[0024] The second construction of the parent Hamiltonian used to determine the first Hamiltonian is based on the general construction described in Perez-Garcia et al., which may be applied whenever the state |Ψ> has the representation known as projected entangled pair state (PEPS). See Perez-Garcia, D., Verstraete, F., Cirac, JI, and Wolf, MM PEPS as unique ground states of local Hamiltonians, Quant. Inf. Comp. 8, 0650 (2008), which is incorporated by reference in its entirety. In some embodiments, the probability distribution p(s) can be a Gibbs distribution of classical spin Hamiltonians with one and two body terms. Such a classical Hamiltonian is

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[0025] Although the embodiments described herein use the above two methods of constructing the parent Hamiltonian, those skilled in the art will understand that other constructions of the parent Hamiltonian may be used in the design of the quantum algorithms described herein based on the present disclosure. The physical realization of the parent Hamiltonian is not limited to the use of the Rydberg state of neutral atoms described below. In general, any local parent Hamiltonian can be efficiently simulated as a quantum circuit. Those skilled in the art will understand how to implement such a circuit on a given platform, including but not limited to superconducting qubits, trapped ion qubits, and neutral atom qubits. The direct physical realization of the parent Hamiltonian as described in the present disclosure has the advantage that it can be implemented with minimal overhead for Hamiltonian simulation, making it suitable for currently available quantum devices. Those skilled in the art will understand that other Hamiltonians may be realized on a given physical platform, with each platform supporting a particular set of naturally realizable Hamiltonians.

[0026] Gibbs distribution In some embodiments, the probability distribution can be a Gibbs distribution for the classical Hamiltonian. Every configuration s of the classical system has a classical energy H c (s). The corresponding Gibbs distribution is

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[0027] The examples described below show that the methods described herein of sampling from classical probability distributions using quantum computers can result in speedups over classical Markov chains. If the examples were simple enough to be analytically and numerically tractable, it would have been possible to rigorously establish this improvement. However, more challenging problems are of greater practical significance. In some embodiments, such problems include sampling from a Gibbs distribution of spin glasses with many spins or sampling from independent sets on large disordered graphs. While it may not be possible to compute the exact phase diagram for such examples, it may nevertheless be possible to establish that the initial state |φ> and the final state |Ψ> belong to different quantum phases. In some embodiments, knowledge of an approximation of the phase diagram may be sufficient to identify candidate pathways for preparing |Ψ>. In some embodiments, the pathways may be further optimized using a hybrid classical-quantum optimization loop, where a parametrized pathway is realized on a quantum computer and its parameters are optimized on a classical computer.

[0028] Time Progression To prepare the state |Ψ>, we prepare a quantum system in the initial state |φ> and subject it to a time-dependent Hamiltonian H(t). The state can be calculated by the Schrödinger equation

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[0029] In some embodiments, the time evolution can be adiabatic. Without being bound by theory, the adiabatic theorem states that if we assume that the gap in H(t) does not go to zero anywhere along the path, this has an independent global phase α,

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[0030] Although aspects of the present disclosure discuss adiabatic time evolution, those skilled in the art will appreciate that the time evolution does not need to be completely adiabatic over the time evolution. In some aspects, it is advantageous for a quantum state to have a non-adiabatic transition to an excited state before jumping back to the ground state along a continuous path. In the following example of weighted independent sets on a star graph, the time evolution of a quantum state includes a non-adiabatic transition.

[0031] Sampling from the 1D Ising model Here, the developed quantum algorithm is illustrated by considering a ferromagnetic Ising model composed of n spins in one dimension (1D). In some embodiments, the classical Hamiltonian is

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[0032] In some embodiments, the quantum phase diagram of the parent Hamiltonian for any value of h, J1, and J2 is obtained by performing a Jordan-Wigner transformation that maps Equation (6) to the free fermion model (see below). The different quantum phases are shown in FIG. 2A. The model simplifies to a transverse field Ising model on the J2 / h = 0 axis, where the phase transition from paramagnetic (PM) to ferromagnetic (FM) occurs at J1 / h = 1. Along the J1 / h = 0 axis, the ground state undergoes a symmetry-protected topological phase transition from a paramagnetic to a cluster state-like (CS) phase at J2 / h = ±1. Note that the tricritical point 250 at (J1 / h, J2 / h) = (2, 1) describes the parent Hamiltonian corresponding to zero temperature (β → ∞).

[0033] To prepare the state |Ψ(β)> for the desired inverse temperature β, we convert the Hamiltonian into its final form H qBefore smoothly varying the parameters (h, J1, J2) to (β), H0 = H q We can start from the ground state |Ψ(β)〉 by a path (ii) 220 that is sufficiently in the paramagnetic phase to be connected to |Ψ(β)〉. In this case, both adiabatic state preparation and Markov chains are efficient. Indeed, the Markov mixing time t m It has previously been shown that there exists a general quantum algorithm with run time on the order of log n for a gapped parent Hamiltonian that is identical to

[0034] The mixing time of the Markov chain is

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[0035] Without being bound by theory, these scalings are a natural consequence of the nature of the phase transition. The dynamical critical exponent at the tricritical point is z = 2, and Δ~1 / n 2 This means that the gap closes with system size as n, which is consistent with the time required along path (ii) 220. As shown below, the kinetic critical exponent for all phase transitions away from the tricritical point 250 is z = 1, and the gap closes when Δ~1 / n. Therefore, the paramagnetic to ferromagnetic phase transition can be crossed adiabatically in a time proportional to n, limited only by the ballistic propagation of domain walls shown in FIG. 1C, as opposed to the diffusive propagation in the Markov chain shown in FIG. 1B. As paths (iii) 230 and (iv) 240 approach the tricritical point 250, there is no second-order slowdown, which is due to the large overlap of the final and ground states in the ferromagnetic phase. Since the gap between the paramagnetic and cluster-state-like phases is exactly zero for certain parameters even in finite-sized systems (see below), path (i) 210 performs worse than path (ii) 220. To further support the statement that the speedup is of quantum mechanical origin, we note that the ground-state half-chain entanglement entropy diverges logarithmically with the number of spins when paths (iii)230 and (iv)240 cross from the paramagnetic to the ferromagnetic phase. Therefore, the local classical Hamiltonian H cIt is not possible to find a representation for the ground state at the phase transition in the form of equation (3) with , since any such representation is a matrix product state with constant bond dimension and bounded entanglement entropy. In some embodiments, this example shows that quantum speedups can be obtained by judiciously choosing a path through the quantum phase transition.

[0036] This example illustrates the mechanism of quantum speedup, but sampling from a Gibbs distribution for large Ising chains is difficult only at zero temperature. Since sampling at zero temperature is equivalent to optimization, there may be more appropriate algorithms to solve this problem. However, the approach described herein is not limited to such special cases, as illustrated below by Gibbs distributions where the Markov chains slowly mix even at finite temperatures. Note also that the parent Hamiltonian in equation (2) above does not have a simple physical realization, but it is a sum of local terms and therefore can be efficiently realized on a quantum computer by Hamiltonian simulation. However, for near-term applications on small quantum devices, it is desirable to identify problems for which the parent Hamiltonian has a natural implementation. Such an example is provided below for sampling from the weighted independent sets problem of the unit circle graph.

[0037] Weighted independent set An independent set of a graph is any subset of vertices such that no two vertices share an edge. An example of an independent set (black vertices 310) is shown in Figure 3A. A vertex i is a set of vertices that is in an independent set and has occupancy number n i A vertex is said to be occupied if it is assigned n = 1. All other vertices are i = 0. In the weighted independent set problem, each vertex is further assigned a weight w i and subject to the independent set constraint

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[0038] In some embodiments, to construct a quantum algorithm, each vertex is assigned a spin variable

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[0039] Projection operator P i contains up to d-field terms, where d is the degree of the graph. Nevertheless, in some embodiments, they can be implemented using, for example, programmable atomic arrays with minimal experimental overhead for certain classes of graphs. In the case of the so-called unit circle graph shown in FIG. 3A, these operators are naturally realizable in quantum systems that contain highly excited Rydberg states of multiple confined neutral atoms (see FIG. 3B and methods below). A unit circle graph is a planar graph in which any two vertices are connected if and only if they are separated by a distance less than some unit distance R. As a simple example of a unit circle graph, a chain of length n is considered, with equal weights w i= 1 is chosen. The resulting parent Hamiltonian has been studied both theoretically and experimentally using Rydberg atoms. Its quantum phase is

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[0040] Hamiltonian H q The set of (β) is shown by curve (i) 320 in FIG. 3C. Note that |Ψ(0)> is not a product state. However, the Hamiltonian H q (0) is Ω / V g= 0, V e / V g > 3, where the ground state is the product state of all unoccupied sites. In the disordered phase where the Hamiltonian has a gap, such paths can be located perfectly well, so adiabatic state preparations for |Ψ(0)> are efficient. Similarly, Markov chains at infinite temperature are efficient when the parent Hamiltonian has a gap. More generally, a gap can be created at high temperatures, e -2 β and at low temperatures e - β / n 2 It can be shown numerically below that the gap is proportional to . In contrast to the Ising chain, even for systems of finite size the gap vanishes as β → ∞. The physical reason is that

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[0041] Quantum speedup can be obtained by choosing a different path (ii) 330 shown in FIG. 3C through the quantum phase transition. For example, an approximately linear scaling t a〜n is then observed along path (ii) 330 as shown in FIG. c = 2log n, while (ii) 330 terminates as β → ∞. The sweep rate is selected according to equation (5), and the threshold fidelity used to compute the adiabatic preparation time is

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[0042] Sampling from difficult graphs As an example where sampling from an independent set is difficult even at nonzero temperature, consider next a graph. The graph takes the shape of a star with b branches and two vertices per branch, as shown in Figure 4A. The weight of the vertex at the center is b, while all other weights are set to 1. The classical model is c = log φ ≒ 0.48, where φ is the golden ratio (see method below). Above the phase transition temperature, the center is unoccupied. b The free energy is dominated by the entropic contribution from the states. Below the transition temperature, the independent set constraint reduces the number of available states by 2 b When limiting to β, it is more favorable to reduce potential energy by occupying centers at the expense of reducing entropy. As shown in FIG. 4B, the system c ≒ 0.48, which shows a discontinuous phase transition 405 (vertical dashed line). The central vertex isc and highly unoccupied otherwise. Curves 410, 420 and 450 were obtained for finite size systems with branches of b = 10, 20 and 50 respectively, while curve 460 shows the thermodynamic limit (see Methods below).

[0043] Since changing the central vertex from unoccupied to occupied requires all adjacent vertices to be unoccupied, the Markov chain on this graph has some dynamical constraints. Assuming that each individual branch is in thermal equilibrium, the probability of undergoing such a move is p0→1 = [(1 + eβ) / (1 + 2eβ)]. b The reverse process is given by p1→0= e -b The central vertex is energetically constrained by the acceptance probability of β. Thus, the central vertex can become trapped in a thermodynamically unfavorable configuration, resulting in a mixing time that grows exponentially with b at any finite temperature. When starting from a random configuration, the probability of a central vertex being initially occupied is exponentially small, so the Markov chain nevertheless samples efficiently at high temperatures. By the same argument, the Markov chain will almost certainly start in the wrong configuration in the low temperature phase, and convergence to the Gibbs distribution will occur in a time

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[0044] The corresponding quantum dynamics is captured by a two-state model formed by |Ψ0(β)〉 and |Ψ1(β)〉, which encode a Gibbs distribution at an inverse temperature β where a fixed central vertex is unoccupied or occupied, respectively (see Fig. 4A and Methods below). The tunneling rate between these states, i.e., the matrix element 〈Ψ0|H q |Ψ1> is

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[0045] Unstructured search problems The unstructured search problem has been important in the development of quantum algorithms. Grover's algorithm provided an early example of provable quantum speedup and remains an essential subroutine in many proposed quantum algorithms. Furthermore, the unstructured search problem played a crucial role in the concept of adiabatic quantum computing. It is shown below that when applied to the unstructured search problem, in some embodiments, the formalism described herein recovers the adiabatic quantum search algorithm (AQS) with its quadratic speedup over any classical algorithm. The nonlocality of the resulting parent Hamiltonian makes it a challenge to implement in practice, but the results highlight the ability of this approach to enable quantum speedup.

[0046] Consider the problem of identifying a single marked configuration m (i.e., a single solution) in the entire space of N elements. In some embodiments, to couple this search problem to the sampling problem, we assign energy −1 to the marked configuration, while all other states have energy 0. This can be expressed as:

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[0047] Now, the search problem is solved by solving H c Given the lack of structure in the problem, a natural choice for the Markov chain is to propose any configuration with equal probability 1 / N. If updates are accepted according to the Metropolis-Hastings law, the parent Hamiltonian according to equation (2) can be of the form

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[0048] Since |Ψ0> is contained in the subspace spanned by [|m>, |m⊥>], the Hamiltonian works trivially for orthogonal subspaces. In fact, all nontrivial dynamics arise from the last two terms in equation (9). Figure 5A shows the parameter space corresponding to the Hamiltonian in equation (9). Curve 510 plots V0(β) and V0(β) given by equations (11) and (12). q (β) with Hamiltonian H q (β). Line 530 corresponds to the adiabatic quantum search algorithm, and line 520 indicates the location of the first-order quantum phase transition. The inset shows an expanded view of the region of parameter space close to the origin. FIG. 5B shows the gap in the Hamiltonian in equation (6) along curves 510 and 530 of FIG. 5A as a function of V. To gain insight into the performance of the Markov chain, (V, V m Consider the quantum phase diagram in the extended two-dimensional parameter space, where any value of |m⊥ is possible. By rewriting equation (9) only in terms of |m> and |m⊥>, we obtain

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[0049] Hamiltonian Hq The path determined by the set of (β) is (V0, V m ) = (1, 0) and ends at (0, 1 / N) as β → ∞. The gap at zero temperature is equal to 1 / N, and the mixing time is

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[0050] The above description applies to V0 and V m This suggests a speedup when a path is chosen that crosses the phase transition at a point where V0, V m One such path 530 is shown in FIG. 5A, which represents a straight line segment connecting β = (1,0) to (0,1). The end point (0,1) is the parent Hamiltonian H qIt can be easily continued to (0, 1 / N) corresponding to (β). Since the Hamiltonian is purely diagonal along V0 = 0 in the computer calculation basis, there is no non-adiabatic transition along the latter segment, and the parameters can change abruptly. Along the former segment connecting (0, 1) to (1, 0), the Hamiltonian is actually identical to the Hamiltonian of the AQS algorithm. The AQS algorithm was derived by interpolating between the projection operators |Ψ0><Ψ0| and |m><m| such that the relevant Hamiltonian parameters are associated with a small time s by V0 = 1 - s and V m = s. By carefully choosing the rate of change of s, it has been shown that it is actually possible to prepare a final basis state with high fidelity at a time

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[0051] Method Adiabatic sweep In the above example, the rate of change of the Hamiltonian parameters was selected using Equation (5) with the goal of optimally satisfying the adiabatic condition at all points along a given path. If the path is parameterized by a general set of parameters λμ, Equation (5) can be written as

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[0052] For the cases tested here, it is shown below that a constant fidelity close to unity is achieved for small values ​​of ε that are nearly independent of the size of the system. Therefore, the parametric dependence of the adiabatic state preparation time on the system size follows from the integral in equation (15). Indeed, it is found that the scaling along the various paths 210, 220, 230 and 240 for the 1D Ising model can be analytically established from the individual properties of gμν at the tricritical point 250. A similar numerical analysis is provided below for both independent set problems.

[0053] Implementation with Rydberg atoms For the unit circle graph, the parent Hamiltonian for the weighted independent sets problem (7) can be implemented efficiently in quantum systems involving highly excited Rydberg states of multiple neutral atoms confined by optical tweezers, for example. As shown in Figure 3B, the ground state |g i > encodes the unoccupied state of a given vertex i. Similarly, an occupied state is the Rydberg state |r i >. The first and last terms of equation (7) are encoded as |g i >From|r i >. e,i The value of is set by the detuning of the drive, while Ω i is the driving ε i is proportional to the amplitude of the projection operator P iarises due to Rydberg blockade, where each of the plurality of confined neutral atoms is configured, when excited to a Rydberg state, to block at least one other of the plurality of confined neutral atoms. When the atoms are excited to a Rydberg state, a strong van der Waals interaction U vdW shifts the Rydberg states of all neighboring atoms off resonance, effectively turning off the drive, thereby enforcing the independent set constraint. The remaining second term in (7) describes the Rydberg blockade and the shift from the ground state to the auxiliary Rydberg state |r' i A similar approach can be used to tailor the AC Stark shift induced by non-resonant driving to > 1 / r with distance r between two atoms. 6 We add additional terms to the Hamiltonian that decay as . Note that these terms are ignored throughout and strategies to weaken their role are proposed in relevant contexts. See preprint at Pichler, H., Wang, S.-T., Zhou, L., Choi, S., and Lukin, MD Computational complexity of the Rydberg blockade in two dimensions, http: / / arxiv.org / abs / 1809.04954 (2018), which is incorporated by reference in its entirety. In an embodiment, initializing the quantum system can include exciting each of a subset of the plurality of confined neutral atoms according to a ground state of the second Hamiltonian. In some embodiments, evolving can include directing a time-varying beam of coherent electromagnetic radiation to each of the plurality of confined neutral atoms. In an embodiment, performing a measurement can include imaging the plurality of confined neutral atoms, for example by quantum gas microscopy.

[0054] In some embodiments, interactions between Rydberg atoms can also be used to implement more complex parent Hamiltonians. For example, Foerster resonance between Rydberg states can result in the simultaneous flip of two adjacent spins. In a chain graph, such an update propagates the deletion of two adjacent unoccupied vertices without an energy barrier. Finally, note that even if the star graph was not a unit circle graph, its parent Hamiltonian could potentially be implemented using anisotropic interactions between Rydberg states.

[0055] Classical phase transition of star graphs Without being bound by theory, it is possible to compute exactly the temperature at which the classical model related to the independent set problem for star graphs undergoes a phase transition. The partition function is

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[0056] A two-state model for star graphs. A star graph has three types of vertices: a central vertex and vertices on each of the internal and external branches. Without being bound by theory, if we restrict the analysis to subspaces that are completely symmetric under permutations of the branches, then the total occupancy

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[0057] Without being bound by theory, permutation symmetry leads to a bosonic algebra, each with occupation number n in , n out and the boson annihilation operator b associated with n0 in , b out and b0 are defined. The Hamiltonian can be divided into blocks whose central vertices are either occupied or unoccupied, and the off-diagonal terms coupling them. Specifically,

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[0058] The Hamiltonian is P(n in = 0) can be nearly diagonalized by perturbative treatment.

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[0059] P(n in = 0), a Schrieffer-Wolf transformation is performed on the subspace spanned by |Ψ0> and |Ψ1>, giving the efficient Hamiltonian

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[0060] Ising chain Parent Hamiltonian In some embodiments, the Grover dynamics defines a Markov chain according to the following provision: we randomly choose a spin and draw its new direction from a Gibbs distribution with all other spins fixed. Thus, starting from configuration s, the probability of flipping spin i in the Ising chain is

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[0061] Spin i The value of the operator

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[0062] Free fermion decomposition Ground state Without being bound by theory, the Hamiltonian in equation (6) or (32) above can be mapped onto a free fermion model using the Jordan-Wigner transformation. Fermion annihilation and creation operators

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[0063] The above formula (6) is

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[0064] momentum space operators

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[0065] The Hamiltonian above can be diagonalized by the standard Bogoliubov transformation, but for the purposes described herein it turns out to be more convenient to map it to non-interacting spins. For 0 < k < π,

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[0066] It is easy to check that these operators satisfy the same commutation relations as the Pauli matrices. Moreover, the operators corresponding to different values ​​of k commute in such a way that they can be considered as independent spin-1 / 2 systems, one for each value of k. The momentum ranges are governed by the multiplicity

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[0067] For concreteness, we may assume that the number of spins n is even. Then, the special cases k = 0 and k = π are both in odd parity subspaces (e i π N = -1). The Hamiltonian of the even parity subspace is

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[0068] The construction of the ground state is similar to the equivalent case, with the additional requirement that either a0 or aπ fermions are occupied, whichever has the lower energy. The resulting energy is given by, if h + J1 + J2 and h - J1 + J2 have the same sign,

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[0069] The adiabatic evolution described above was considered to start from the ground state with J1 = J2 = 0. According to the above arguments, this state is part of the even subspace. Since the time evolution preserves parity, we can restrict the description to the even subspace, dropping down all the relevant labels.

[0070] Excitation spectrum and phase diagram Excited states can be constructed by flipping any of the τ spins. Since any spin rotation commutes with the parity operator, a singly excited state can be constructed by flipping any of the τ spins.

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[0071] 6, cases 1 and 2 delineate the ferromagnetic phase 610 from the paramagnetic 620 and cluster state phases 630 in accordance with the above identification. Case 3 separates the paramagnetic phase 620 from the cluster state phase 630.

[0072] The dispersion is linear in all three cases except for two special points simultaneously satisfying h = J2 and h = ±J1. These are case 3 and the tricritical points where the dispersion minima from either case 1 or 2 merge into a single minimum with quadratic dispersion. Thus, the kinetic critical exponent is z = 1 for all phase transitions except for the tricritical point where z = 2. The gap is ~n, as can be easily seen by considering the value of k closest to the dispersion minimum for a finite size system. -zNote that in case 3 the gap exhibits oscillatory behavior and can even be exactly zero as a function of system size for fixed (h, J1, J2). Nevertheless, the envelope follows the expected approximately 1 / n scaling.

[0073] Numerical details To compute the fidelity after evolution under a time-dependent Hamiltonian following a given path, we solve the Schrödinger equation for each spin τ k Integrate numerically with respect to the energy ±2E k have

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[0074] where λ1 = J1, λ2 = J2, and we set h = 1 throughout. The total evolution time t tot To change s, the value of ε is simply adjusted. n+1 = s n + Δs n Before increasing the interval

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[0075] Adiabatic path length The above numerical observations indicate that the adiabatic conditioning time is proportional to the quantity referred to herein as the adiabatic path length, l.

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[0076] By considering the adiabatic metric close to the tricritical point, an analytical understanding of the adiabatic path length can be obtained. From equations (43) and (46),

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[0077] In the thermodynamic limit, the momentum sums turn into integrals and can be evaluated analytically. Setting h = 1 and parameterizing J1 = 2 + ηcosα, J2 = 1 + ηsinα, the result expands near the tricritical point (small η) to

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[0078] The first case corresponds to the ferromagnetic phase, while the second applies to the paramagnetic and cluster state regimes. In both cases, the adiabatic metric follows a power law, G~nη - ρ diverges as ρ, with ρ = 5 / 2 in the ferromagnetic phase and ρ = 5 elsewhere.

[0079] For finite size systems, it can be shown that at the critical point, G~nσ, exactly, where σ = 6 for any direction not parallel to the J1 axis. Based on the finite size scaling argument, the metric is expected to follow the expression for infinite systems as it approaches the tricritical point until it saturates to a final value, G~nσ. So,

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[0080] The above predictions are in good agreement with the numerical results for paths (ii)-(iv) shown in FIG. 8, but fail for path (i), where third-order scaling is observed. FIGS. 9A and 9B show G as a function of distance from the critical point when approached from the ferromagnetic phase 610 and when approached from the paramagnetic phase 620, respectively. Lines 910, 920, and 930 correspond to the system sizes shown in the legend to FIG. 9B. Curve 940 shows the analytical results for an infinite chain. FIG. 9B shows that in the latter case the scaling assumption breaks down and an oscillating term in G contributes significantly to the path length l. The oscillating term can be attributed to the fact that the variance minimum is neither k = 0 nor k = π when the tricritical point is approached from the ferromagnetic phase, resulting in an insufficient length scale that destroys the scale invariance.

[0081] Those skilled in the art will appreciate that a similar analysis can be performed for the transition between paramagnetic and ferromagnetic phases away from the tricritical point: the adiabatic path length is always found to scale linearly with the system size.

[0082] Weighted Independent Set Problem Parent Hamiltonian In some embodiments, for the weighted independent set problem, the Metropolis-Hastings update law is used instead of Grover dynamics. The motion is accept = min(1, e - βΔ E ), where ΔE is the change in energy. The probability of changing the occupancy of vertex i by a single-site update is

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[0083] Chain graph The parent Hamiltonian for the (unweighted) independent set problem on chains has been discussed previously in different contexts. For general parameters, the quantum phase diagram of the Hamiltonian can be assumed using a numerical diagonalization of the chain of finite size. The complexity of the problem is simplified such that it is only necessary to consider a subspace of independent sets. This is further restricted to states that are themselves invariant under translation (zero momentum). Assuming that the initial state satisfies this condition, the states will remain within this subspace at all times, since the Hamiltonian is translationally invariant.

[0084] Hamiltonian H as a function of β q The set gap of (β) is shown in Figures 10A and 10B. Figure 10A shows the gap for four different system sizes as a function of β. Lines 1050 and 1060 show the function e up to a constant pre-factor. -2 β and e - β, respectively. FIG. 10B shows the dependence of the gap on the number of vertices n for β = 8. Line 1070 follows the power law 1 / n 2 At high temperatures, the gap is e -2 β. Therefore, at low temperatures, the gap is e - β / n 2It is observed that as the temperature increases, the dependence on both the temperature and the system size increases. The quadratic dependence on n is due to the domain walls, i.e.

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[0085] Due to the exponentially vanishing gap, it is not possible to adiabatically achieve a state that encodes the Gibbs distribution at zero temperature along path (i) 320 discussed above. To that end, we have a β c = 2 log n. c This reflects the fact that the correlation length at is a fixed ratio of the system sizes. c It is possible to increase the overlap by adding a constant to . To determine the state preparation fidelity, we numerically integrate the Schrödinger equation by exactly diagonalizing the Hamiltonian at discrete time steps Δt. The steps are Δt|d|0> / dt| = 10 -3 where |0> denotes the instantaneous ground state. The formula is given by the identity

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[0086] The fact that the fidelity at a fixed value of ε is approximately independent of n again indicates that the adiabatic preparation time t a The adiabatic path length l along paths (i) 320 and (ii) 330 from β = 0 to β = 10 are plotted in Figures 11A and 11B, respectively. The dashed line shows the fit to the value of l at β = 10. For (i), the function Aeβ / 2 n 3 (ii) was fitted with a linear function Bn + C using the formula: For large β, the adiabatic path length is eβ along path (i) as shown in FIG. 11A. / 2 n 3 This diverges as β = β c = 2 log n, the scaling t a ~n 4 In contrast, l converges to a finite value along path (ii) as β → ∞, as shown in Figure 11B. The value is a It grows linearly with n such that ~n.

[0087] Star Graph Numerical evaluation of f In the method section above, the quantity f is

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[0088] Adiabatic Pathway Before describing the different adiabatic paths, we note that states encoding the Gibbs distribution with β = 0 can be efficiently prepared. For example, V e,i = 1 and V g,i = Ω i = 0. Then, V g,i Before doing the same for i are simultaneously increased by the same constant factor to 1. The Hamiltonian remains gapped along this path so that adiabatic state preparation proceeds efficiently, so a state encoding a Gibbs distribution with β = 0 can be used, as the first point for all adiabatic paths.

[0089] Next, the Hamiltonian H q Consider adiabatic evolution along a path defined by the set of (β). Figure 13A shows the adiabatic evolution along a path defined by the set of (β) from β = 0 to β = 2β c Hamiltonian H q Infidelity as a function of the parameter ε along the path defined by the set of (β)

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[0090] According to the above discussion, Ω cen By increasing the value of Ω, the adiabatic preparation time is expected to improve. cen is held constant at 1, while all other parameters are q A path can be considered that varies according to (β) from β = 0 to the desired final value of β (assumed to be below the phase transition). The time required to cross the phase transition is again t a Approximately 1 / J, where J is evaluated at the phase transition. The term in the efficient Hamiltonian

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[0091] Slight modifications of the pathway achieve final-state infidelity that is not only polynomially but also exponentially small in b: First, V e,cen is reduced from its initial value 1 to -1, which is the same as before at time t a Approximately (3 / 2) b / 2 All other parameters can then be calculated using the Hamiltonian H q The numerical results for these two steps are shown in Figures 14A and 14B. e,cen Before H is decreased from 1 to -1 q Non-fidelity along the path starting at (0)

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[0092] The speedup of this path over the Markov chain is quadratic [

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[0093] In some embodiments, the approach to quantum sampling algorithms described herein reveals the relationship between computational complexity and phase transitions, providing physical insight into the origin of quantum speedup. Assuming that the Gibbs distribution belongs to a local classical Hamiltonian and that the Markov chain updates are local, the quantum Hamiltonians seen in the construction are guaranteed to be local. As a result, the time evolution under these quantum Hamiltonians can be performed using Hamiltonian simulations. Furthermore, hardware-efficient implementation in near-term devices may be possible for certain cases such as the independent set problem. Although the proposed realization using Rydberg blockade is limited to unit circle graphs, a broader class of graphs may become accessible, for example, using anisotropic interactions and individual addressing with multiple atomic sublevels.

[0094] In various embodiments discussed above, the quantum computer includes a plurality of trapped neutral atoms. However, various physical embodiments of quantum computers are suitable for use according to the present disclosure. Although qubits are characterized herein as mathematical objects, each corresponds to a physical qubit that can be implemented using several different physical implementations, such as trapped ions, optical cavities, individual elementary particles, molecules, or aggregates of molecules that exhibit qubit behavior. Thus, in some embodiments, the quantum computer includes a nonlinear optical medium. In some embodiments, the quantum computer includes a cavity quantum electrodynamic device. In some embodiments, the quantum computer includes an ion trap. In some embodiments, the quantum computer includes a nuclear magnetic resonance device. In some embodiments, the quantum computer includes a superconducting device. In some embodiments, the quantum computer includes a solid-state device.

[0095] 16, a schematic of an example computing node is shown. Computing node 10 is just one example of a suitable computing node and is not intended to suggest any limitation as to the scope of use or functionality of the embodiments described herein. Regardless, computing node 10 may execute and / or perform any of the functions described above.

[0096] Among the computing nodes 10 are computing systems / servers 12 that operate in conjunction with many other general purpose or special purpose computing system environments or configurations. Examples of well-known computing systems, environments and / or configurations that may be suitable for use with the computing systems / servers 12 include, but are not limited to, personal computing systems, server computing systems, thin clients, thick clients, handheld or laptop devices, multiprocessor systems, microprocessor-based systems, set-top boxes, programmable consumer electronics, network PCs, minicomputer systems, mainframe computing systems, and distributed cloud computing environments that include any of the above-mentioned systems or devices.

[0097] The computer system / server 12 may be described in the general context of computer system executable instructions, such as program modules, executed by a computer system. Generally, program modules may include routines, programs, objects, components, logic, data structures, etc. that perform particular tasks or implement particular abstract data types. The computer system / server 12 may be practiced in a distributed cloud computing environment where tasks are performed by remote processing devices that are linked through a communications network. In a distributed cloud computing environment, program modules may be located in both local and remote computer system storage media, including memory storage devices.

[0098] 16, computer system / server 12 in computer computing node 10 is shown in the form of a general purpose computer computing device. Components of computer system / server 12 may include, but are not limited to, one or more processors or processing units 16, a system memory 28, and a bus 18 that couples various system components such as system memory 28 to processor 16.

[0099] Bus 18 represents any one or more of several types of bus structures, including a memory bus or memory controller, a peripheral bus, an accelerated graphics port, and a processor or local bus using any of a variety of bus architectures, including, by way of example and without limitation, Industry Standard Architecture (ISA) bus, MicroChannel Architecture (MCA) bus, Enhanced ISA (EISA) bus, Video Electronics Standards Association (VESA) local bus, Peripheral Component Interconnect (PCI) bus, Peripheral Component Interconnect Express (PCIe), and Advanced Microcontroller Bus Architecture (AMBA).

[0100] Computer system / server 12 typically includes a variety of computer system readable media. Such media may be any available media that is accessible by computer system / server 12 and includes both volatile and nonvolatile media, removable and non-removable media.

[0101] The system memory 28 may include computer system readable media in the form of volatile memory such as random access memory (RAM) 30 and / or cache memory 32. The computer system / server 12 may further include other removable / non-removable, volatile / non-volatile computer system storage media. By way of example only, the storage system 34 may be provided for reading from and writing to a non-removable, non-volatile magnetic medium (not shown, typically referred to as a "hard drive"). Although not shown, a magnetic disk drive may be provided for reading from and writing to a removable, non-volatile magnetic disk (e.g., a "floppy disk"), and an optical disk drive may be provided for reading from or writing to a removable, non-volatile optical disk, e.g., a CD-ROM, DVD-ROM, or other optical media. In such an example, each may be coupled to the bus 18 by one or more data media interfaces. As further shown and described below, the memory 28 may include at least one program product having a set (e.g., at least one) program module configured to perform the functions of aspects of the present disclosure.

[0102] A program / utility 40 having a set (at least one) program module 42, as well as, by way of example and not limitation, an operating system, one or more application programs, other program modules and program data, may be stored in memory 28. Each of the operating system, one or more application programs, other program modules and program data, or some combination thereof, may include execution in a network environment. The program modules 42 generally perform the functions and / or methodologies of the embodiments described herein.

[0103] The computer system / server 12 may also communicate with one or more external devices 14, such as, for example, a keyboard, a pointing device, a display 24; one or more devices that allow a user to interact with the computer system / server 12; and / or any devices (e.g., network cards, modems, etc.) that allow the computer system / server 12 to communicate with one or more other computing devices. Such communication may occur via an input / output (I / O) interface 22. Still further, the computer system / server 12 may communicate with one or more networks, such as a local area network (LAN), a general wide area network (WAN), and / or a public network (e.g., the Internet) via a network adapter 20. As shown, the network adapter 20 communicates with other components of the computer system / server 12 via a bus 18. Although not shown, it should be understood that other hardware and / or software components may be used with the computer system / server 12. Examples include, but are not limited to: microcode, device drivers, redundant processing units, external disk drive arrays, RAID systems, tape drives, and data archival storage systems.

[0104] Referring to Figure 17, a schematic diagram of a combined classical and quantum system according to an embodiment of the present disclosure is provided. In this example, a quantum computer 1701 is connected to a computing node 10, for example, via an I / O interface 22 or a network adapter 20. It is understood that additional means of connectivity between quantum computers and classical computing nodes are known in the art. In this example, the quantum computer 1701 includes a plurality of trapped neutral atoms 1702. However, as described further herein, the quantum computer 1701 may use alternative quantum systems.

[0105] In various embodiments, the computing node 10 receives a description of the probability distribution, for example from a remote node via the network adapter 20 or from an internal or external storage such as the storage system 34. The computing node 10 determines a first Hamiltonian having a ground state that encodes the probability distribution, and determines a second Hamiltonian, the second Hamiltonian being continuously convertible to the first Hamiltonian via a path through at least one quantum phase transition. The computing node 10 provides instructions to the quantum computer 1701, for example via the I / O interface 22 or the network adapter 20. The instructions indicate initializing the quantum system according to a ground state of the second Hamiltonian and evolving the quantum system from the ground state of the second Hamiltonian to the ground state of the first Hamiltonian via a path through at least one quantum phase transition. As described above, the instructions may indicate parameters of a time-varying beam of coherent electromagnetic radiation to be directed to each of the plurality of confined neutral atoms 1702. However, it is understood that alternative physical implementations of the quantum computer 1702 have platform-specific instructions for preparing and evolving the quantum system. The computing node 10 receives measurements on the quantum system from the quantum computer, thereby obtaining samples from the probability distribution.

[0106] The present disclosure may be embodied as a system, method, and / or computer program product. The computer program product may include a computer-readable storage medium(s) having computer-readable program instructions thereon for causing a processor to perform aspects of the present disclosure.

[0107] A computer readable storage medium may be a tangible device that can hold and store instructions for use by an instruction execution device. A computer readable storage medium may be, for example, but not limited to, an electronic storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or any suitable combination of the above. A non-exhaustive list of more specific examples of computer readable storage medium includes the following: portable computer diskettes, hard disks, random access memories (RAM), read-only memories (ROM), erasable programmable read-only memories (EPROM or flash memory), static random access memories (SRAM), portable compact disk read-only memories (CD-ROM), digital versatile disks (DVD), memory sticks, floppy disks, punch cards or mechanically encoded devices such as raised structures in grooves with instructions recorded on the grooves, and any suitable combination of the above. As used herein, a computer-readable storage medium is not to be understood as a transitory signal per se, such as electric waves or other freely propagating electromagnetic waves, electromagnetic waves propagating through a waveguide or other transmission medium (e.g., light pulses through a fiber optic cable), or electrical signals transmitted through wires.

[0108] The computer readable program instructions described herein may be downloaded from a computer readable storage medium to the respective computing / processing device or to an external computer or storage device via a network, such as the Internet, a local area network, a wide area network, and / or a wireless network. The network may include copper transmission cables, optical fiber transmission, wireless transmission, routers, firewalls, switches, gateway computers, and / or edge servers. A network adapter card or network interface in each computing / processing device receives the computer readable program instructions from the network and transfers the computer readable program instructions for storage in a computer readable storage medium within the respective computing / processing device.

[0109] The computer readable program instructions for carrying out the operations of the present disclosure may be either source code or object code written in any combination of assembler instructions, instruction set architecture (ISA) instructions, machine instructions, machine dependent instructions, microcode, firmware instructions, state setting data, or one or more programming languages, such as object-oriented programming languages ​​such as Smalltalk, C++, and traditional procedural programming languages ​​such as the "C" programming language or similar programming languages. The computer readable program instructions may run entirely on the user's computer, partially on the user's computer as a stand-alone software package, partially on the user's computer and partially on a remote computer, or entirely on a remote computer or server. In the latter scenario, the remote computer may be connected to the user's computer through any type of network, including a local area network (LAN) or a wide area network (WAN), or the connection may be to an external computer (e.g., through the Internet using an Internet service provider). In some embodiments, electronic circuitry including, for example, a programmable logic circuit, a field programmable gate array (FPGA), or a programmable logic array (PLA), can execute computer readable program instructions using state information of the computer readable program instructions to personalize the electronic circuitry to perform aspects of the present disclosure.

[0110] Aspects of the present disclosure are described herein with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems) and computer program products according to embodiments of the present disclosure. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer readable program instructions.

[0111] These computer readable program instructions may be provided to a processor of a general purpose computer, special purpose computer or other programmable data processing device to manufacture a machine, and the instructions executing via the processor of the computer or other programmable data processing device create means for performing the functions / acts identified in the flowchart and / or block diagram blocks or blocks. These computer readable program instructions may also be stored in a computer readable storage medium that can direct a computer, programmable data processing device and / or other device to function in a particular manner, and a computer readable storage medium having instructions stored therein includes an article of manufacture that includes instructions for performing aspects of the functions / acts identified in the flowchart and / or block diagram blocks or blocks.

[0112] The computer readable program instructions may also be loaded into a computer, other programmable data processing apparatus or other device to cause a sequence of operational steps to be executed on the computer, other programmable apparatus or other device to create a computer-implemented process, the instructions executing on the computer, other programmable apparatus or other device performing the functions / acts identified in the flowchart and / or block diagram blocks or blocks.

[0113] The flowcharts and block diagrams in the figures illustrate the architecture, functions and possible implementation operations of the systems, methods and computer program products according to various aspects of the disclosure. In this regard, each block in the flowchart or block diagram may represent a module, segment or portion of instructions that includes one or more executable instructions for performing a specific logical function(s). In some alternative implementations, the functions noted in the blocks may occur out of the order noted in the figures. For example, two blocks shown in succession may in fact be executed substantially simultaneously, or the blocks may sometimes be executed in reverse order depending on the functionality involved. It is also noted that each block of the block diagrams and / or flowchart representations, and combinations of blocks in the block diagrams and / or flowchart representations, may be executed by a special purpose hardware-based system that performs a specific function or act or a combination of special purpose hardware and computer instructions.

[0114] Although some exemplary aspects are thus described, it is understood that various variations, modifications, and improvements will be readily apparent to those skilled in the art. Such variations, modifications, and improvements are intended to form part of the present disclosure and are intended to be within the spirit and scope of the present disclosure. Although some examples shown herein include specific combinations of functions or structural elements, it should be understood that these functions and elements can be combined in other ways according to the present disclosure to achieve the same or different purposes. In particular, acts, elements, and features discussed with respect to one embodiment are not intended to be excluded from similar or other roles in other embodiments. Furthermore, the elements and components described herein may be further divided into additional components or combined together to form fewer components for performing the same functions. Accordingly, the foregoing description and accompanying drawings are intended to be illustrative only and not limiting. The aspects of the present invention include the following. Item 1 receiving a description of the probability distribution; determining a first Hamiltonian having a basis state that encodes a probability distribution; determining a second Hamiltonian, where the second Hamiltonian is continuously transformable into the first Hamiltonian via a path through at least one quantum phase transition; initializing the quantum system according to a ground state of the second Hamiltonian; evolving the quantum system from a ground state of the second Hamiltonian to a ground state of the first Hamiltonian according to a path through at least one quantum phase transition; and performing a measurement on the quantum system to thereby obtain a sample from the probability distribution; Methods for sampling from probability distributions, including: Section 2 Item 2. The method of item 1, wherein determining the first Hamiltonian includes deriving the first Hamiltonian from a projected entangled pair state (PEPS) representation of a ground state of the first Hamiltonian. Section 3 2. The method of claim 1, wherein the description of the probability distribution comprises a description of a Markov chain whose stationary distribution is the probability distribution. Section 4 The method described in section 3, in which the Markov chain satisfies detailed balance. Section 5 Item 5. The method of item 4, wherein determining the first Hamiltonian includes constructing the first Hamiltonian from a Markov chain. Section 6 4. The method of claim 3, wherein the description of the Markov chain includes a generator matrix. Section 7 The method of paragraph 3, wherein the Markov chain comprises a single site update. Section 8 2. The method of claim 1, wherein a ground state of the second Hamiltonian is a product state. Section 9 2. The method of claim 1, wherein the progression is adiabatic. Section 10 2. The method of claim 1, wherein the quantum system comprises a plurality of confined neutral atoms. Section 11 Item 11. The method of item 10, wherein each of the plurality of confined neutral atoms is configured to sequester at least one other of the plurality of confined neutral atoms when excited to a Rydberg state. Section 12 Item 11. The method of item 10, wherein initializing the quantum system includes exciting each of a subset of the plurality of confined neutral atoms according to a ground state of the second Hamiltonian. Section 13 11. The method of claim 10, wherein the advancing comprises directing a time-varying beam of coherent electromagnetic radiation at each of the plurality of confined neutral atoms. Item 14 11. The method of claim 10, wherein the plurality of trapped neutral atoms are trapped by optical tweezers. Item 15 2. The method of claim 1, wherein the probability distribution comprises a classical Gibbs distribution. Section 16 16. The method of claim 15, wherein the paths are distinct from paths along a first set of Hamiltonians associated with Gibbs distributions at different temperatures. Section 17 17. The method of claim 16, wherein the Gibbs distribution is a Gibbs distribution of weighted independent sets of the graph. Section 18 Item 18. The method of item 17, wherein the graph is a unit pie graph. Section 19 20. The method of claim 18, wherein the graph is a chain graph. Section 20 Item 18. The method of item 17, wherein the graph is a star graph with two vertices per branch. Section 21 Item 17. The method according to item 16, wherein the Gibbs distribution is a Gibbs distribution of an Ising model. Section 22 Item 22. The method of item 21, wherein the Gibbs distribution is a zero-temperature Gibbs distribution and the Ising model is a ferromagnetic 1D Ising model. Section 23 17. The method of claim 16, wherein the Gibbs distribution is a Gibbs distribution of a classical Hamiltonian that encodes an unstructured search problem. Section 24 24. The method of claim 23, wherein the Gibbs distribution is a zero-temperature Gibbs distribution and the unstructured search problem has a single solution. Section 25 Item 11. The method of item 10, wherein the step of performing a measurement includes imaging a plurality of confined neutral atoms. Section 26 26. The method of claim 25, wherein imaging the plurality of confined neutral atoms comprises quantum gas microscopy. Section 27 receiving a description of the probability distribution; determining a first Hamiltonian having a basis state that encodes a probability distribution; determining a second Hamiltonian, where the second Hamiltonian is continuously transformable into the first Hamiltonian via a path through at least one quantum phase transition; to quantum computers; to initialize the quantum system according to the ground state of the second Hamiltonian, and providing instructions to evolve the quantum system from a ground state of the second Hamiltonian to a ground state of the first Hamiltonian according to a path through at least one quantum phase transition; and receiving measurements on the quantum system from the quantum computer, thereby obtaining samples from the probability distribution; A method for configuring a quantum computer to sample from a probability distribution, comprising: Section 28 28. The method of claim 27, wherein determining the first Hamiltonian comprises deriving the first Hamiltonian from a projected entangled pair state (PEPS) representation of a ground state of the first Hamiltonian. Section 29 28. The method of claim 27, wherein the description of the probability distribution comprises a description of a Markov chain whose stationary distribution is the probability distribution. Section 30 30. The method of claim 29, wherein the Markov chain satisfies detailed balance. Section 31 31. The method of paragraph 30, wherein determining the first Hamiltonian comprises constructing the first Hamiltonian from a Markov chain. Section 32 30. The method of claim 29, wherein the description of the Markov chain includes a generator matrix. Item 33 30. The method of paragraph 29, wherein the Markov chain comprises a single site update. Section 34 28. The method of claim 27, wherein a ground state of the second Hamiltonian is a product state. Section 35 28. The method of paragraph 27, wherein the progression is adiabatic. Section 36 28. The method of claim 27, wherein the quantum system comprises a plurality of confined neutral atoms. Section 37 37. The method of claim 36, wherein each of the plurality of confined neutral atoms is configured to sequester at least one other of the plurality of confined neutral atoms when excited to a Rydberg state. Section 38 37. The method of claim 36, wherein initializing the quantum system includes exciting each of a subset of the plurality of confined neutral atoms according to a ground state of the second Hamiltonian. Section 39 37. The method of claim 36, wherein the advancing comprises directing a time-varying beam of coherent electromagnetic radiation at each of the plurality of confined neutral atoms. Section 40 37. The method of claim 36, wherein the plurality of trapped neutral atoms are trapped by optical tweezers. Section 41 28. The method of paragraph 27, wherein the probability distribution comprises a classical Gibbs distribution. Section 42 42. The method of claim 41, wherein the paths are distinct from paths along a first set of Hamiltonians associated with Gibbs distributions at different temperatures. Section 43 43. The method of claim 42, wherein the Gibbs distribution is a Gibbs distribution of weighted independent sets of the graph. Section 44 44. The method of claim 43, wherein the graph is a unit pie graph. Section 45 45. The method of claim 44, wherein the graph is a chain graph. Section 46 44. The method of claim 43, wherein the graph is a star graph with two vertices per branch. Section 47 Item 43. The method of item 42, wherein the Gibbs distribution is a Gibbs distribution of an Ising model. Section 48 Item 48. The method of item 47, wherein the Gibbs distribution is a zero-temperature Gibbs distribution and the Ising model is a ferromagnetic 1D Ising model. Section 49 43. The method of claim 42, wherein the Gibbs distribution is a Gibbs distribution of a classical Hamiltonian that encodes an unstructured search problem. Section 50 50. The method of claim 49, wherein the Gibbs distribution is a zero-temperature Gibbs distribution and the unstructured search problem has a single solution. Section 51 37. The method of claim 36, wherein the step of performing a measurement comprises imaging the plurality of trapped neutral atoms. Section 52 52. The method of paragraph 51, wherein imaging the plurality of confined neutral atoms comprises quantum gas microscopy. Section 53 1. A computer program product for configuring a quantum computer to sample from a probability distribution, the computer program product comprising a computer readable storage medium having program instructions embodied thereby, the program instructions being executable by a processor to cause the processor to perform a method, the method comprising: receiving a description of the probability distribution; determining a first Hamiltonian having a basis state that encodes a probability distribution; determining a second Hamiltonian, where the second Hamiltonian is continuously transformable into the first Hamiltonian via a path through at least one quantum phase transition; For quantum computers: to initialize the quantum system according to the ground state of the second Hamiltonian, and providing instructions to evolve the quantum system from a ground state of the second Hamiltonian to a ground state of the first Hamiltonian according to a path through at least one quantum phase transition; and receiving, from the quantum computer, measurements on the quantum system to thereby obtain samples from the probability distribution; 2. A computer program product comprising: Section 54 54. The computer program product of paragraph 53, wherein determining the first Hamiltonian comprises deriving the first Hamiltonian from a projected entangled pair state (PEPS) representation of a ground state of the first Hamiltonian. Section 55 54. The computer program product of clause 53, wherein the description of the probability distribution comprises a description of a Markov chain whose stationary distribution is the probability distribution. Section 56 56. The computer program product of clause 55, wherein the Markov chain satisfies the detailed balance. Section 57 57. The computer program product of paragraph 56, wherein determining the first Hamiltonian comprises constructing the first Hamiltonian from a Markov chain. Section 58 56. The computer program product of clause 55, wherein the description of the Markov chain includes a generator matrix. Section 59 56. The computer program product of paragraph 55, wherein the Markov chain comprises a single site update. Item 60 54. The computer program product of paragraph 53, wherein a ground state of the second Hamiltonian is a product state. Section 61 54. The computer program product of paragraph 53, wherein the evolution is adiabatic. Section 62 54. The computer program product of paragraph 53, wherein the quantum system comprises a plurality of confined neutral atoms. Section 63 63. The computer program product of paragraph 62, wherein each of the plurality of confined neutral atoms is configured to sequester at least one other of the plurality of confined neutral atoms when excited to a Rydberg state. Section 64 63. The computer program product of paragraph 62, wherein initializing the quantum system includes exciting each of a subset of the plurality of confined neutral atoms according to a ground state of the second Hamiltonian. Section 65 63. The computer program product of paragraph 62, wherein the step of propagating includes directing a time-varying beam of coherent electromagnetic radiation to each of the plurality of confined neutral atoms. Section 66 63. The computer program product of paragraph 62, wherein the plurality of trapped neutral atoms are trapped by optical tweezers. Section 67 54. The computer program product of paragraph 53, wherein the probability distribution comprises a classical Gibbs distribution. Section 68 70. The computer program product of paragraph 67, wherein the paths are distinct from paths along a first ensemble of Hamiltonians associated with Gibbs distributions at different temperatures. Section 69 70. The computer program product of paragraph 68, wherein the Gibbs distribution is a Gibbs distribution of weighted independent sets of the graph. Section 70 70. The computer program product of paragraph 69, wherein the graph is a unit pie graph. Section 71 71. The computer program product of paragraph 70, wherein the graph is a chain graph. Section 72 70. The computer program product of paragraph 69, wherein the graph is a star graph with two vertices per branch. Section 73 70. The computer program product of paragraph 68, wherein the Gibbs distribution is a Gibbs distribution of an Ising model. Section 74 64. The computer program product of paragraph 63, wherein the Gibbs distribution is a zero-temperature Gibbs distribution and the Ising model is a ferromagnetic 1D Ising model. Section 75 70. The computer program product of paragraph 68, wherein the Gibbs distribution is a Gibbs distribution of a classical Hamiltonian that encodes the unstructured search problem. Section 76 76. The computer program product of paragraph 75, wherein the Gibbs distribution is a zero-temperature Gibbs distribution and the unstructured search problem has a single solution. Section 77 63. The computer program product of paragraph 62, wherein performing the measurement comprises imaging the plurality of confined neutral atoms. Section 78 80. The computer program product of paragraph 77, wherein imaging the plurality of confined neutral atoms comprises quantum gas microscopy. Section 79 Quantum computers; and To receive a description of the probability distribution; determining a first Hamiltonian having a ground state that encodes a probability distribution; determining a second Hamiltonian, where the second Hamiltonian is continuously transformable into the first Hamiltonian via a path through at least one quantum phase transition; For quantum computers: to initialize the quantum system according to the ground state of the second Hamiltonian, and providing instructions to evolve the quantum system from a ground state of the second Hamiltonian to a ground state of the first Hamiltonian according to a path through at least one quantum phase transition; and Receive measurements on the quantum system from the quantum computer, thereby obtaining samples from the probability distribution. The computer computing node that will be built Including, the system. Section 80 80. The system of claim 79, wherein determining the first Hamiltonian includes deriving the first Hamiltonian from a projected entangled pair state (PEPS) representation of a ground state of the first Hamiltonian. Section 81 80. The system of claim 79, wherein the description of the probability distribution includes a description of a Markov chain whose stationary distribution is the probability distribution. Section 82 82. The system of claim 81, wherein the Markov chain satisfies detailed equilibrium. Section 83 83. The system of claim 82, wherein determining the first Hamiltonian includes constructing the first Hamiltonian from a Markov chain. Section 84 82. The system of claim 81, wherein the description of the Markov chain includes a generator matrix. Section 85 82. The system of claim 81, wherein the Markov chain comprises a single site update. Section 86 80. The system of claim 79, wherein a ground state of the second Hamiltonian is a product state. Section 87 80. The system of paragraph 79, wherein the progression is adiabatic. Section 88 80. The system of claim 79, wherein the quantum system comprises a plurality of confined neutral atoms. Section 89 90. The system of claim 88, wherein each of the plurality of confined neutral atoms is configured to sequester at least one other of the plurality of confined neutral atoms when excited to a Rydberg state. Section 90 89. The system of claim 88, wherein initializing the quantum system includes exciting each of a subset of the plurality of confined neutral atoms according to a ground state of the second Hamiltonian. Section 91 90. The system of claim 88, wherein the step of: directing a time-varying beam of coherent electromagnetic radiation to each of the plurality of confined neutral atoms. Section 92 89. The system of paragraph 88, wherein the plurality of trapped neutral atoms are trapped by optical tweezers. Section 93 80. The system of claim 79, wherein the probability distribution comprises a classical Gibbs distribution. Section 94 94. The system of claim 93, wherein the paths are distinct from paths along a set of first Hamiltonians associated with Gibbs distributions at different temperatures. Section 95 95. The system of clause 94, wherein the Gibbs distribution is a Gibbs distribution of weighted independent sets of the graph. Section 96 96. The system of claim 95, wherein the graph is a unit pie graph. Section 97 97. The system of claim 96, wherein the graph is a chain graph. Section 98 96. The system of claim 95, wherein the graph is a star graph with two vertices per branch. Section 99 95. The system of claim 94, wherein the Gibbs distribution is an Ising model Gibbs distribution. Item 100 99. The system of claim 99, wherein the Gibbs distribution is a zero-temperature Gibbs distribution and the Ising model is a ferromagnetic 1D Ising model. Section 101 95. The system of paragraph 94, wherein the Gibbs distribution is a Gibbs distribution of a classical Hamiltonian that encodes an unstructured search problem. Section 102 102. The system of claim 101, wherein the Gibbs distribution is a zero-temperature Gibbs distribution and the unstructured search problem has a single solution. Section 103 89. The system of claim 88, wherein performing the measurement includes imaging the plurality of trapped neutral atoms. Section 104 104. The system of claim 103, wherein imaging the plurality of confined neutral atoms includes quantum gas microscopy.

Claims

1. receiving a description of a probability distribution; determining a first Hamiltonian having basis states that encode a probability distribution; determining a second Hamiltonian, where the second Hamiltonian is continuously transformable into the first Hamiltonian via a path through at least one quantum phase transition; initializing the quantum system according to a ground state of the second Hamiltonian; evolving the quantum system from a ground state of the second Hamiltonian to a ground state of the first Hamiltonian according to a path through at least one quantum phase transition; and performing measurements on the quantum system to thereby obtain samples from the probability distribution Methods for sampling from probability distributions, including:

2. receiving a description of a probability distribution; determining a first Hamiltonian having basis states that encode a probability distribution; determining a second Hamiltonian, where the second Hamiltonian is continuously transformable into the first Hamiltonian via a path through at least one quantum phase transition; For quantum computers; To initialize the quantum system according to the ground state of the second Hamiltonian, and Following a path through at least one quantum phase transition, the second Hamiltonian We advance the quantum system from the ground state of the first Hamiltonian to the ground state of the first Hamiltonian. Let it unfold providing instructions; and receiving measurements on the quantum system from the quantum computer, thereby obtaining samples from the probability distribution; A method for configuring a quantum computer to sample from a probability distribution, comprising:

3. 3. The method of claim 1, wherein determining the first Hamiltonian comprises deriving the first Hamiltonian from a projected entangled pair state (PEPS) representation of a ground state of the first Hamiltonian.

4. 3. A method according to claim 1 or 2, wherein the description of the probability distribution comprises a description of a Markov chain whose stationary distribution is the probability distribution.

5. The method of claim 4, wherein the Markov chain satisfies detailed balance.

6. 6. The method of claim 5, wherein determining the first Hamiltonian comprises constructing the first Hamiltonian from a Markov chain.

7. The method of claim 4 , wherein the description of the Markov chain includes a generator matrix.

8. The method of claim 4 , wherein the Markov chain comprises a single site update.

9. 3. The method of claim 1 or 2, wherein the ground state of the second Hamiltonian is a product state.

10. 3. The method of claim 1 or 2, wherein the progression is adiabatic.

11. The method of claim 1 or 2, wherein the quantum system comprises a plurality of confined neutral atoms.

12. 12. The method of claim 11, wherein each of the plurality of confined neutral atoms is configured to sequester at least one other of the plurality of confined neutral atoms when excited to a Rydberg state.

13. 12. The method of claim 11, wherein initializing the quantum system comprises exciting each of a subset of the plurality of confined neutral atoms according to a ground state of the second Hamiltonian.

14. The method of claim 11 , wherein the advancing comprises directing a time-varying beam of coherent electromagnetic radiation at each of the plurality of confined neutral atoms.

15. The method of claim 11 , wherein the plurality of trapped neutral atoms are trapped by optical tweezers.

16. 3. The method of claim 1 or 2, wherein the probability distribution comprises a classical Gibbs distribution.

17. 17. The method of claim 16, wherein the paths are distinct from paths along a first set of Hamiltonians associated with Gibbs distributions at different temperatures.

18. 20. The method of claim 17, wherein the Gibbs distribution is a Gibbs distribution of weighted independent sets of the graph.

19. 20. The method of claim 18, wherein the graph is a unit pie graph.

20. 20. The method of claim 19, wherein the graph is a chain graph.

21. 20. The method of claim 18, wherein the graph is a star graph with two vertices per branch.

22. 20. The method of claim 17, wherein the Gibbs distribution is an Ising model Gibbs distribution.

23. 23. The method of claim 22, wherein the Gibbs distribution is a zero-temperature Gibbs distribution and the Ising model is a ferromagnetic 1D Ising model.

24. 20. The method of claim 17, wherein the Gibbs distribution is a Gibbs distribution of a classical Hamiltonian that encodes an unstructured search problem.

25. 25. The method of claim 24, wherein the Gibbs distribution is a zero-temperature Gibbs distribution and the unstructured search problem has a single solution.

26. The method of claim 11 , wherein performing a measurement comprises imaging the plurality of confined neutral atoms.

27. 27. The method of claim 26, wherein imaging the plurality of confined neutral atoms comprises quantum gas microscopy.

28. 1. A computer program for configuring a quantum computer to sample from a probability distribution, the computer program having program instructions executable by a processor to cause the processor to perform a method, the method comprising: receiving a description of a probability distribution; determining a first Hamiltonian having basis states that encode a probability distribution; determining a second Hamiltonian, where the second Hamiltonian is continuously transformable into the first Hamiltonian via a path through at least one quantum phase transition; For quantum computers: To initialize the quantum system according to the ground state of the second Hamiltonian, and Following a path through at least one quantum phase transition, the second Hamiltonian We advance the quantum system from the ground state of the first Hamiltonian to the ground state of the first Hamiltonian. Let it unfold providing instructions; and receiving, from the quantum computer, measurements on the quantum system to thereby obtain samples from the probability distribution; A computer program comprising:

29. Quantum computers; and a computing node, the computing node comprising: to receive a description of a probability distribution; Determine the first Hamiltonian whose ground states encode the probability distribution. To determine; To determine the second Hamiltonian, where the second Hamiltonian is via a path through at least one quantum phase transition, can be continuously converted to For quantum computers: Initialize the quantum system according to the ground state of the second Hamiltonian. and A quantum system can be created by following a path through at least one quantum phase transition. From the ground state of the second Hamiltonian to the ground state of the first Hamiltonian, to the ground state of To provide instructions; and Receive measurements on the quantum system from the quantum computer and So that we can sample from a probability distribution To be constructed, Including, the system.

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