Direct quantum simulation of heterogenous catalysis

Abstract

Quantum computing devices, systems and methods for performing a quantum simulation of a chemical reaction. An initial quantum state related to a chemical reaction is received, and a model Hamiltonian for the chemical reaction is determined. A user specification is received specifying one or more parameters of a quantum simulation of the chemical reaction. The initial quantum state is encoded within a plurality of qubits of a quantum computing system. Time evolution is performed on the plurality of qubits using the model Hamiltonian and the user specification to produce an output quantum state. Measurements are performed on the plurality of qubits to obtain classical measurement results. The classical measurement results are stored in a non-transitory computer-readable memory medium.

Description

DIRECT QUANTUM SIMULATION OF HETEROGENOUS CATALYSIS Technical Field

[0001] Embodiments herein relate generally to quantum computational methods, systems and devices, such as photonic devices (or hybrid electronic / photonic devices) for emulating physical systems. 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 may be an extremely complex and computationally intensive procedure, particularly for more complex systems. Accordingly, improvements in the field of quantum computational systems and methods are desired to increase the efficiency and reduce the complexity of quantum simulation. Summary

[0004] Some embodiments described herein include quantum computing devices, systems and methods for performing a quantum simulation of a chemical reaction, such as a catalysis reaction.

[0005] In some embodiments, an initial quantum state related to a chemical reaction is received. Determining the initial quantum state may include determining a respective first electronic state for a plurality of respective electrons of one or more molecules of the chemical reaction, and determining a respective set of vibrational modes for a plurality of nuclei of the one or more molecules. A respective thermal density matrix may be created based on each respective set of vibrational modes, and purified thermal states may be determined using the thermal density matrices. The initial quantum state may be determined based on a combination of the first electronic states and the purified thermal states.

[0006] In some embodiments, a model Hamiltonian may be determined for the chemical reaction. The model Hamiltonian may be determined based on an exact Hamiltonian by approximating nuclei as reduced nuclei having reduced charges. The model Hamiltonian may include kinetic energy terms for each non-core electron and each reduced nucleus, as well as pseudo-interaction energy terms that are functions (e.g., matrix functions) of the positions of the non-core electrons of the reduced nuclei and / or the positions of the reduced nuclei.

[0007] In some embodiments, a user specification is received specifying one or more parameters of a quantum simulation of the chemical reaction.

[0008] In some embodiments, the initial quantum state is encoded within a plurality of qubits of a quantum computing system.

[0009] In some embodiments, time evolution is performed on the plurality of qubits using the model Hamiltonian and the user specification to produce an output quantum state.

[0010] In some embodiments, measurements are performed on the plurality of qubits to obtain classical measurement results. The measurements may include an application of a point group symmetry operator, an expectation value of a difference in respective positions of two or more nuclei of the one or more reactant molecules, and / or a difference in joint correlation functions of various nuclei in the chemical reaction.

[0011] In some embodiments, the classical measurement results are stored in a non-transitory computer-readable memory medium.

[0012] 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.

[0013] 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

[0014] 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.

[0015] Figure 1A is a system diagram illustrating a quantum computing system, according to some embodiments;

[0016] Figures 1B-G illustrate the utilization of surface codes to constructed an error-corrected fault- tolerant logical qubit, according to some embodiments;

[0017] Figure 1H illustrates a qubit fusion system interfacing with a classical computing system, according to some embodiments;

[0018] Figure 2 is a flowchart diagram illustrating a method for simulating a chemical reaction, according to some embodiments;

[0019] Figure 3 is a quantum circuit diagram illustrating a decomposition of a time evolution operator into a plurality of qubitization operators, according to some embodiments;

[0020] Figure 4 is a quantum circuit diagram illustrating the component subroutines of a qubitization operator, according to some embodiments;

[0021] Figure 5 is a quantum circuit diagram illustrating a qubitization circuit for phase estimation, according to some embodiments;

[0022] Figure 6 is a high-level flowchart illustrating a method for simulating a chemical reaction, according to some embodiments; and

[0023] Figure 7 is a flowchart diagram illustrating classical preprocessing for the reactant and catalyst molecules and quantum processing for performing state preparation, according to some embodiments.

[0024] 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

[0025] 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.

[0026] 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

[0027] To facilitate understanding of the disclosure, an overview of relevant concepts and terminology is provided in the following paragraphs.

[0028] 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.

[0029] 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.

[0030] 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.

[0031] 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 ofmolecules, 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

[0032] 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.

[0033] 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).

[0034] 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 input device 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. Theprocessor 102 may store that output back into the memory 104 and / or provide the output to the QPU over the channel 112.

[0035] 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

[0036] 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 occupied by 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.

[0037] 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.

[0038] 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 a nonzero probability of being occupied, e.g., a state. In 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. The amplitudes a1 and a2 depend on the reflectivity (or transmissivity) of the beam splitters and on any phase shifts that are introduced.

[0039] A single physical qubit (e.g., such as the 2-level physical qubit illustrated in Figure 1B with aquantum statemay 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.

[0040] 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.

[0041] 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.

[0042] 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 of physical 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.

[0043] 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.

[0044] 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.

[0045] 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,andi.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 stabilizer measurements in CBQC, or the number of layers of resource states in FBQC. may determine thenumber of stabilizer checks in the protocol from which information may be gathered for post-selection. A minimal depth ofmay 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.

[0046] 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-q4 as 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 3-5.

[0047] Figure 1H shows an illustrative example for one way to implement a fusion site as part of a photonic quantum computer architecture, according to some embodiments. Figure 1H shows one possible example of a fusion site 1501 as configured to operate with a fusion controller 140 to provide measurement outcomes to a decoder for fault tolerant quantum computation, in accordance with some embodiments. In this example, fusion site 1501 may be an element of a fusion array that contains multiple fusion sites, and although only one instance is shown for purposes of illustration, the fusion array may include any number of instances of fusion sites 1501 connected to the fusion controller.

[0048] The qubit fusion system 1505 may receive two or more qubits (qubit 1 and qubit 2) that are to be fused. Qubit 1 is one qubit that may be entangled with one or more other qubits (not shown) as part of a first resource state and qubit 2 is another qubit that may be entangled with one or more other qubits (not shown) as part of a second resource state. The fusion operations that take place at the fusion sites are fully destructive joint measurements between qubit 1 and qubit 2 such that classical information remains after the measurement is performed representing the measurement outcomes on the detectors, e.g., detectors 1503, 1505, 1507, 1509. Quantum information contained within qubits 1 and / or 2 may be transferred to the remaining (i.e., unmeasured) qubits of their respective resource states, to which they entangled. The classical information is decoded by a decoder 146 and may be used in subsequent steps of the described embodiments.

[0049] In this example, qubit 1 and qubit 2 are dual rail encoded photonic qubits. Accordingly, qubit 1 and qubit 2 are input on waveguides 1521, 1523 and 1525, 1527, respectively. An interferometer1524, 1528 may be placed in line with each qubit, and within one arm of each interferometer 1524, 1528 a programmable phase shifter 1530, 1532 may be applied to affect the basis in which the fusion operation is applied, e.g., XX, XY, YY, ZZ, etc.). The programmable phase shifters 1530, 1532 may be coupled to the fusion controller 1519 via control line 1529 and 1531 such that signals from the fusion controller 1519 may be used to set the basis in which the fusion operation is applied to the qubits. For example, the programmable phase shifters may be programmable to either apply or not apply a Hadamard gate to their respective qubits, altering the basis (e.g., x vs. z) of the type II fusion measurement. In some embodiments the basis may be hard-coded within the fusion controller 1519, or in some embodiments the basis may be chosen based upon external inputs, e.g., instructions provided by the fusion pattern generator 144. Additional mode couplers, e.g., mode couplers 1533 and 1532 may be applied after the interferometers followed by single photon detectors 1503, 1505, 1507, 1509 to provide a readout mechanism for performing the joint measurement. In the example shown in Figure 1H, the fusion site implements an un-boosted Type II fusion operation on the incoming qubits. One of ordinary skill will appreciate that any type of fusion operation may be applied (and may be boosted or un-boosted) without departing from the scope of the present disclosure. In some embodiments, the fusion controller 1519 may also provide a control signal to the detectors 1503, 1505, 1507, 1509. A control signal may be used, e.g., for gating the detectors or for otherwise controlling the operation of the detectors. Each of the detectors 1503, 1505, 1507, 1509 provides one bit of information (representing a “photon detected” or “no photon detected” state of the detector), and these four bits may be preprocessed at the fusion site 1501 to determine a measurement outcome (e.g., fusion success or not) or passed directly to the decoder 146 for further processing. Figure 2 – Flowchart for Direct Simulation of a Chemical Reaction

[0050] Figure 2 is a flowchart diagram illustrating a method for simulating a chemical reaction, such as a catalysis reaction. The catalysis reaction may involve one or more reactants and one or more catalysts, and it may produce one or more product molecules. The reactants are generally atoms or molecules of one or more different types. The catalysts may be catalyst atoms or molecules, but they also may be material substrates, e.g., with a particular crystalline structure or geometry (e.g., a copper or titanium plate). Some of the described method steps are specific to embodiments where the chemical reaction is a heterogeneous catalysis reaction involving reactant and catalyst molecules, but it may be appreciated by one of skill in the art how the described methods may be applied to more general types of chemical reactions between other atoms, molecules and / or materials.

[0051] In the following description, a specific example of a CO2reduction catalysis reaction - a reverse water-gas shift (RWGS) reaction - may be referenced which includes carbon dioxide (CO2) and hydrogen gas (H2) as reactants, a copper (Cu) plate as a catalyst, and water (H2O) and carbon monoxide (CO) as potential product molecules. This specific example of a CO2reduction catalysis reaction is exemplary only to facilitate understanding, and is not intended to be limiting to the scope of the present disclosure.

[0052] 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 processing unit 105. In some embodiments, a quantum controller 106 may apply a sequence of gates to the qubits 110 under the direction of the classical computing system 103 to manipulate the qubits according to a desired quantum computational method. A measurement system 108 may be used to measure at least a subset of the qubits to obtain classical measurement results, which may be provided to the quantum processing unit and / or the classical computing system for decoding and / or processing.

[0053] 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.

[0054] At 202, an initial quantum staterelated to a chemical reaction is determined. The initialquantum state may be determined by a classical processor (such as the CPU 102 shown in Figure 1A) in a classical pre-processing stage prior to running the quantum computation. The initial quantum state describes the initial state of the reactants and / or catalysts that are involved in the chemical reaction (e.g., the state of the reactants and / or catalysts at an initial time,. The determined classical information that describes the initial quantum state may be used to encode the qubits in the initial quantum state at step 208.

[0055] In some embodiments, determining the initial quantum state includes determining an electronic state for electrons of each of the reactant(s) and / or catalyst(s) involved in the chemical reaction. The determined electronic states are typically electronic ground states of the reactant(s)and / or catalyst(s), and will be described as ground states below for clarity. However, more generally the determined states may also be electronic excited states, depending on the specifics of the chemical reaction. For example, in certain conditions the initial state of one or more electrons in a chemical reaction may be an excited state (or have a substantial probability of being in an excited state), although the initial state is commonly the ground state.

[0056] A separate ground state may be determined for each of the different species of reactants and catalysts involved in the chemical reaction. For example, in the RWGS reduction catalysis reaction, an electronic ground state may be separately determined for each of CO2, H2, and the Cu catalyst. The electronic ground states may be determined with the assistance of classical-computational techniques such as density-functional theory (DFT), coupled-cluster methods, etc.

[0057] In some embodiments, determining the initial quantum state further includes determining a set of vibrational modes, i.e., collective motional modes of all the nuclei of a molecule, for each of the reactants and / or catalysts of the chemical reaction. For a catalyst material with an extended structure (e.g., a copper plate), the collective nuclear-motional modes of the catalyst material are typically referred to as “phonon modes” rather than “vibrational modes”. For simplicity, we refer to herein all nuclear motional degrees of freedom of the reactants and / or catalysts as vibrational modes.

[0058] In some embodiments, a thermal density matrix over vibrational modes is created for each reactant and / or catalyst. The thermal density matrices over vibrational modes for each reactant and / or catalyst may be independent of the electronic ground states (i.e., the full state of a single reactant / catalyst may be a tensor product of its electronic ground state and its thermal density matrix over vibrational modes). This may be appropriate when it is determined that the temperature of the reactants and / or catalysts multiplied by Boltzmann’s constant is much less than an energy gap between the electronic ground states and the electronic first excited states of electrons of the molecules. In other words, when the thermal energy scale of the reactants and / or catalysts is much less than the energy gap between their respective electronic ground and first excited states, a thermal density matrix may be constructed that accurately describes the thermal dynamics of the nuclei while assuming that the electrons are entirely (or predominantly) in the ground state.

[0059] In some embodiments, the thermal density matrices are approximate thermal density matrices that incorporate mode truncation. For example, when creating the thermal density matrices, mode truncation may be utilized to limit the number of excitations of the vibrational modes that are considered based on physical conditions of the chemical reaction. For example, vibrational modes that are associated with a thermal energy that is much larger than the energy scale of the operating temperature of the chemical reaction may be truncated. Said another way, vibrational modes may be truncated when they have a Boltzmann weight or probability that is less than a predeterminedthreshold, e.g., as represented in a partition function of the system (e.g., the canonical partition function, the grand canonical partition function, etc.). The truncated modes may be removed from (or suppressed in) the thermal density matrix. For example, at T=400K we may only have a few excitation quanta for the lowest vibrational modes for CO2 and H2. For the Cu catalyst, in the same example, we may choose to have all quanta for any phonon modes up to a cutoff chosen by a threshold probability of say 1%. At 1%, even a single quantum of a sufficiently high energy phonon mode is below the threshold probability and so these and higher energy modes may be omitted. For all other phonon modes, quanta may be kept until the 1% threshold for their Boltzmann probability is reached.

[0060] In some embodiments, a respective purified thermal state is determined based on each thermal density matrix. The initial quantum state is then determined from a combination of the electronic ground states and the purified thermal states. 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 purified thermal state is such a pure quantum state, i.e., a single vector representing the quantum state, that encodes all the information of the thermal 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 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.

[0061] To finish the specification of the initial state, the initial state of the center-of-mass may be specified for each of the reactant molecules and catalysts. In one embodiment, a launched Gaussian wavepacket center-of-mass state is constructed for each reactant molecule, such that they impinge on a catalyst surface (at rest). For example, CO2and H2may be launched in a Gaussian wavepacket onto the surface of Cu at rest.

[0062] At 204, a model Hamiltonian for the chemical reaction is determined. The model Hamiltonian may be determined by a classical processor (such as the CPU 102 shown in Figure 1A) in a classical pre-processing stage prior to running the quantum computation. The model 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 of the chemical reaction.

[0063] In embodiments where the chemical reaction is a catalysis reaction involving reactants and catalysts, the model Hamiltonian may be determined for some of the reactants and / or catalysts, andan exact Hamiltonian may be used for other ones of the reactants and / or catalysts. For example, the model Hamiltonian may be determined for the electrons and nuclei of the catalyst materials, and an exact Hamiltonian may be used for electrons and nuclei of the reactant molecules. This may be desirable, for example, when one or more of the reactants and / or catalysts are composed of a large number of atoms and / or are composed of atoms with large atomic weights. This may introduce a large computational burden to evolve these types of molecules or materials with an exact Hamiltonian, and determining an approximate Hamiltonian may significantly expedite the simulation. In some embodiments, particular details of one or more of the reactants and / or catalysts may be known a priori to have a modest effect on the dynamics of the chemical reaction, and determining a simpler model Hamiltonian may reduce the computational cost of the simulation without significantly affecting the accuracy of the results. More generally, fundamental physics analysis, heuristics, and / or experimental data may be used to inform the decision of which reactants and / or catalysts to evolve with an exact Hamiltonian, and which to evolve with a simpler model Hamiltonian.

[0064] In some embodiments, the model Hamiltonian is determined based on an exact Hamiltonian for the catalysis reaction that includes kinetic energy terms and interaction energy terms for electrons and nuclei of the initial quantum state. A description of the exact Hamiltonian may be received by the quantum computing system as classical input. The model Hamiltonian may be determined by approximating each nucleus of the exact Hamiltonian as a reduced nucleus having a reduced charge that is a difference between a charge of the nucleus and a charge of core electrons that exist within a predetermined critical distance of the nucleus with at least a threshold probability. In other words, the reduced nucleus has a reduced charge that subtracts the charges of the core electrons that are within a predetermined distance of the nucleus to within some probability. In contrast, the non-core electrons of each nucleus are the remaining subset of electrons of the nucleus that exist within the critical distance of the nucleus with less than the threshold probability. Alternatively, electrons that are strongly bound to the nucleus (i.e., that will be ionized by an excitation energy that is much higher than the energy scales of the chemical reaction, e.g., where the excitation energy is greater than the energy scales of the chemical reaction by more than a threshold amount), and so do not affect the chemical dynamics greatly, may be designated as core electrons, with only the remaining valence electrons designated as non-core electrons. Each species of molecule in the chemical reaction may have its own critical distance, threshold probability, and / or appropriately chosen set of valence electrons to delineate between its core and non-core electrons.

[0065] In some embodiments, the model Hamiltonian includes a pseudo-interaction energy that is a function of the positions of the non-core electrons of the reduced nucleus rnce and the position of thereduced nucleus rn. Advantageously, using the pseudo-interaction energy to evolve the initial state may treat the position vectors of the reduced nuclei as dynamic variables in the quantum simulation. In prior art, most classical simulation of quantum chemical / material systems invokes the “Born- Oppenheimer” (BO) approximation, in which electrons and nuclei have largely separated characteristic energy scales (mass – atomic masses are ~10000x the electron mass and so move much more slowly than electrons), and so to good approximation, one can treat electrons as dynamical degrees of freedom over an effectively fixed nuclear background. In effect, this approximation “freezes” out the nuclear degrees of freedom with their collective effect captured as a static potential experienced by the electrons. The “exact” BO Hamiltonian considers all electrons as dynamical experiencing static Coulomb potentials from all of the nuclei. To further reduce the complexity of the simulation, the nuclei and suitably-chosen core electrons may be combined into reduced frozen nuclei that contribute an effective static potential, described herein as a “pseudopotential.” There are many variations of pseudopotentials and most are developed by proposing an ansatz with fitting parameters that matched to simulation data of exact Hamiltonian (“all-electron”) results.

[0066] In contrast to the above, embodiments herein take the reduced nuclei as bona-fide dynamical particles, with the pseudo-interaction energy between the reduced nuclei and the non-core electrons defined by augmenting the pseudopotential (which contains local and non-local electron terms) with a spatially diagonal operator in the reduced nuclei coordinate. Interactions between reduced nuclei are given by a Coulomb interaction with reduced charges. This is one embodiment of dynamical reduced nuclei, termed “pseudoion,” and how to meaningfully define interactions between pseudoions and between pseudoions and electrons.

[0067] The pseudo-interaction energy may include local interaction terms and nonlocal interaction terms between non-core electrons and the reduced nuclei. A local interaction term is a function of the positions (rnce ,rn). A non-local interaction term is a function of two sets of the positions, (rnce, rn) and(r'nce,r'n). In other words, the non-local interaction terms are non-diagonal matrices in the positions of the non-core electrons and the reduced nuclei. The pseudo-interaction energy may include a Coulomb interaction potential term for the reduced charge and one or more non-Coulomb interaction potential terms. As one example, the non-Coulomb potential terms may include a Coulomb interaction potential modified by an error function erf (rnce,rn). The model Hamiltonian may also include a kinetic energy term for each non-core electron and each reduced nucleus.

[0068] At 206, a user specification is received that specifies one or more parameters of a quantum simulation of the catalysis reaction. The parameters may include precision parameters for the quantum simulation, temperature(s) of the reactant and / or catalyst molecules, geometric parameters such as collision geometry for the chemical reaction, a basis for constructing the initial quantumstate, run time for the quantum simulation, a type of measurement to be performed on the output quantum state, or parameters of the pseudo-interaction energies, among other possibilities. More generally, the user specification may include any parameters or details that are related to how the chemical reaction is to be simulated. The user specification may be received as classical data by the CPU 102.

[0069] In some embodiments, the chemical reaction is a catalysis reaction involving one or more reactants and one or more catalysts, and the user specification indicates a first temperature of the reactant(s) and a second temperature of the catalyst(s). In these embodiments, the thermal density matrices for the reactant(s) and catalyst(s) may be separately created based on the first temperature and the second temperature. For example, the thermal density matrix for each of the reactant molecules and catalyst materials may be constructed based on their respective temperatures.

[0070] At 208, the initial quantum stateis encoded within a plurality of qubits of a quantumcomputing system. For example, a plurality of qubits such as the qubits 110 may be prepared in an initial state (e.g., the null state or another initial state), and a controller such as the controller 106 may apply a series of gates to the qubits to encode them with the initial quantum state that was determined at step 202.

[0071] In some embodiments, a symmetrization operation is performed on the combined first electronic states and purified thermal states. For first quantization simulations where each particle is given a single-particle basis set (e.g. plane wave, Gaussian orbital, etc.), multiple particles of the same species, e.g. electrons, may be symmetrized / anti-symmetrized explicitly as per their particle statistics. For example, indistinguishable bosons (like phonons or certain atoms) may be symmetrized and indistinguishable fermions (like electrons) may be anti-symmetrized. This is indicated by quantum mechanics to represent a physically valid multi-particle state.

[0072] In some embodiments, a self-thermalization process is performed on the combination of the first electronic states and the purified thermal states for each of the molecules. The self- thermalization process causes the states of reactant molecules to self-thermalize (i.e., evolve in time) without incorporating interactions between the different types of reactants and catalysts. In other words, the reactants and / or catalysts are evolved in time in isolation from each other based on their individual kinetic energies and self-interaction terms (i.e. the full Hamiltonian may be restricted to the individual reactant(s) and / or catalyst(s)).

[0073] At 210, time evolution is performed on the plurality of qubits based at least in part on the model Hamiltonian and the user specification to produce an output quantum state. For example, a time evolution operator such as the one shown in Figure 3 may be applied to the plurality of qubits in the state and a plurality of block-encoded Hamiltonian qubits prepared in the Lagrangian state| , to apply the time evolution operator to the quantum state qubits and simulate timeevolution for a time t with the Hamiltonian H.

[0074] In some embodiments, to apply the time evolution operator, the model Hamiltonian is block- encoded using a plurality of qubits. The block-encoding of the Hamiltonian encodes the Hamiltonian in qubits to a predetermined level of precision, and this block-encoding of the Hamiltonian may then be called repeatedly by the time evolution operator in order to applyto the initial quantum state.

[0075] In some embodiments, to block-encode the model Hamiltonian, a Fourier transform may be performed on the model Hamiltonian to transform the model Hamiltonian to a momentum-space representation, and the model Hamiltonian may be block encoded in the momentum-space representation. In these embodiments, performing time evolution on the plurality of qubits using the model Hamiltonian is performed using the block-encoded model Hamiltonian in the momentum- space representation. In some embodiments, the model Hamiltonian is transformed from Cartesian to spherical coordinates, the Fourier transform is performed in SO(3), and the model Hamiltonian is block-encoded in terms of spherical harmonic operators.

[0076] Block-encoding can be done in several ways, in various embodiments. In some embodiments, block-encoding the model Hamiltonian can be directly done in momentum space (e.g. in the plane wave basis) or position space (e.g. in the Gaussian orbital basis), or partially done in one space along with the utilization of a quantum Fourier transformation to switch between the two. If part of the model Hamiltonian is expressed in terms of spherical harmonics, then again block-encoding may be done directly in the basis of interest by expanding the spherical harmonics (either in position or momentum space), or by utilizing a quantum Fourier transformation on SO(3) to switch between positional angular coordinatesand spherical harmonics as appropriate in theconstruction of the block-encoding. In all cases, the final result may be a block-encoding of the Hamiltonian in the basis of interest in the simulation. For example, if the plane wave basis is chosen, the block-encoding is the model Hamiltonian written in the plane wave basis. Note that block- encodings are non-unique and so there are myriad options available for a given model Hamiltonian, with varying levels of efficiency / complexity, as long as they finally yield the desired unitarity property (the purpose of the block-encoding) in the chosen simulation basis.

[0077] In some embodiments, time evolution of the model Hamiltonian is modelled as a plurality of qubitization operators, as shown in Figure 3. The time evolution of the physical system may be modelled by operating a sequence of qubitization operators on the modelled system. Thedecomposition of each qubitization operator into SELECT, prepare inverse (PREP†), energy cutoff ,and prepare (PREP) subroutines is shown in Figure 4.

[0078] At 212, one or more measurements are performed on the plurality of qubits in the output quantum state to obtain classical measurement results. The classical measurement results may indicate one or more aspects of the output quantum state. The plurality of qubits may be entirely or partially measured, in various embodiments. A variety of specific examples are provided below, but in general the measurements may be performed to probe various aspects of the output quantum state, such as the presence and / or responses of molecular intermediates, transition states, and / or chemical products at the final simulation time. For example, symmetry properties, position and / or momentum correlations between particles, bond lengths, and other aspects of the output quantum state may be directly or indirectly measured to deduce the composition of particles described in the output quantum state.

[0079] In some embodiments, the chemical reaction is a catalysis reaction involving one or more reactant molecules and one or more catalyst materials, and the classical measurement results describe product molecule(s) and / or properties of product molecule(s) of the catalysis reaction, as described in greater detail below. The classical measurement results may also describe the reactant(s) and / or properties of the reactant(s), for example, when the catalysis reaction did not proceed to completion and some proportion of the reactant(s) remain in the output quantum state. The classical measurement results may provide information about the electronic and reduced-nuclear state of the system, which may be helpful to deduce physically- and / or chemically-salient information from, e.g. which products were produced after the reaction proceeds for a period of time.

[0080] In some embodiments, measuring the qubits includes application of a point group symmetry operator. In these embodiments, the classical measurement results may indicate a molecular structure of the output quantum state based on one or more symmetries of the output quantum state indicated by the classical measurement results. The point group symmetry operator may be applied either in the case of distinguishable or indistinguishable particles in the output quantum state. The symmetry properties of the product molecule(s) may be known, such that the measured symmetry of the output quantum state may indicate whether the product molecule(s) are present in the output quantum state (and potentially how many product molecule(s) are present). For example, a reflection symmetry across an orthogonal plane (to the linear CO2 molecule) containing the C in CO2 will return the same CO2 molecule but the same reflection (around the C atom) in a state containing CO+H2O will not return the same CO or H2O. Another example of point group symmetry is the rotational symmetry of a product molecule that may be absent in the reactants, e.g. the initial reactant molecule may have a 4-fold rotational symmetry about some axis and the product molecule may have areduced 2-fold rotational symmetry about the same axis. In another embodiment, the reactants may have a reflection symmetry that is present about some plane that is lost in the product molecules. Other examples are also possible.

[0081] In some embodiments, the point group symmetry operator may be applied based on orientation information of the output quantum state. For example, a symmetry axis may be determined for performing a rotation or reflection. In some embodiments, classical information, such as the thresholds of certain bond lengths, or the axis of rotation or plane of reflection for a given symmetry, may be pre-computed (alternatively or additionally, procedural information on how to compute the axes / planes may be provided as classical information) and used to direct the application of the point group symmetry operator. In some embodiments, the output quantum state may be evolved under a Hamiltonian including an imposed external field (e.g., an electric or magnetic field), and the point group symmetry operator may be applied with an axis that is related to an orientation of the imposed external field.

[0082] In some embodiments, measuring the qubits includes measuring an expectation value of the unsigned positional difference of nuclei in the output quantum state. The expectation value of the difference in positions of the nuclei of the output quantum state may indicate whether bond lengths of the nuclei of the output quantum state indicate products of the catalysis reaction. For example, the C-O bond lengths in CO and CO2are different with the CO bond length being smaller by virtue of being a triple bond vs. the double bonds in CO2.

[0083] In some embodiments, measuring the qubits includes measuring a difference in joint correlation functions of a reactant molecule and a product molecule of the catalysis reaction. For example, if one wants to distinguish carbon monoxide (CO) and carbon dioxide (CO2), in the reverse water gas shift reaction (CO2+ H2-> CO + H2O), the connected 3-nuclear positional correlation function <COO>-<CO><O> will show a non-zero value with CO2and a zero value with CO since only CO2has both oxygens fully correlated with the carbon, distinct from only one oxygen correlated with the carbon in the case of CO.

[0084] In some embodiments, measuring the qubits of the output quantum state includes performing a projection onto hyper-spherical harmonic functions, where the hyper-spherical harmonics are derived from angles subtended by lines connecting the locations of nuclei in a user-specified manner.

[0085] In some embodiments, after performing time evolution on the plurality of qubits (but before measuring the qubits), the model Hamiltonian may be modified by removing interactions between the reactant molecules and the catalyst materials and adding an electromagnetic field. Time evolution may then be performed on the plurality of qubits using this modified model Hamiltonian. The applied electromagnetic field may be selected to probe particular aspects of the reactants and / orproducts of the reaction. For example, in the CO2reduction catalysis reaction, the product molecule carbon monoxide CO is known to have a resonant response with an electromagnetic field of a particular frequency. An electromagnetic field of this frequency may be added to the model Hamiltonian, to probe whether the output quantum state exhibits the resonant response. Additionally or alternatively, an electromagnetic field of another frequency may be added to probe a resonant response in one or more of the reactants. Measuring the qubits may then include measuring a response function of the qubits after performing time evolution using the modified model Hamiltonian.

[0086] In some embodiments, after performing time evolution on the plurality of qubits (but before measuring the qubits), the model Hamiltonian may be modified by removing interactions between the reactant molecules and the catalyst materials and adding an interaction with a new reactant molecule that interacts with at least one of the product and / or reactant molecules with at least a predetermined threshold interaction strength. Time evolution may then be performed on the qubits using the modified model Hamiltonian, and the classical measurement results may indicate whether the new reactant molecule has interacted with the one or more product molecules. This process may serve as a proxy to determine whether and to what degree the chemical reaction has produced the product molecules.

[0087] At 214, the one or more classical measurement results are stored in a non-transitory computer-readable memory medium. The stored classical measurement results may indicate the degree to which the chemical reaction proceeded to produce product molecules, and they may also specify properties (e.g., chemical composition, temperature, geometrical information, etc.) of the one or more product molecules. The quantum simulation may be designed (e.g., through the user specification) to evolve for a duration of time that partially completes the chemical reaction, such that the classical measurement results provide a mechanistic understanding of intermediate dynamics of the reaction. For example, the classical measurement results may indicate intermediate products that are produced in the reaction, and / or what the state of the system is after a predetermined time, among other possibilities. In some embodiments, the classical measurement results may be used to identify a rate-determining step of the chemical reaction, e.g., by identifying a bottleneck that is indicated by the chemical composition of the quantum state at an intermediate time. First Quantization Hamiltonian

[0088] In a quantum simulation, the system of interest (e.g., including atoms, molecules, a crystal lattice, etc.) may be described in different ways, e.g., in "first quantization" or "second quantization". In first quantization, the space that is being simulated is partitioned into grid cells, such that eachqubit register is associated with a particle and a grid coordinate. In second quantization, the space is partitioned into a small number of "orbitals" that are relevant to the simulated system. Each qubit is associated with an orbital, where the qubit state (0 or 1) indicates the occupation of the orbital, i.e., whether a particle is found in the orbital. The number of qubits utilized to describe a specific system is typically lower in second quantization. Therefore, it is typically the preferred method when minimizing the number of qubits. However, the description of the system is substantially simpler in first quantization than in second quantization. In first quantization, the Hamiltonian describing the system in many cases is just the kinetic energy term p2 / 2m plus the Coulomb potential energy. On the other hand, in second quantization, the Hamiltonian is a sum of a large number (often millions) of Hamiltonian terms with different coefficients that are computed on a classical computer before the simulation is executed on a quantum computer. These numbers (of which there are potentially millions) are accessed by the quantum computer in every step of the simulation using a data loading (QROM) circuit. As one example, a 100-orbital Hamiltonian may contain ~1004= 100 million Hamiltonian terms of the formand c are fermionic creation and annihilation operators and i, j, k, and l are orbital labels between 1 and 100. Each one of those terms comes with a different pre-factor that would be loaded into the quantum computer in each step of the simulation in a second quantization computational method. Contrariwise, the only Hamiltonian coefficients in a first-quantized Hamiltonian are the two pre-factors of the kinetic and potential energy terms.

[0089] In some embodiments, Hamiltonians of systems of particles with two-body interactions are block encoded in the first quantization usingoperations for time evolution with time or phase estimation of an energyHere, is thenumber of orbitals and is a per-particle high-energy cutoff describing the maximum per-particle kinetic and potential energy below which the simulation is accurate (such that isthe total cutoff energy). The per-particle cutoff energy may be chosen independently from other parameters and generally may not scale with η . Advantageously, the described embodiments provide a polynomial improvement in the scaling with η and an exponential improvement in the scaling with N compared to existing first-quantization qubitization methods, which scale asFurthermore, the described methods are parallelizable, as each qubitization step has a depth ofAdvantageously, the described methods use only simple arithmetic operations and have the potential to be applied in a wide variety of simulations and physical systems, including the simulation of Coulomb-interacting electrons and nuclei, Coulomb-interacting ions with frozen core electrons, and interacting particles coupled to a heat bath, among other possibilities.Hamiltonian for Coulomb-interacting Particles

[0090] The Hamiltonian describing a physical system of Coulomb-interacting particles may consist of two types of terms:

[0091]

[0092] From this expression, one would expect that all it takes to encode this Hamiltonian on a quantum computer are addition, multiplication, switching between the position and momentum bases (e.g., a Fourier transform), and applying the function

[0093] However, methods in the literature typically use much more complicated operations. For example, methods that use second quantization lose the simplicity of the Hamiltonian by translating it into a sum of many are creation and annihilationoperators, respectively, that require a large quantity of numbers to describe the coefficients. Other methods that use first quantization may have a higher complexity because they operate either in only the position or momentum basis and accordingly, involve loading many classical numbers into the quantum computer.

[0094] In some embodiments, a method is described that utilizes standard arithmetic operations to block-encode a Hamiltonian of a system of interacting particles. These particles may be electrons and nuclei, or the method may be applied more generally to other types of particles and combinations of types of particles. For example, two non-limiting possibilities are 1) the replacement of nuclei and core electrons with ions that have frozen core electrons and 2) the addition of a heat bath for the simulation of open system dynamics (for example, in the preparation of finite-temperature states or the measurement of dissipative effects).

[0095] The proposed methods exhibit improved asymptotic scaling compared to previous implementations. These improvements may be facilitated by introducing a high-energy cutoff to the Hamiltonian. Instead of simulating the Hamiltonian , a different Hamiltonian isHsimulated. The Hamiltonianis equivalent to H for states that are superpositions of momentum eigenstates in which each particle has a kinetic energy below , and superpositions of positioneigenstates in which each particle has a total potential energy below

[0096] For example, a 3D system consisting of interacting electrons and nuclei may have a Hamiltonian of the following form:

[0097]

[0098]

[0099]

[0100] where are the kinetic and potential energy of particle The state of theelectron system will be referred to herein as

[0101] Instead of block-encoding the Hamiltonianspecified above, a slightly different Hamiltonian is block encoded that acts on a slightly larger system described by a state is a register, which is referred toherein as the energy register. The "double number" operator is defined as:

[0102]

[0103] The double number operator is also referred to herein as the energy register operator, as itacts to extract a scalar value for the energy from the energy register. Note that c does not have units of energy, it is simply an integer, hence each full model Hamiltonian term additionally includes the multiplicative factor to obtain the correct energy magnitude. This is why the operator andthe register may also be referred to as a double number operator and a number register, respectively. To avoid confusion, we refer to them herein as the energy register operator and the energy register, respectively. If we write as a register of qubits

[0104]

[0105] and assume that the integer is encoded in bits using the two’s complement representation, then may be written as a sum of Z Pauli operators on the bequbits as:

[0106]

[0107] Furthermore, we define the operators

[0108]

[0109] and

[0110]

[0111] Hereare momentum and position eigenstates of thesystem. is a function that discretizes the energy and applies a cutoff:

[0112]

[0113] where expresses the energy in integer units ofand applies a high-energy cutoff, if A kinetic energy operator with a high-energy cutoff may then be defined as

[0114]

[0115] and a potential energy operator with a high-energy cutoff as

[0116]

[0117] Finally, we define the Hamiltonian

[0118]

[0119] This new Hamiltonianapproximates the original Hamiltonian as

[0120]

[0121] With a finite cutoff energy,will approximate the original Hamiltonian as long asis a superposition of momentum eigenstates with per-particle kinetic energies below , and asuperposition of position eigenstates with per-particle potential energies below This may bethe case, e.g., for sufficiently low-momentum wave packets that are sufficiently localized such that the total Coulomb attraction of each particle (balanced by the Coulomb repulsion) is not too large.

[0122] may be written as a sum of Pauli operators with equal coefficients as follows:

[0123]

[0125] Here, is a sum of Pauli terms, each with a coefficient Noticethat the eigenenergies of are between . They are not necessarily asbig as, but they are found within this interval. For qubitization, we may block-encode operators that have eigenvalues between -1 and +1. Therefore, in the following, we will be block- encoding the rescaled Hamiltonian

[0126] In other embodiments, a phase estimation circuit may operate on the state of the system to estimate a ground state energy of the physical system, as shown in the quantum circuit illustrated in Figure 5. During phase estimation, the qubitization operators W may be phase independent. Additional Technical Detail

[0127] The following numbered paragraphs provide additional technical detail and description regarding embodiments herein. Qubitization to Emulate Time Evolution

[0128] Qubitization is a framework for developing a variety of types of quantum computational methods. For embodiments described herein, qubitization is employed exclusively in the context of emulating time evolution and phase estimation. With qubitization (or more accurately, quantum signal processing), a time evolution operator may be written as a sequence of operators , as shownin Figure 3, where the time evolution of the state as shown in 1802 is decomposed as a sequence of operators as shown in 1804.

[0129] are referred to herein as "qubitization operators". The circuit shown in Figure 3 time- evolves the state with a Hamiltonian which is a linear combination of unitaryoperators with real coefficientsIn addition, it makes use of a statewhere the real numbernormalizes the state. Time evolution with timeand target error may be performed using a sequence ofqubitization operators, where thephases (i.e., angles) depend on the Hamiltonian (and time) and may be computed beforehand. Thetotal number of phasesin a particular computation generally scales linearly with both the duration of time t in the time evolution, and it also increases as the energy cutoff is increased. The operators may be further decomposed into SELECT and PREPARE subroutines, as shown in Figure 4. The SELECT subroutine applies the unitary operatorsregister (or "system register") controlled on theregister (or "control register"), wherePREP is a subroutine that prepares the state, whereoperator 1906 applies a Z-rotation 1908 of magnitude when each of the input qubits to is zero.Figure 5 - Phase Estimation Circuit

[0130] In some embodiments, the circuit for phase estimation (e.g., to estimate the ground-state energy of a Hamiltonian) is similar in some respects to the qubitization circuit for emulating time evolution, and is shown in Figure 5. Estimating an energy with an error of phase estimation of anenergy with an additive error controlled-operations for a relative errorare qubitization operators where the subscripthas been removed and W implements a constant phase ofIn other words, rather than implementing qubitization operators with a unique sequence of phasesa sequence of equivalent qubitization operators with is implemented. The circuit for the operator is illustrated at2012 in Figure 5.

[0131] To summarize, implementing time evolution or phase estimation via qubitization, involves implementing the qubitization operatorand then repeating it times for the fullcomputation. Figures 6-7 – Method Flowcharts for Simulating a Chemical Reaction

[0132] Figure 6 is a high-level flowchart illustrating a method for simulating a chemical reaction, according to some embodiments. Figure 7 is a flowchart diagram illustrating classical preprocessing for the reactant and catalyst molecules and quantum processing for performing state preparation, according to some embodiments. The methods shown in Figure 6 and 7 may be used in conjunction, e.g., with the methods described in reference to Figure 2.

[0133] 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.

[0134] 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.

[0135] 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.

[0136] 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 andencompasses any and all possible combinations of one or more of the associated listed items. It will be 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.

[0137] 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.

[0138] 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.

[0139] 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: determining an initial quantum state related to a catalysis reaction (502a, 502b); determining a model Hamiltonian for the catalysis reaction (504); receiving a user specification specifying one or more parameters of a quantum simulation of the catalysis reaction (506); encode the initial quantum state within a plurality of qubits of a quantum computing system (508); perform time evolution on the plurality of qubits based at least in part on the model Hamiltonian and the user specification to produce an output quantum state (510); perform one or more measurements on one or more qubits of the plurality of qubits to obtain one or more respective classical measurement results (512); and store the one or more classical measurement results in a non-transitory computer- readable memory medium (514).

2. A method, comprising: determining an initial quantum state related to a chemical reaction, wherein determining the initial quantum state comprises: determining a respective first electronic state for a plurality of respective electrons of one or more molecules of the chemical reaction; determining a respective set of vibrational modes for each respective nucleus of a plurality of nuclei of the one or more molecules; creating a respective thermal density matrix based on each respective set of vibrational modes, wherein the respective thermal density matrices are not based on the first electronic states; determining, based on each respective thermal density matrix, a respective purified thermal state; and determining the initial quantum state based at least in part on a combination of the first electronic states and the purified thermal states; determining a model Hamiltonian for the chemical reaction; receiving a user specification specifying one or more parameters of a quantum simulation of the chemical reaction;encode the initial quantum state within a plurality of qubits of a quantum computing system; perform time evolution on the plurality of qubits based at least in part on the model Hamiltonian and the user specification to produce an output quantum state; perform one or more measurements on one or more qubits of the plurality of qubits to obtain one or more respective classical measurement results; and store the one or more classical measurement results in a non-transitory computer-readable memory medium.

3. The method of claim 2, wherein the thermal density matrix comprises an approximate thermal density matrix that incorporates mode truncation, wherein incorporating mode truncation comprises limiting a number of excitations of the vibrational modes based on physical conditions of the chemical reaction.

4. The method of claim 2, wherein determining the initial quantum state further comprises performing a symmetrization operation on the combined first electronic states and purified thermal states.

5. The method of claim 2, wherein determining the initial quantum state further comprises performing a self- thermalization process on the combination of the first electronic states and the purified thermal states for each of the one or more molecules.

6. The method of claim 2, wherein the chemical reaction comprises a catalysis reaction involving one or more reactant molecules and one or more catalyst materials, wherein the user specification comprises a first temperature of the one or more reactant molecules and a second temperature of the one or more catalyst materials; and wherein creating the respective thermal density matrices based on the respective sets of vibrational modes is performed based at least in part on the first temperature and the second temperature.

7. The method of claim 2,wherein creating the respective thermal density matrices based on the respective sets of vibrational modes and not based on the first electronic states is performed responsive to a determination that a temperature of at least one of the one or more molecules multiplied by Boltzmann’s constant is much less than an energy gap between the respective first electronic states and respective first excited states of electrons of the at least one of the one or more molecules.

8. The method of claim 2, wherein the respective first electronic states comprise electronic ground states of the respective electrons.

9. The method of claim 2, wherein the chemical reaction comprises a catalysis reaction involving one or more reactant molecules and one or more catalyst materials, and wherein the one or more classical measurement results comprise one or more product molecules of the catalysis reaction.

10. A method, comprising: determining an initial quantum state related to a catalysis reaction; determining a model Hamiltonian for the catalysis reaction; receiving a user specification specifying one or more parameters of a quantum simulation of the catalysis reaction; encode the initial quantum state within a plurality of qubits of a quantum computing system; perform time evolution on the plurality of qubits based at least in part on the model Hamiltonian and the user specification to produce an output quantum state; perform one or more measurements on one or more qubits of the plurality of qubits to obtain one or more respective classical measurement results, wherein the one or more measurements comprise one or more of: an application of a point group symmetry operator, wherein the one or more classical measurement results indicate a molecular structure based on one or more symmetries of the output quantum state indicated by the one or more classical measurement results; an expectation value of a difference in respective centers-of-mass of two or more nuclei of the one or more reactant molecules; and a difference in joint correlation functions of a reactant molecule and a product molecule of the catalysis reaction; andstore the one or more classical measurement results in a non-transitory computer-readable memory medium.

11. The method of claim 10, wherein the one or more symmetries of the output quantum state indicate whether the output quantum state is unchanged by swapping locations of two nuclei of the one or more reactant molecules.

12. The method of claim 10, wherein the expectation value of the difference in respective centers-of-mass of the two or more nuclei of the one or more reactant molecules indicates whether one or more bond lengths of the two or more nuclei indicate one or more products of the catalysis reaction.

13. The method of claim 10, further comprising: after performing time evolution on the plurality of qubits based at least in part on the model Hamiltonian: modifying the model Hamiltonian by: removing interactions between the one or more reactant molecules and the one or more catalyst materials, and adding an electromagnetic field; and performing time evolution on the plurality of qubits using the modified model Hamiltonian, wherein performing the one or more measurements on the one or more qubits of the plurality of qubits comprises measuring a response function of the one or more qubits after performing time evolution using the modified model Hamiltonian.

14. The method of claim 10, further comprising: after performing time evolution on the plurality of qubits based at least in part on the model Hamiltonian: modifying the model Hamiltonian by: removing interactions between the one or more reactant molecules and the one or more catalyst materials, andadding a first reactant molecule, wherein the first reactant molecule interacts with one or more product molecules of the catalysis reaction with an interaction strength that is greater than a predetermined threshold; and performing time evolution on the plurality of qubits using the modified model Hamiltonian, wherein the classical measurement results indicate whether the first reactant molecule has interacted with the one or more product molecules.

15. The method of claim 10, wherein performing the one or more measurements comprises applying one or more hyper- spherical harmonic functions to the output quantum state, wherein the hyper-spherical harmonic functions are parametrized to measure one or more angles between locations of three or more nuclei of the output quantum state.

16. A method, comprising: determining an initial quantum state related to a chemical reaction; determining a model Hamiltonian for the reaction, wherein determining the model Hamiltonian comprises: receiving an exact Hamiltonian for the catalysis reaction, the exact Hamiltonian comprising a plurality of kinetic energy terms and a plurality of interaction energy terms for a plurality of electrons and nuclei of the initial quantum state; determining the model Hamiltonian based at least in part on the exact Hamiltonian, wherein determining the model Hamiltonian comprises: approximating each respective nucleus of the plurality of nuclei as a respective reduced nucleus having a respective reduced charge that comprises a difference between a respective charge of the respective nucleus and a respective charge of respective core electrons of the plurality of electrons that exist within a critical distance of the respective nucleus with at least a threshold probability; wherein the model Hamiltonian comprises: for each respective reduced nucleus, a respective pseudo-interaction energy that is a function of one or more respective distances between one or more respective non-core electrons of the respective reduced nucleus and a respective position of the respective reduced nucleus; anda respective kinetic energy term for each non-core electron and each reduced nucleus; encode the initial quantum state within a plurality of qubits of a quantum computing system; perform time evolution on the plurality of qubits based at least in part on the model Hamiltonian to produce an output quantum state; perform one or more measurements on one or more qubits of the plurality of qubits to obtain one or more respective classical measurement results; and store the one or more classical measurement results in a non-transitory computer-readable memory medium.

17. The method of claim 16, wherein each respective pseudo-interaction energy comprises one or more respective local interaction terms and one or more respective nonlocal interaction terms between the respective non- core electrons and the respective reduced nucleus.

18. The method of claim 16, wherein each respective pseudo-interaction energy comprises a respective Coulomb interaction potential term for the respective reduced charge and one or more respective non-Coulomb interaction potential terms.

19. The method of claim 16, wherein the non-core electrons of each reduced nucleus comprise a subset of the plurality of electrons that exist within the critical distance of the nucleus with less than the threshold probability.

20. The method of claim 16, further comprising: block encoding the model Hamiltonian by: performing a Fourier transform on the model Hamiltonian to transform the model Hamiltonian to a momentum-space representation; and block encoding the model Hamiltonian in the momentum-space representation, wherein performing time evolution on the plurality of qubits based at least in part on the model Hamiltonian is performed using the block encoded model Hamiltonian in the momentum- space representation.

21. The method of claim 16, further comprising:transforming the model Hamiltonian from Cartesian to spherical coordinates, wherein the Fourier transform is performed in SO(3), and wherein the model Hamiltonian is block encoded in terms of spherical harmonic operators.

22. The method of claim 16, wherein the chemical reaction comprises a catalysis reaction involving one or more reactant molecules and one or more catalyst materials, wherein the plurality of electrons and nuclei comprise electrons and nuclei of the one or more reactant molecules and the one or more catalyst materials, wherein the model Hamiltonian is determined for the electrons and nuclei of the catalyst materials, and wherein performing time evolution on the plurality of qubits is further based at least in part on the exact Hamiltonian for the reactant molecules.

23. The method of claim 16, further comprising: receiving a user specification specifying one or more parameters of a quantum simulation of the catalysis reaction, wherein performing the time evolution on the plurality of qubits to produce the output quantum state is performed further based at least in part on the user specification.

24. A quantum computing system, comprising: a non-transitory computer-readable memory medium; a plurality of physical qubits; and a controller, wherein the quantum computing system is configured to perform the method steps of any of claims 1-23.

25. A non-transitory computer-readable memory medium storing program instructions which, when executed by a processor, cause a quantum computing system to perform the method of any of claims 1-23.

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