State preparation for quantum computing

Abstract

Quantum computing devices, systems and methods for performing quantum state preparation of a purified target quantum state. A first quantum register is prepared in a first state that includes, for each of a plurality of collective modes, a summation over energy label states weighted based on a Gibbs probability distribution. A second quantum register is prepared in a second state that includes, for each respective collective mode, a summation over respective eigenstates. A third quantum register is prepared in a third state by applying a plurality of third unitary operators to the second and third quantum registers to transform the collective mode coordinates of the second state to physical coordinates, and adding this transformed state into the third quantum register. The state of the second quantum register is then un-computed to return the second quantum register to the null state and the first and third quantum registers are prepared in the purified target quantum state.

Description

STATE PREPARATION FOR QUANTUM COMPUTING Technical Field

[0001] Embodiments herein relate generally to quantum computational methods, systems and devices for performing quantum state preparation. Background

[0002] Quantum computing can be distinguished from “classical” computing by its reliance on structures referred to as “qubits.” At the most general level, a qubit is a quantum system that may exist in one of two orthogonal states (denoted as |0 and |1 in the conventional bra / ket notation) or in a superposition of the two states By operating on a system (or ensemble) ofqubits, a quantum computer may quickly perform certain categories of computations that would require impractical amounts of time in a classical computer.

[0003] One application of quantum computing is the simulation of physical quantum systems. The quantum system may include a plurality of particles and / or fields of different types, with differing properties and interactions. Quantum simulation involves preparing qubits into an initial quantum state, which may be a complex and computationally intensive procedure, particularly for more complex systems. Accordingly, improvements in the field of quantum state preparation are desired to increase the efficiency and reduce the complexity of quantum simulation and other quantum computational methods. Summary

[0004] Embodiments described herein include quantum computing devices, systems, quantum circuits, and methods for performing quantum state preparation to prepare quantum registers into a target quantum state.

[0005] In some embodiments, a first quantum register is prepared in a first state. This first state includes, for each of a plurality of collective modes, a summation over energy label states weighted based on a Gibbs probability distribution.

[0006] In some embodiments, a second quantum register is prepared in a second state, conditional on the state of the first quantum register. Preparing the second quantum register in the second state conditioned on the first state of the first quantum register entangles the second state with the first state. The resultant joint state of the first and second registers includes, for each of the plurality of collective modes, a summation over eigenstates of a respective collective mode.

[0007] In some embodiments, a third quantum register is prepared in a third state. The third register is prepared in the third state by applying a plurality of third unitary operators to thesecond and third quantum registers. In some embodiments, applying the plurality of third unitary operators to the second and third quantum registers transforms the collective mode coordinates of the second state to physical coordinates, and adds this transformed state (i.e., the third state) into the third quantum register. After this operation, the third state will be entangled with the first and second states.

[0008] In some embodiments, the second quantum register is prepared in a null state. The second quantum register is prepared in the null state by applying a plurality of fourth unitary operators 14a-b) to the second and third quantum registers. Applying the plurality of fourth unitary operators to the second and third quantum registers rotates the physical coordinates of the third state back into the collective mode coordinates, and subtracts the result from the second register. Returning the second quantum register to the null state removes the entanglement between the second quantum register and the first and third quantum registers, and prepares the first and third quantum registers in an entangled purified state that is the target quantum state.

[0009] In some embodiments, a quantum simulation using a full Hamiltonian may be performed using the purified quantum state. In some embodiments, after performing the quantum simulation, measurements are performed on one or more of the quantum registers to obtain classical measurement results. In some embodiments, the classical measurement results are stored in a non- transitory computer-readable memory medium.

[0010] The techniques described herein may be implemented in and / or used with a number of different types of devices, including but not limited to photonic quantum computing devices and / or systems, hybrid quantum / classical computing systems, and any of various other quantum computing systems.

[0011] This Summary is intended to provide a brief overview of some of the subject matter described in this document. Accordingly, it will be appreciated that the above-described features are merely examples and should not be construed to narrow the scope or spirit of the subject matter described herein in any way. Other features, aspects, and advantages of the subject matter described herein will become apparent from the following Detailed Description, Figures, and Claims. Brief Description of the Drawings

[0012] For a better understanding of the various described embodiments, reference should be made to the Detailed Description below, in conjunction with the following drawings in which like reference numerals refer to corresponding parts throughout the Figures.

[0013] Figure 1A is a system diagram illustrating a quantum computing system, according to some embodiments;

[0014] Figures 1B-G illustrate the utilization of surface codes to constructed an error-corrected fault- tolerant logical qubit, according to some embodiments;

[0015] Figure 2 is a flowchart diagram illustrating a method for performing quantum state preparation, according to some embodiments;

[0016] Figure 3 is a quantum circuit diagram configured to perform quantum state preparation, according to some embodiments;

[0017] Figure 4 is a quantum circuit diagram illustrating a decomposition of a time evolution operator into a plurality of qubitization operators, according to some embodiments;

[0018] Figure 5 is a quantum circuit diagram illustrating the component subroutines of a qubitization operator, according to some embodiments; and

[0019] Figure 6 is a quantum circuit diagram illustrating a qubitization circuit for phase estimation, according to some embodiments.

[0020] While the features described herein may be susceptible to various modifications and alternative forms, specific embodiments thereof are shown by way of example in the drawings and are herein described in detail. It should be understood, however, that the drawings and detailed description thereto are not intended to be limiting to the particular form disclosed, but on the contrary, the intention is to cover all modifications, equivalents and alternatives falling within the spirit and scope of the subject matter as defined by the appended claims. DETAILED DESCRIPTION

[0021] Disclosed herein are examples (also referred to as “embodiments”) of systems and methods for simulating a physical quantum system using various quantum computing systems, including photonic systems.

[0022] Although embodiments are described with specific detail to facilitate understanding, those skilled in the art with access to this disclosure will appreciate that the claimed invention may be practiced without these details. Reference will now be made in detail to embodiments, examples of which are illustrated in the accompanying drawings. In other instances, well-known methods, procedures, components, circuits, and networks have not been described in detail so as not to unnecessarily obscure aspects of the embodiments.Overview of Quantum Computing

[0023] To facilitate understanding of the disclosure, an overview of relevant concepts and terminology is provided in the following paragraphs.

[0024] Quantum computing relies on the dynamics of quantum objects, e.g., photons, electrons, atoms, ions, molecules, nanostructures, and the like, which follow the rules of quantum theory. In quantum theory, the quantum state of a quantum object is described by a set of physical properties, the complete set of which is referred to as a mode. In some embodiments, a mode is defined by specifying the value (or distribution of values) of one or more properties of the quantum object. For example, in the case where the quantum object is a photon, modes may be defined by the frequency of the photon, the position in space of the photon (e.g., which waveguide or superposition of waveguides the photon is propagating within), the associated direction of propagation (e.g., the k- vector for a photon in free space), the polarization state of the photon (e.g., the direction (horizontal or vertical) of the photon’s electric and / or magnetic fields), a time window in which the photon is propagating, the orbital angular momentum state of the photon, and the like.

[0025] Persons of ordinary skill in the art will be able to implement examples using any of a variety of types of quantum systems, including but not limited to photonic systems, solid state system, topological quantum computing systems, hybrid quantum computing systems, and superconducting systems, among other possibilities.

[0026] As used herein, a “qubit” (or quantum bit) is a quantum system with an associated quantum state that may be used to encode information. A quantum state may be used to encode one bit of information if the quantum state space can be modeled as a (complex) two-dimensional vector space, with one dimension in the vector space being mapped to logical value 0 and the other to logical value 1. In contrast to classical bits, a qubit may have a state that is a superposition of logical values 0 and 1. More generally, a “qudit” describes any quantum system having a quantum state space that may be modeled as a (complex) n-dimensional vector space (for any integer n), which may be used to encode log2(n) bits of information. For the sake of clarity of description, the term “qubit” is used herein, although in some embodiments the system may also employ quantum information carriers that encode information in a manner that is not necessarily associated with a binary bit, such as a qudit or a plurality of qubits encoded to form an error-corrected logical qubit. For example, embodiments herein for quantum computational methods and circuits that utilize fault-tolerant quantum computing schemes use the term “qubit” to refer to an error-corrected logical qubit that contains a plurality of physical qubits entangled together in an error-correcting code. Embodiments herein use the term “quantum register” to refer to a set of one or more qubits used in a quantumcomputational method. Typically, distinct quantum registers are separated at the logical layer, i.e., different quantum registers serve distinct logical purposes in a quantum computing method or circuit.

[0027] Qubits (or qudits) may be implemented in a variety of quantum systems. Examples of qubits include: polarization states of photons; presence of photons in waveguides; or energy states of molecules, atoms, ions, nuclei, or photons. Other examples include other engineered quantum systems such as flux qubits, phase qubits, or charge qubits (e.g., formed from a superconducting Josephson junction); topological qubits (e.g., Majorana fermions); or spin qubits formed from vacancy centers (e.g., nitrogen vacancies in diamond). Figure 1A – Quantum Computing System

[0028] Figure 1A is a system diagram of a quantum computing system 101 that may be utilized to implement method steps of embodiments described herein. As illustrated, the system includes a classical computing system 103 coupled to a quantum processing unit (QPU) 105 over a classical channel 112. The classical channel may relay classical information between the classical computing system and the QPU.

[0029] In some embodiments, the classical computing system 103 includes one or more non- transitory computer-readable memory media 104, one or more central processing units (CPUs) or processor(s) 102, a power supply, an input / output (I / O) subsystem, and a communication bus interconnecting these components. The processor(s) 102 may execute modules, programs, and / or instructions stored in memory 104 and thereby perform processing operations. The processor(s) may additionally or alternatively perform operations based on information and / or instructions received from the QPU 105 over the channel 112. The processor may comprise a dedicated processor, or it may be a field programmable gate arrays (FPGA), an application specific integrated circuit (ASIC), or a “system on a chip” that includes classical processors and memory, among other possibilities. In some embodiments, memory 104 stores one or more programs (e.g., sets of instructions) and / or data structures and is coupled to the processor(s).

[0030] The classical computing system may be classical in the sense that it operates computer code represented as a plurality of classical bits that may take a value of 1 or 0. Programs may be written in the form of ordered lists of instructions and stored within the classical (e.g., digital) memory 104 and executed by the classical (e.g., digital) processor 102 of the classical computer. The memory 104 is classical in the sense that it stores data and / or program instructions in a storage medium in the form of bits, which have a single definite binary state at any point in time. The processor may read instructions from the computer program in the memory 104 and / or write data into memory, and may optionally receive input data from a source external to the computer 103, such as from a user inputdevice such as a mouse, keyboard, or any other input device. The processor 102 may execute program instructions that have been read from the memory 104 to perform computations on data read from the memory 104 and / or input from the QPU, and generate output from those instructions. The processor 102 may store that output back into the memory 104 and / or provide the output to the QPU over the channel 112.

[0031] The QPU 105 may include a plurality of qubits and a controller 106 configured to interface with a plurality of qubits 110. In some embodiments, the qubits are divided into one or more independent qubit modules, where each qubit module includes a self-contained plurality of fault- tolerant qubits, and different qubit modules may be interchangeably used for various steps within a quantum computation. The controller 106 may include physical hardware to interact with and / or perform operations on the qubits, e.g., to apply quantum gates or perform other operations. In some embodiments, the controller further includes a classical processor, potentially coupled to its own dedicated non-transitory (classical) memory, that is configured to direct the physical hardware to interact with the qubits and communicate with the processor of the classical computing system 103 over the channel 112. Alternatively, the classical processor of the classical computing system 103 may directly communicate with the hardware of the controller to provide instructions for interacting with and manipulating the qubits. The qubits may be configured to evolve in time under the directed influence of the controller, and a measurement system 108 may at times perform quantum measurements on all or a subset of the qubits to obtain quantum measurement results in the form of classical data bits (e.g., ones and zeros). The classical data from the measurement results may be intermediate results that inform behavior of the classical computing system and / or the quantum controller 106 during a quantum computation, and they may additionally include classical results of the quantum computation. In some embodiments, the QPU further includes one or more decoders configured to receive and decode the classical measurement results, and the decoded measurement results may be provided to the classical computing system for processing. The measurement results may be communicated to the classical computing system and / or the controller 106, and further the classical computing system may provide directions and / or instructions to the controller 106 and the measurement system 108 to guide the behavior of the QPU while performing a quantum computation. Figures 1B-J – Surface Codes and Physical implementations

[0032] Qubits (and operations on qubits) may be implemented using a variety of physical systems. In some embodiments, qubits are provided in an integrated photonic system employing waveguides, beam splitters, photonic switches, and single photon detectors, and the modes that may be occupiedby photons are spatiotemporal modes that correspond to presence of a photon in a waveguide. Modes may be coupled using mode couplers, e.g., optical beam splitters, to implement transformation operations, and measurement operations may be implemented by coupling single- photon detectors to specific waveguides. One of ordinary skill in the art with access to this disclosure will appreciate that modes defined by any appropriate set of degrees of freedom, e.g., polarization modes, temporal modes, and the like, may be used without departing from the scope of the present disclosure. For instance, for modes that only differ in polarization (e.g., horizontal (H) and vertical (V)), a mode coupler may be any optical element that coherently rotates polarization, e.g., a birefringent material such as a waveplate. For other systems such as ion trap systems or neutral atom systems, a mode coupler may be any physical mechanism that couples two modes, e.g., a pulsed electromagnetic field that is tuned to couple two internal states of the atom / ion.

[0033] In some embodiments of a photonic quantum computing system using dual-rail encoding, a qubit may be implemented using a pair of waveguides. In some embodiments, a photon in a first waveguide of the pair and no photon in a second waveguide of the pair (also referred to as a vacuummode) may correspond to the |0 state of a photonic qubit. Alternatively, a state with a photon in thesecond waveguide and no photon in the first waveguide may correspond to the |1 state of thephotonic qubit. To prepare a photonic qubit in a known logical state, a photon source may be coupled to one end of one of the waveguides. The photon source may be operated to emit a single photon into the waveguide to which it is coupled, thereby preparing a photonic qubit in a known state. Photons travel through the waveguides, and by periodically operating the photon source, a quantum system having qubits whose logical states map to different temporal modes of the photonic system may be created in the same pair of waveguides. In addition, by providing multiple pairs of waveguides, a quantum system having qubits whose logical states correspond to different spatiotemporal modes may be created. It should be understood that the waveguides in such a system need not have any particular spatial relationship to each other. For instance, they may be but need not be arranged in parallel.

[0034] Some embodiments described below relate to physical implementations of unitary operations that couple modes of a quantum system, which may be understood as transforming the quantum state of the system. For instance, if the initial state of the quantum system (prior to mode coupling) is one in which one mode is occupied with probability 1 and another mode is unoccupied with probability 1(e.g., a state |10 in Fock notation), mode coupling may result in a state in which both modes have anonzero probability of being occupied, e.g., a state |10 |01 , where |some embodiments, operations of this kind may be implemented by using beam splitters to couple modes together and variable phase shifters to apply phase shifts to one or more modes. Theamplitudes a1and a2depend on the reflectivity (or transmissivity) of the beam splitters and on any phase shifts that are introduced.

[0035] A single physical qubit (e.g., such as the 2-level physical qubit illustrated in Figure 1B with aquantum state may be used for quantum computation in principle. However,individual physical qubits are generally highly susceptible to noise and decoherence. Fault-tolerant quantum computing utilizes a plurality of entangled physical qubits to encode a single logical qubit to mitigate the frailty and / or short coherence times of individual physical qubits. In fault-tolerant quantum computing schemes, a plurality of physical qubits is entangled together according to a specific error-correcting code (e.g., using fusion measurements on resource states) to produce a single logical qubit that is less susceptible to noise and decoherence, such as is shown in Figure 1C.

[0036] Figure 1C illustrates one example for constructing a fault-tolerant logical qubit using a circuit-based approach. In the illustrated example, the light shaded circles are data qubits (e.g., qubits 125-131) that encode quantum information. The data qubits are entangled with adjacent measure qubits, illustrated as dark shaded circles (such as measure qubit 123). The measure qubits may be measured to determine aspects of the quantum information encoded in the data qubits. The example illustrated in Figure 1C has a code length of 12. Fusion-based approaches to encoding fault- tolerant logical qubits may also be used for embodiments described herein. Encoding qubits in this manner causes the resultant logical qubit to be less sensitive to error and noise, and resultant errors may be fixed via quantum error correction. Encoding a logical qubit may itself be vulnerable to errors, which may likewise be corrected and / or tolerated.

[0037] In some quantum computing methodologies, such as fusion-based quantum computing, a logical qubit is encoded from a plurality of physical qubits using a sequence of specific measurements (e.g., stabilizer measurements). The measurement sequence may be constructed where a subset of the physical qubits is measured (e.g., collapsing the quantum state and producing classical information, i.e., the measurement result) in such a way that the remaining unmeasured / un-collapsed degrees of freedom (e.g., a 2-dimensional subspace which has support over all the physical qubits) form the desired encoded logical qubit. Accordingly, the processes of performing stabilizer measurements and / or encoding a fault-tolerant logical qubit may receive a plurality of physical qubits as input and as output may produce both the encoded logical qubit and classical information (e.g., syndrome graph data) resulting from the measurement sequence.

[0038] In some embodiments, a logical qubit may be a component of a quantum error-correcting code where an operation (for example, a quantum gate acting on the logical qubit) may be performed on encoded logical information. For example, a logical qubit may include multiple resources states that are entangled with one another in a specific way. Resource states are defined as a plurality ofphysical qubits prepared in a specific entangled manner. In some embodiments, 6-qubit resource states may be used, or other types of resource states may be used.

[0039] If the above-described surface code measurement schedule is applied for numerous time steps, the system effectively acts as a fault-tolerant quantum memory for the logical qubit encoded by the underlying surface code or, viewed another way, as a fault-tolerant logical identity gate on the logical qubit that is encoded by the underlying surface code. Viewed yet another way, this process operates as a fault-tolerant logical channel.

[0040] Figure 1D illustrates a 3-dimensional graphical depiction of such a fault-tolerant logical identity gate. The surface labeled 114 is the input to the gate and includes an arbitrary logical state encoded in a surface code, represented as the input checkerboard surface. Likewise, the surface labeled 118 identifies the output qubits after the identity gate I has been applied to it. As one example, in a circuit-based implementation the fault-tolerant logical qubit shown in Figure 1C may be utilized as the input surface 114, which may be operated on within the illustrated volume and output as the surface 118. The input and output surfaces, which may be associated with either the physical or relational arrangement of qubits, are connected to each other via an intervening volume that represents the unique set of measurements to be applied over time. Accordingly, in Figure 1D, time flows from left to right and the lighter shaded (front and back) and darker shaded (top and bottom) sides of the boundaries of the volume depict whether the primal or dual plaquettes are disposed on that boundary. Figure 1E represents the same concept but written in a more familiar quantum circuit notation illustrating the analogy between the more familiar quantum circuit. While Figure 1D shows the logical identity gate, any gate can be depicted in this manner and such a depiction is one example of a logical block that specifies a set of instructions to be performed on the underlying surface code qubits to perform a logical operation (the identity gate in this example) on the logical qubit that is encoded by surface code. Other examples of such gates are the S gate, the Hadamard gate, and the CX gate, among other possibilities. This combination of gates may be used, for example, to implement the quantum circuits described in various embodiments.

[0041] The protocol for preparing an encoded logical state may contain two parameters, and . Here is referred to as the “distance” of the scheme, which corresponds to the length and width of the cross section shown in Figure 1D – it determines the code distance of the surface code state being prepared. In some embodiments, L may be separated into two parameters, Lxand Ly,i.e., the code distance may be different in the two spatial directions. This may be desirable, for example, when there is an asymmetery in the noise model or logical error rates in the X and Z directions, and the code distance may be separately tuned in the two spatial directions. is referred to as the “depth” of the scheme – it can be thought of as simulated time, i.e. the number of rounds of stabilizermeasurements in CBQC, or the number of layers of resource states in FBQC. may determine the number of stabilizer checks in the protocol from which information may be gathered for post- selection. A minimal depth of 2 may be chosen, however, longer depths may also be used (using more overhead) to allow for more information to be collected in order to better predict logical errors on the output state.

[0042] Figures 1F and 1G illustrate a specific example in fusion-based quantum computing (FBQC) of an arrangement of physical qubits that may be used to perform a (Z2, Z3) measurement on four logical qubits q1-q4. The individual circles shown in the rectangular sheet 120 in the top half of Figure 1G represent individual physical qubits, and the lines connecting adjacent qubits indicate entanglement (e.g., via fusion measurements). In the stack of d = 9 layers shown at 122 of Figure 1G, the vertical direction represents the depth of the logical qubit (i.e., time), which is a sequence of nine entangling measurements performed on the 9x9 grid of physical qubits representing each of the qubits q1-q4as well as a portion of the auxiliary qubits 121. While Figures 1F and 1G illustrate an arrangement of physical qubits that may be used to perform a simple dual-qubit measurement, it is understood by those of ordinary skill in the art how more complex arrangements of qubits may be utilized to perform the operations shown in the quantum circuit diagrams shown in Figures 3-6. Quantum State Preparation

[0043] Embodiments herein describe quantum computing systems, quantum circuits, and methods for performing quantum state preparation. The quantum state describes the state of a plurality of particles of a physical system, which may include one or more molecules, atoms, substances, etc. In some embodiments, the particles include reactants and / or catalysts of a chemical reaction, and the state describes an initial state for the chemical reaction. In some embodiments, the quantum state is a thermal quantum state that describes electronic and / or thermal (i.e., phononic or vibratory) degrees of freedom of the particles of the physical system. In preparing the quantum state, the Hamiltonian of the physical system may be approximated as a quadratic function and diagonalized in the collective modes of the system. After preparing the state in the collective mode basis (i.e., with collective mode coordinates), a rotation may be performed to rotate the state into the physical basis (i.e., the spatial basis, with physical coordinates).

[0044] In some embodiments, the electronic and vibrational / phononic degrees of freedom (d.o.f.) of reactants and catalyst may be assumed to be in a tensor product, with the former prepared in the ground state and the latter prepared in a thermal state with respect to a mean-field nuclear Hamiltonian. This assumption may be taken to hold at the initial time 0, while the system is evolved under a non- Born-Oppenheimer (non-BO) Hamiltonian during a quantum simulation.

[0045] In these embodiments, the full Hamiltonian of the system may be represented as:

[0046]

[0047] As described above, the full complexity of the quantum Hamiltonian may be utilized when performing time evolution, but approximations may be incorporated to prepare the initial state. For example, it may be approximated that the electronic degrees of freedom are taken to be in a pure state, i.e., not entangled with the ionic degrees of freedom. This approximation ignores electron-phonon correlations at the initial time 0. At this initial time, an effective nuclear Hamiltonian may be defined and a thermal state is prepared with respect to the effective nuclear Hamiltonian. For 0, this initial state may evolve under the influence of the full Hamiltonian, so the problem may be set up so that electron-phonon correlations have time to build up before the reaction of interest occurs, in at least some embodiments.

[0048] More specifically, in the Born-Oppenheimer (BO) or adiabatic approximation one may compute an approximation to the electronic ground state when the nuclei are fixed at equilibrium positionsThat is, we may take the semiclassical Hamiltonian

[0049]

[0050] neglecting the nuclei kinetic energy term and their potential energy (the latter is just a constant), and define as the ground state. We may then consider an effective, mean-field phonon Hamiltonian obtained as

[051]

[0052] describing the nuclei oscillating around their equilibrium positionin the average potential generated by the electrons. Since the average potential has been computed for the nuclei fixed at position , this approximation is appropriate only when these oscillations are sufficiently small. The approximation is that we may take the starting state of the electronic and phononic / vibrational d.o.f. to be of the form

[053]

[0054] whereas the time evolution is computed through the full (non-BO) Hamiltonian. Note thatwhere k is Boltzmann’s constant and T is the temperature.

[0055] Thermal state preparation for a dynamical, non-BO Hamiltonian presents considerable computational difficultly and complexity. For example, existing methods exhibit exponential complexity overheads or involve very complex subroutines and / or unknown parameters that remove any potential quantum computational advantage, in particular while simulating more complex physical systems. Embodiments herein improve on these methods by approximating the Hamiltonian as a quadratic and bosonic Hamiltonian for the initial time (t = 0) when performing initial state preparation,whereas the full Hamiltonian may potentially be used when simulating time evolution. Advantageously, only a few modes of the system are appreciably populated under reasonable working conditions, and exploiting this fact may result in an exponential speedup in computational complexity. Once a quadratic Hamiltonian is obtained, it may be diagonalized in normal modes (i.e., the collective modes of the Hamiltonian) efficiently by a quantum computational method, as described above. Advantageously, this may reduce the problem to that of preparing uncoupled harmonic oscillators, which may be performed via quantum rejection sampling or other methods, in some embodiments.

[0056] Diagonalizing the Hamiltonian and preparing the uncoupled harmonic oscilllators may utilize as inputs the results of appropriate classical pre-computations (e.g., of energies, eigenstates, a dynamical matrix, and / or a normal mode rotation matrix) via standard density functional theory (DFT) calculations, in some embodiments. Figure 2 – Flowchart for Quantum State Preparation

[0057] Figure 2 is a flowchart diagram illustrating a method for preparing a plurality of qubits into a target quantum state, according to some embodiments. The method shown in Figure 2 may be used in conjunction with any of the computer systems or devices shown in the above Figures, among other devices. For example, the method shown in Figure 2 may be performed by a quantum computing device or system 101 as illustrated in Figure 1A. The quantum computing system may be configured to direct the described method steps, and may include (or be coupled to) a classical computing system 103 for processing classic information and directing operations of the quantum computing device. In some embodiments, a quantum controller 108 may apply a sequence of gates to the qubits 110 under the direction of the classical computing system 103 to prepare the qubits into the target quantum state.

[0058] In some embodiments, the methods described in Figure 2 may be implemented using the quantum circuit illustrated in Figure 3. In the description of Figure 2, reference numerals in parentheses are used to refer to corresponding elements of the quantum circuit diagram shown in Figure 3.

[0059] In some embodiments, the described quantum circuits may be implemented in any of a variety of types of quantum computing systems, including but not limited to photonic, semiconductor, superconducting and / or topological quantum computing systems. It is to be understood this method may be used by any of a variety of types of quantum computing architectures, and these other types of systems should be considered within the scope of the embodiments described herein. As illustrated, the method shown in Figure 2 may proceed as follows.

[0060] At 202, a first quantum register (302) is prepared in a first state (308). The first quantum register is labelled as the “purification register” 302 in the quantum circuit shown in Figure 3, andincludes a separate sub-register for each of the modesThe first state includes, for eachof a plurality of collective modes, a summation over energy label states weighted based on a Gibbs probability distribution. To prepare the first quantum register in the first state, the first quantum register may be first prepared in an initial state (e.g., the null state), followed by application of a firstunitary operator to prepare the firstquantum register in the first state. An example of the first unitary operator is shown in Equation 98,where the energy eigenstates are labelled with the energy label states denotes the Gibbsprobability distribution, and the summation is over the plurality of collective modes. Note that may take the values {0, 1, 2, …} to denote the ground state, first excited state, etc. In some embodiments, the summation over the energy label states is weighted by a square root of the Gibbs probability distribution. An example expression for the Gibbs probability distribution is shown in Eq.95.

[0061] In some embodiments, the collective modes are vibratory collective modes (e.g., thermal modes). The first state may be determined by a classical processor (such as the processor(s) 102 shown in Figure 1A) in a classical pre-processing stage prior to running the quantum computation. Aspects of the first state (such as the Gibbs probability distribution of each collective mode) may be determined using various classical-computational techniques such as density-functional theory (DFT), coupled-cluster methods, etc. The classical information determined by the classical processor that describes the first state may be used to prepare the first quantum register in the first state at step 202 (e.g., it may be used to determine the unitary operators to be applied to the first quantum register). For example, after preparing the first quantum register in the null state, a controller such as the controller 106 may apply a series of gates to the first quantum register to apply the unitary operators and prepare the first quantum register in the first state.

[0062] In some embodiments, the Gibbs probability distribution is a truncated distribution that incorporates a maximum mode cutoff on the collective modes based on physical conditions of a chemical reaction involving a plurality of particles. For example, the state preparation may be intended for use in a quantum simulation of a physical and / or chemical process involving the plurality of particles, and a maximum mode cutoff may be employed on modes that are of a sufficiently high energy such that their probability of being populated is below a threshold probability (e.g., based on the ambient temperature and / or other factors). For example, when creating the Gibbs probability distribution, mode truncation may be utilized to limit the number of excitations of the collective modes that are considered based on physical conditions of a chemical reaction being simulated. Vibrational modes that are associated with a thermal energy that is much larger than theenergy scale of the operating temperature of the chemical reaction may be truncated. An example expression for a maximum mode cutoff is shown in Equation 97, although other mode cutoffs may also be used.

[0063] In some embodiments, to determine the collective modes and eigenenergies, a Hamiltonian is approximated as a quadratic function. In some embodiments, approximating the Hamiltonian utilizes a Born-Oppenheimer approximation that incorporates a harmonic approximation of nuclei of the plurality of particles about their equilibrium positions. An example quadratic Hamiltonian is shown in Eq. 9. The approximate Hamiltonian may then be diagonalized with a unitary transformation to obtain a diagonalized Hamiltonian that is a quadratic function of the plurality of collective modes. Diagonalization of the Hamiltonian is described in Eqs.36-76 and the associated description. The collective modes may be collective vibratory modes of a physical system that may include a plurality of particles. In these embodiments, the Gibbs probability distribution, the eigenstates, and the plurality of controlled unitary operators may be determined based at least in part on the diagonalized Hamiltonian.

[0064] The diagonalized Hamiltonian may be determined by a classical processor (such as the processor(s) 102 shown in Figure 1A) in a classical pre-processing stage prior to running the quantum computation. Advantageously, the diagonalized Hamiltonian may be computationally simpler to block-encode in the quantum computing system (e.g., utilizing fewer qubits, gates, operations, etc.) than the exact Hamiltonian.

[0065] In some embodiments, the plurality of particles is arranged in a lattice with a discrete translational symmetry. In these embodiments, approximating the Hamiltonian may include performing a discrete Fourier transform on an interatomic force matrix for each primitive cell of the lattice. This procedure is described in Eqs.5-35 and the associated description.

[0066] At 204, a second quantum register (304) is prepared in a second state (310a-b). The second quantum register is labelled as the “normal mode” register 304 in the quantum circuit shown inFigure 3, and includes a separate logical qubit for each of the modes. The second stateincludes, for each of the plurality of collective modes, a summation over eigenstates of a respective collective mode. To prepare the second quantum register in the second state, the second quantum register may be first prepared in aninitial state (e.g., the null state), followed by application of a controlled second unitary operator (one for each of the plurality of collective modes, labelled inFigure 3 as PREPi(1) 310a-b for to the first and second quantum registers, controlled onthe state of the first quantum register, to prepare the second quantum register in the second state. Note that “second” is used simply to distinguish the second state and unitary operator from the “first” state and unitary operator described above. The controlled second unitary operators may beapplied to qubits of the second quantum register, controlled on the energy eigenstates of the qubits in the first quantum register. An example of the controlled unitary operator is shown in Equation 99, where | | illustrates the control over the energy label states in the first quantum register,denotes the (uncontrolled) second unitary operator to be applied to the second quantum register, and the summation is over the plurality of collective modes. Note that the “controlled second unitary operator” includes the summation over the collective modes and acts on both the first and second quantum registers, whereas the “second unitary operator” is a unitary operator for a particular collective mode l that acts on the second quantum register only, and it does not include the sum over collective modes. The controlled application of the second unitary operator entangles the second state with the first state. For example, the entanglement between the first and second state may be obtained by applying the preparation operators on qubits of the second quantum register controlled on corresponding qubits of the first quantum register.

[0067] The second unitary operator is derived and discussed in greater detail in Eqs.89-94 and the associated description. As shown in Eq.94, application of a second unitary operator to a qubit in the null state prepares the state in an eigenstate | of the collective mode l. In someembodiments, preparing the second quantum register in the second state is performed using quantum rejection sampling. Quantum rejection sampling prepares a qubit in a target state by first preparing the qubit in a simpler state, and coherently rejecting samples from this state that differ from the target state until the qubit approaches the target state. Other methods may also be used to prepare the second quantum register in the second state, in various embodiments.

[0068] At 206, a third quantum register (306) is prepared in a third state. The third quantum register is labelled as the “ion” quantum register 306 in the quantum circuit shown in Figure 3. The third quantum register is prepared in the third state by applying a plurality of third unitary operators to the second and third quantum registers. Applying the plurality of third unitaryoperators|to the second and third quantum registers transforms the collective mode coordinates of the second state to physical coordinates, and adds this transformed state (i.e., the third state) into the third quantum register. Said another way, the third quantum register is prepared in the third state by adding to the third quantum register the result of applying this transformation from collective coordinates to physical coordinates to the second state. “Physical coordinates” in this context refers to coordinates in physical space. Expressed mathematically, applying the third unitary operators performs the transformation, wheredenotes the state of the second quantum register, |0 |0 denotes the initial (null) stateof the third quantum register, and denotes the third quantum register afterapplying the third unitary operators (i.e., prepared in the third state). Note that the state of the second quantum register is unchanged except that it has become entangled with the third quantum register (e.g., via coupling between respective modes j).

[0069] Note that after applying the plurality of third unitary operators, the first, second and third quantum registers will be entangled with each other. An expression for the pre-rotated state is shown in Eq.105, applying the plurality of third unitary operators performs the rotation shown in Eq.106, and the rotated state is shown in Eq.107, in at least some embodiments. As shown in the quantum circuit illustrated in Figure 3, the result of applying the unitary|to the second state is added intothe third quantum register for each of the modesThis procedure is described in greaterdetail in Eqs.105-108 and the associated description.

[0070] At 208, the second quantum register is returned to the null state (316) (or potentially to another initial state of the second quantum register). The second quantum register is returned to the null state by applying a plurality of fourth unitary operators U | (314a-b) to the second and third quantum registers. Applying the plurality of unitary fourth operators to the second and third quantum registers rotates the physical coordinates of the third state back into the collective mode coordinates, and subtracts the result from the second register. Expressed mathematically, applying the fourthunitary operators performs the transformation |denotes the state of thesecond quantum register before applying the fourth unitaries, |denotes the thirdquantum register prepared in the third state, |0 |0 denotes the second quantum register returnedto the null state, and | | denotes the desired purified states. This will un-compute the state ofthe second quantum register, returning the second quantum register to the initial (null) state and removing the entanglement between the second quantum register and the first and third quantum registers. This procedure is described in greater detail in Eqs.109-111 and the associated description.

[0071] After returning the second quantum register to the null state, the first and third quantum registers will be entangled. For example, for each collective mode, the respective summation of the first state in the first quantum register will be entangled with the corresponding term in the third state in the third quantum register. The combined state of all three quantum registers after this operation is shown at 322 in Figure 3.

[0072] Advantageously, after performing step 208, the combination of the first and third quantum registers are entangled together in a “pure state” corresponding to the desired target quantum state. Every mixed quantum state (a general density matrix) can be viewed as a reduction of a pure (single vector) quantum state on an enlarged system. A pure state may be represented by a single vectorrepresenting the quantum state that encodes all the information of the density matrix. This may be accomplished by artificially increasing the degrees of freedom (which is a non-unique choice) and preparing the coefficients of the enlarged system state such that by tracing out (averaging out) the artificial degrees of freedom, one recovers the density matrix, which in some embodiments is a thermal density matrix. In other words, if we desire a thermal state on system A, we introduce an additional non-unique auxiliary system B and a pure quantum state (single vector) on the joint system A+B, such that tracing out system B yields the thermal state on A. The first quantum register serves the role of this auxiliary system, and is hence called the purification register in Figure 3.

[0073] In some embodiments, after preparing the first and third quantum registers in the purified target quantum state, the method may further include performing a quantum simulation on the purified target quantum state using a full Hamiltonian (or an approximate Hamiltonian) of a physical system. In some embodiments, evolving these registers in time may simulate a self-thermalization process for the purified target quantum state. More broadly, the purified target quantum state may be used in any subsequent step of a quantum computation.

[0074] In some embodiments, before or after performing the quantum simulation, one or more measurements are performed on one or more of the quantum registers to obtain classical measurement results. The classical measurement results may indicate one or more aspects of an output quantum state. Qubits of the first and third quantum registers may be entirely or partially measured, in various embodiments. In some embodiments, the one or more classical measurement results are stored in a non-transitory computer-readable memory medium. Figure 3 – Quantum Circuit for State Preparation

[0075] Figure 3 is a quantum circuit configured to perform state preparation, according to some embodiments. The quantum circuit may be utilized to implement the method described in reference to Figure 2, in some embodiments. As illustrated, the quantum circuit in Figure 3 includes three quantum registers, a first purification register 302, a second normal mode register 304, and a third ion register 306. Each of the three quantum registers includes J (logical) qubits that are initiallyprepared in the null state |0 . As shown, a respective PREPweights operator 308 is applied to each ofthe J qubits in the first register. After application of the PREPweights operators, the combined state of all three quantum registers is shown in the equation labelled at 318.

[0076] A respective controlled unitary operator PREPi(j)(310a-b) is then applied to each of the J qubits in the normal mode register, controlled on a respective qubit in the purification register. After application of the controlled PREPi(j)operators, the combined state of all three quantum registers is shown in the equation labelled at 320.

[0077] A rotation into the physical coordinate basis is then performed by the unitary operator, which add the result of applying to the normal mode register into a respective qubit of the ion registerfor each mode q. Finally, a subtraction operation is performed to uncompute the state of the normal mode registers. To accomplish this, the result of applying the inverse operator to the state of theion register is subtracted from a respective qubit of the normal mode register for each mode q. After performing these operations, the state of all three qubits registers is shown in the equation labelled at 322. The purification and ion registers are now prepared in a purified target state, and may be utilized in a subsequent quantum computation. Additional Technical Detail

[0078] The following numbered paragraphs provide additional technical detail, mathematical derivations, and description regarding embodiments herein. Diagonalization of the phonon Hamiltonian via real normal modes

[0079] In some embodiments, the phonon Hamiltonian for the nuclei may be defined on a finite, discrete lattice (a supercell) with periodic boundary conditions. The supercell may be made of a repetition of primitive cells with atoms each. Exploiting the discrete lattice symmetry, theproblem of diagonalizing this Hamiltonian (which involves 3 coordinates) may be almostreduced to diagonalizing a 3 3 dynamical matrix. A residual coupling within 2 2 blocksmay be taken care of via a further diagonalization, with some extra treatment for the resulting normal modes coordinates to satisfy the canonical commutation relation. BvK Supercell, Harmonic Approximation, Translation Invariance

[0080] In some embodiments, the lattice structure of the system may be exploited to simplify the approximation of the Hamiltonian. In some embodiments, the approach of the Born-von Kármán (BvK) supercell is utilized. Specifically, a primitive lattice cell may be defined the nuclei within it labeled by m. C primitive cells may be considered, which define a supercell. The primitive cells maybe labeled by the direct lattice vectorsruns over:

[0081]

[0082]

[0083]

[0084] with primitive lattice vectors. Periodic boundary conditions may be imposed at the boundary of the supercell. The nuclei may then be labelled with indexes , , wherelabels the primitive cells andlabels the nuclei within it.

[0085] In some embodiments, the harmonic approximation is utilized to expand(the operator in the position coordinate basis) around the 3 equilibrium positionsas:

[0086]

[087] where 1 2 3 denote three orthogonal directions, , , are displacements of nucleusfrom its equilibrium position in the direction and

[0088]

[0089] The coefficients are known as interatomic force constants and are assumed tobe known, e.g. via standard DFT calculations. The constant shall be dropped in the followingdescription, to simplify the expressions.

[0090] Working in a frame where the centre of mass of the slab is at rest we get that the phonon Hamiltonian is

[0091]

[0092] One can use convenient mass-weighted coordinates

[0093]

[0094] and mass-weighted interatomic force constants

[0095]

[0096] In these coordinates the Hamiltonian becomes

[0097]

[0098] The following description works in these coordinates, but the bar will be dropped for simplicity in the following.

[0099] Discrete lattice symmetry and final form. In some embodiments, it is assumed that the interatomic force matrix along the lattice directions satisfies a discrete translational symmetry, capturing the lattice structure of the catalyst and the periodic boundary conditions. The final problem at hand in mass-weighted coordinates is then

[0100]

[0101] where the discrete translational symmetry reads

[0102] Change Of Coordinates via Discrete Fourier Transform on the Lattice

[0103] Coordinate changes. In some embodiments, to exploit the discrete symmetry along the lattice, a coordinate transformation is performed in which the interatomic force matrix is block-diagonal. For example, in some embodiments the coordinate Consider alinear transformation coordinates

[0104]

[0105]

[0106]

[0107]

[0108]

[0109] Weighted discrete Fourier transform on the lattice. As suggested by the discrete translational symmetry on the lattice, in our case the coordinate change is a discrete, weighted Fourier transform on the lattice defined by the direct lattice vectors:

[0110]

[0111] t rs

[0112]

[0113] with the primitive vectors of the reciprocal lattice which satisfy the duality condition

[0114]

[0115] To avoid any confusion the kronecker delta is indicated using the identity matrix . The vectors are to be fixed later, but they are chosen so that the matrix of change of coordinates,

[0116]

[0117] is unitary. For this to be the case, it suffices to take the vectors to be, forevery fixed , an orthonormal basis.

[0118] y

[0119]

[0120]

[0121] Proof. First note that the vectors taken as columns of a matrix, give a3 3 unitary, and the transpose of said unitary is itself a unitary, which means the followingalso holds:o is ly w

[0231] Note that is a unitary matrix, since was unitary and the move to real normal modes is a simple 2 rotation. It then follows that the real normal modes satisfy canonical commutationrelations. Thermal State Preparation

[0232] The following paragraphs present methods to prepare a thermal state for a phonon Hamiltonian, or more generally for any quadratic bosonic Hamiltonian, according to some embodiments. As input, the following information may be utilized.

[0235] We shall first present a subroutine that prepares truncated thermal states in normal modes, and then discuss how to change coordinates back to the physical coordinates. Creating Excited States of Uncoupled Harmonic Oscillators

[0236] Excited states in the momentum basis. First consider the problem of adding excitations to a set of uncoupled harmonic oscillators, working in the momentum basis. The excited states of uncoupled oscillators with unit mass and frequency read in the position basis as

[0240] and are Hermite polynomials. Note that we consider all oscillators to have the same frequency. In some embodiments, the quantum computation is performed in momentum space, so the following moves to that representation. From the canonical commutation relations it follows that in position and momentum space we have, respectively,

[0286] In some embodiments, for error analysis, the error due to the truncation, the state-prep error, and / or the unitary synthesis error may be summed together. Figures 4-6 - Quantum Circuits to Emulate Time Evolution

[0293] It should be understood that all numerical values used herein are for purposes of illustration and may be varied. In some instances, ranges are specified to provide a sense of scale, but numerical values outside a disclosed range are not precluded.

[0294] It should also be understood that all diagrams herein are intended as schematic. Unless specifically indicated otherwise, the drawings are not intended to imply any particular physical arrangement of the elements shown therein, or that all elements shown are necessary. Those skilled in the art with access to this disclosure will understand that elements shown in drawings or otherwise described in this disclosure may be modified or omitted and that other elements not shown or described may be added.

[0295] This disclosure provides a description of the claimed invention with reference to specific embodiments. Those skilled in the art with access to this disclosure will appreciate that the embodiments are not exhaustive of the scope of the claimed invention, which extends to all variations, modifications, and equivalents.

[0296] The terminology used in the description of the various described embodiments herein is for the purpose of describing particular embodiments only and is not intended to be limiting. As used in the description of the various described embodiments and the appended claims, the singular forms “a”, “an” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will also be understood that the term “and / or” as used herein refers to and encompasses any and all possible combinations of one or more of the associated listed items. It willbe further understood that the terms “includes,” “including,” “comprises,” and / or “comprising,” when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and / or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and / or groups thereof.

[0297] It will also be understood that, although the terms first, second, etc., are, in some instances, used herein to describe various elements, these elements should not be limited by these terms. These terms are only used to distinguish one element from another. For example, a first switch could be termed a second switch, and, similarly, a second switch could be termed a first switch, without departing from the scope of the various described embodiments. The first switch and the second switch are both switches, but they are not the same switch unless explicitly stated as such.

[0298] As used herein, the term “if” is, optionally, construed to mean “when” or “upon” or “in response to determining” or “in response to detecting” or “in accordance with a determination that,” depending on the context.

[0299] The foregoing description, for purpose of explanation, has been described with reference to specific embodiments. However, the illustrative discussions above are not intended to be exhaustive or to limit the scope of the claims to the precise forms disclosed. Many modifications and variations are possible in view of the above teachings. The embodiments were chosen in order to best explain the principles underlying the claims and their practical applications, to thereby enable others skilled in the art to best use the embodiments with various modifications as are suited to the particular uses contemplated.

Claims

Claims What is claimed is:

1. A method, comprising: preparing a first quantum register in a first state comprising, for each of a plurality of collective modes, a summation over energy label states weighted based on a Gibbs probability distribution; preparing a second quantum register in a second state comprising, for each of the plurality of collective modes, a summation over eigenstates of a respective collective mode of the plurality of collective modes, wherein the second state is entangled with the first state; applying a plurality of first unitary operators to the second quantum register and a third quantum register to add a third state into the third quantum register, wherein the third state comprises a rotation of the second state from collective mode coordinates to physical coordinates; and applying a plurality of second unitary operators to the second and third quantum registers to prepare the second quantum register in a null state, wherein preparing the second quantum register in the null state comprises subtracting from the second quantum register a rotation of the third state from the physical coordinates to the collective mode coordinates.

2. The method of claim 1, further comprising: approximating a Hamiltonian for a plurality of particles as a quadratic function; diagonalizing the approximate Hamiltonian with a unitary transformation to obtain a diagonalized Hamiltonian comprising a quadratic function of the plurality of collective modes, wherein the plurality of collective modes comprises a plurality of collective vibratory modes of the plurality of particles; and determining the Gibbs probability distribution, the eigenstates, and the pluralities of first and second unitary operators based at least in part on the diagonalized Hamiltonian.

3. The method of claim 2, wherein the plurality of particles is arranged in a lattice comprising a discrete translational symmetry, and wherein approximating the Hamiltonian comprises performing a discrete Fourier transform on an interatomic force matrix for each primitive cell of the lattice.

4. The method of claim 2,wherein approximating the Hamiltonian utilizes a Born-Oppenheimer approximation that incorporates a harmonic approximation of nuclei of the plurality of particles about their equilibrium positions.

5. The method of claim 1, wherein the summation over the energy label states is weighted by a square root of the Gibbs probability distribution.

6. The method of claim 1, wherein the Gibbs probability distribution comprises a truncated distribution that incorporates a maximum mode cutoff on the collective modes based on physical conditions of a chemical reaction involving a plurality of particles.

7. The method of claim 1, wherein preparing the second quantum register in the second state comprises performing quantum rejection sampling.

8. The method of claim 1, further comprising: after applying the pluralities of first and second unitary operators, time evolving the third quantum register using a quantum Hamiltonian simulation of a Hamiltonian for a plurality of particles.

9. The method of claim 1, wherein the plurality of collective modes comprises a plurality of vibratory collective modes.

10. The method of claim 1, wherein, after applying the pluralities of first and second unitary operators, the first and third quantum registers are entangled in a purified target quantum state.

11. The method of claim 10, further comprising: performing a quantum simulation on the purified target quantum state using a full Hamiltonian of a physical system.

12. The method of claim 1,wherein applying the plurality of second unitary operators to the second and third quantum registers removes entanglement between the second quantum register and the first and third quantum registers.

13. The method of claim 1, after applying the pluralities of first and second unitary operators, measuring at least a portion of the first and third quantum registers to produce classical measurement results; and storing the classical measurement results in a non-transitory computer-readable memory medium.

14. A quantum computing system, comprising: a plurality of quantum registers; and a controller, wherein the controller is configured to: prepare a first quantum register of the plurality of quantum registers in a first state comprising, for each of a plurality of collective modes, a summation over energy label states weighted based on a Gibbs probability distribution; prepare a second quantum register of the plurality of quantum registers in a second state comprising, for each of the plurality of collective modes, a summation over eigenstates of a respective collective mode of the plurality of collective modes, wherein the second state is entangled with the first state; apply a plurality of first unitary operators to the second quantum register and a third quantum register of the plurality of quantum registers to add a third state into the third quantum register, wherein the third state comprises a rotation of the second state from collective mode coordinates to physical coordinates; and apply a plurality of second unitary operators to the second and third quantum registers to prepare the second quantum register in a null state, wherein preparing the second quantum register in the null state comprises subtracting from the second quantum register a rotation of the third state from the physical coordinates to the collective mode coordinates.

15. The quantum computing system of claim 14, further comprising: measurement circuitry; a non-transitory computer-readable memory medium; anda classical processor coupled to the non-transitory computer-readable memory medium and the controller, wherein the classical processor is configured to execute program instructions stored in the non-transitory computer-readable memory medium to direct operation of the controller.

16. The quantum computing system of claim 15, wherein the classical processor is further configured to: approximate a Hamiltonian for a plurality of particles as a quadratic function; diagonalize the approximate Hamiltonian with a unitary transformation to obtain a diagonalized Hamiltonian comprising a quadratic function of the plurality of collective modes, wherein the plurality of collective modes comprises a plurality of collective vibratory modes of the plurality of particles; and determine the Gibbs probability distribution, the eigenstates, and the pluralities of first and second unitary operators based at least in part on the diagonalized Hamiltonian.

17. The quantum computing system of claim 14, wherein preparing the second quantum register in the second state comprises performing quantum rejection sampling.

18. A non-transitory computer-readable memory medium storing program instructions which, when executed by a classical processor, cause a quantum computing system to: prepare a first quantum register of a plurality of quantum registers in a first state comprising, for each of a plurality of collective modes, a summation over energy label states weighted based on a Gibbs probability distribution; prepare a second quantum register of the plurality of quantum registers in a second state comprising, for each of the plurality of collective modes, a summation over eigenstates of a respective collective mode of the plurality of collective modes, wherein the second state is entangled with the first state; apply a plurality of first unitary operators to the second quantum register and a third quantum register of the plurality of quantum registers to add a third state into the third quantum register, wherein the third state comprises a rotation of the second state from collective mode coordinates to physical coordinates; and apply a plurality of second unitary operators to the second and third quantum registers to prepare the second quantum register in a null state, wherein preparing the second quantum register inthe null state comprises subtracting from the second quantum register a rotation of the third state from the physical coordinates to the collective mode coordinates.

19. The non-transitory computer-readable memory medium of claim 18, wherein the classical processor is further configured to: approximate a Hamiltonian for a plurality of particles as a quadratic function; diagonalize the approximate Hamiltonian with a unitary transformation to obtain a diagonalized Hamiltonian comprising a quadratic function of the plurality of collective modes, wherein the plurality of collective modes comprises a plurality of collective vibratory modes of the plurality of particles; and determine the Gibbs probability distribution, the eigenstates, and the pluralities of first and second unitary operators based at least in part on the diagonalized Hamiltonian.

20. The non-transitory computer-readable memory medium of claim 18, wherein preparing the second quantum register in the second state comprises performing quantum rejection sampling.

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