Schedules for differential equation quantum computational tasks

Abstract

In some implementations, a computation system may include identifying a computational task for processing on a quantum information processing system, the computational task having one or more differential equations. In addition, the computation system may include generating a scheduling plan for execution of the differential equation on the quantum information processing system, the scheduling plan having a query count specifying a quantity of queries to be performed against a block-encoding of the differential equation to generate results data for the computational task. A quantum information processing system can process the queries to generate a quantum state, such as a history state or final state that corresponds to the differential equation.

Description

PSIQ-558WO1 / 6224.009WO1 SCHEDULES FOR DIFFERENTIAL EQUATION QUANTUM COMPUTATIONAL TASKS Claim of Priority

[0001] This application claims the benefit of priority to U.S. Patent Application Serial No. 63 / 538,456, filed on September 14, 2023, which is incorporated by reference herein in its entirety. Technical Field

[0002] Embodiments herein relate generally to quantum information processing and generating computational task scheduling. Background

[0003] 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 oneof two orthogonal states (denoted as |0 and |1 in the conventionalbra / ket notation) or in a superposition of the two states (e.g., (|0 + |1 ). By operating on a system (or ensemble) of qubits, aquantum computer may quickly perform certain categories of computations that would require impractical amounts of resources (e.g., processing power, memory, electricity) on classical computer systems. While qubit-based processing can provide efficient solutions to some tasks, it can be difficult to identify and implement quantum processing steps in an efficient approach that maintains practical resource overheads. Brief Description of the Drawings

[0004] 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 likePSIQ-558WO1 / 6224.009WO1 reference numerals refer to corresponding parts throughout the Figures.

[0005] FIG. 1 shows an example processing architecture for implementing quantum task scheduling, in accordance with some example embodiments.

[0006] FIG. 2 shows a quantum information processing architecture, in accordance with some example embodiments.

[0007] FIG. 3 shows an example scheduling plan, in accordance with some example embodiments.

[0008] FIG. 4 shows a flow diagram of a method for determining query counts for results of a differential equation of a quantum computational task, in accordance with some example embodiments.

[0009] FIGs. 5A and 5B show example flow diagrams for implementing quantum scheduling, in accordance with some example embodiments.

[0010] FIGs. 6A-6H show utilization of surface codes to construct an error-corrected fault-tolerant logical qubit, in accordance with some example embodiments.

[0011] FIGs. 7A-7C show example quantum computer systems, in accordance with some example embodiments.

[0012] FIG. 8 shows an example classical computer system, in accordance with some example embodiments.

[0013] 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 scope of the subject matter as defined by the appended claims.PSIQ-558WO1 / 6224.009WO1 DETAILED DESCRIPTION

[0014] Disclosed herein are examples (also referred to as “embodiments”) of systems and methods for performing fault-tolerant post-selection using various quantum computing systems.

[0015] 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.

[0016] Quantum information processing systems can be configured to solve different classes of computational tasks that are difficult for classical computational systems to efficiently solve. Examples include quantum transform based tasks (e.g., quantum Fourier related tasks, period finding), quantum search tasks (e.g., Grover’s algorithm), and quantum simulation tasks (e.g., fluid dynamics simulations, quantum chemistry simulations). Generally, in some approaches, such as quantum simulation tasks, a quantum state is prepared (e.g., collection of qubits is prepared for a specified state using gate while maintaining unitary evolution ) to simulate a quantum system (e.g., Hubbard model, Ising model systems) where the evolution of the system is governed or otherwise evolves according to one or more differential equations (e.g., Schrodinger’s equation, or sets of differential equations that correspond a physical system to be modeled). Operations can be performed on the quantum state system (e.g., gates, measurements) to output results, such as the final state of a given simulated system at a later time, a different place, or a different configuration.

[0017] It can be difficult to convert a given system to be simulated into a format that a quantum information system can process (e.g., into aPSIQ-558WO1 / 6224.009WO1 quantum simulation task). One approach is to approximate a given system for simulation as a data representation (e.g., digital representation, matrices), and then discretize the given system across cycles (e.g., iterative steps) to provide a history state and / or final state.

[0018] However, not all dynamical systems can be approximated and is it difficult to determine which systems can be simulated at all. Further difficulty arises when determining whether such systems can be simulated efficiently on a quantum computing system with available resources (e.g., qubits, physical resources such as electrical power, cooling power).

[0019] Some quantum computational tasks utilize queries or calls to oracles (e.g., unitaries, block box states), which can be queried one or more times to yield an output (e.g., a quantum state, output data). However, determining the amount of operations (e.g., number of query calls, queries) to a unitary that corresponds the quantum state is difficult. Implementing an incorrect of number of query calls can yield inaccurate results, and / or exceed the practical resources of a quantum information processing system, such as an intractable task (e.g., a task requiring more resources --physical qubits, logical qubits-- than are available in a given quantum computing system).

[0020] As such, specifying an accurate amount of query calls in a given quantum computational task can ensure accurate results and reduce resource overhead in quantum computation systems (e.g., reduced processing, reduced runtime, reduced footprint, reduced resource consumption).

[0021] For the forgoing reasons, a quantum scheduler system can be implemented (e.g., on a classical computing system) to generate an exact query count for a quantum computational task involving one or more differential equations to be processed by a quantum information processing system (e.g., photonic quantum computer, ion trap quantum computer, superconducting circuit-based quantumPSIQ-558WO1 / 6224.009WO1 computer). The quantum scheduler system can generate block- encodings for one or more differential equation of a quantum computational task and can generate a query count for a number of iterations or queries required to be performed against the block- encodings to yield a quantum state, which can be measured to output results data (e.g., data describing the simulated system at a different time). The quantum scheduler system can further generate and store scheduling data.

[0022] In some example embodiments, a plurality of block-encodings for a plurality of differential equations of a given task are generated and corresponding query counts for each is generated to enable accurate resource estimation and scheduling of tasks on a quantum information processing system. For example, and in accordance with some example embodiments, a scheduling plan can specify a quantity of queries to be performed against a block-encoded matrix, where after the specified the quantity of queries is performed, results data is read-out (e.g., qubits measured) and stored in memory, or displayed on display screen as results data. In some example embodiments, according to a specified quantum algorithm involving a plurality of differential equations, different query counts for different differential equation tasks are ordered or sequenced based on the algorithm complexity or available quantum computer resources (e.g., available qubits of a given system, available computing throughput).

[0023] FIG. 1 is a functional data flow diagram 100 for implementing quantum scheduling, according to some example embodiments. A computational task 105 can be input (e.g., by a user) for processing by a quantum information processing architecture 201, discussed in FIG. 2. The computational task can include one or more ordinary differential equations (ODEs) for processing on the quantum information processing architecture 201 (e.g., quantum simulation task). A plurality of qubits can be implemented to represent a quantum state 110 upon which operations can be performed toPSIQ-558WO1 / 6224.009WO1 implement evolution of the quantum state according to task being performed (e.g., simulation being performed). A quantum scheduler system 205 can implement a classical computing system 203 to precompute data (e.g., query count, unitary encodings) and output a scheduling plan 115 that can order operations to be performed on the quantum information processing architecture 201. In some example embodiments, one or more steps to be performed in the scheduling plan are performed on a quantum information processing system that manages the qubits of the quantum state 110. In some example embodiments, encodings 130 (e.g., one more encoded data structures, matrices) are generated for each of the one or more differential equations of the computational task 105.

[0024] In some example embodiments, the scheduling system 115 is configured to generated an exact query count 125 for the number of query calls to be made against the encodings 130 such that the runtime 120 (e.g., runtime instance of operations being performed on the quantum information processing architecture 201) can be implemented in an efficient manner (e.g., scheduling of tasks based on available resources, avoidance of intractable task, avoidance of tasks that require an impractical or infeasible amount query calls and / or physical resources), and further ensure that the quantity of calls in the query count 125 will lead to an accurate result in the results data 135 (e.g., output data, accurate final state data for a given simulation).

[0025] FIG. 2 illustrates a quantum information processing architecture 201, according to some example embodiments. As illustrated, the architecture 201 includes a classical information processing system 203 (e.g., classical computer comprising cores formed from electrical transistors) coupled to a quantum information processing system 205 (e.g., quantum computer, quantum computing chips) over a channel 212 (e.g., electrical channels, optical channels).PSIQ-558WO1 / 6224.009WO1 The classical channel 212 may relay classical information between the classical and quantum computing systems.

[0026] In some embodiments, the classical computing system 203 includes one or more tangible computer-readable memory media 204, one or more central processing units (CPUs) or processor(s) 202, a power supply, an input / output (I / O) subsystem, and a communication bus or interconnecting these components. The processor(s) 202 may execute modules, programs, and / or instructions stored in memory 204 and thereby perform processing operations. 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 204 stores one or more programs (e.g., sets of instructions) and / or data structures and is coupled to the processor(s).

[0027] In some embodiments, the classical computing system 203 may have installed thereon a dedicated module acting as a fault-tolerant post-selection (FTPS) controller. In some embodiments, the FTPS controller may include its own dedicated memory medium and / or processor(s), which may be a dedicated processor, an FPGA, or an ASIC, among other possibilities. In some embodiments, the FTPS controller may be implemented as software and may share processing resources with other control aspects of the classical computing system.

[0028] The classical computing system 203 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 204 and executed by the classical (e.g., digital) processor 202 of the classical computer.

[0029] The memory 204 is classical in the sense that it stores data and / or program instructions in a storage medium in the form of bitsPSIQ-558WO1 / 6224.009WO1 (rather than as qubits containing quantum information), which have a single definite binary state at any point in time. The processor may read instructions from the computer program in the memory 204 and / or write data into memory, and may optionally receive input data from a source external to the computer 203, such as from a user input device such as a mouse, keyboard, or any other input device. The processor 202 may execute program instructions that have been read from the memory 204 to perform computations on data read from the memory 204 and / or input from the quantum computing system, and generate output from those instructions. The processor 202 may store that output back into the memory 204.

[0030] In the example of FIG. 2, the classical computing system 203 stores a computational task 267 in the memory 204. As an example, the computational task 267 can include a quantum simulation task comprising one or more differential equations for modeling a physical system (e.g., quantum chemistry modeling). As further illustrated, the classical computing system 203 comprises a quantum scheduler that can generate scheduling data 257 (e.g., scheduling plan 115, FIG. 1) from the differential equations and further generate query counts and a scheduling flow for processing on a quantum computing system, such as quantum information processing system 205.

[0031] The quantum information processing system 205 may include a plurality of qubits and a controller 206 configured to interface with the plurality of qubits 210 to control, direct and / or measure the qubits within the quantum circuit. The qubits may be configured to evolve in time under the directed influence of the controller, and a measurement system 208 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 203 and / or the quantum controller 206 during a quantumPSIQ-558WO1 / 6224.009WO1 computation, and they may additionally include classical results of the quantum computation. The measurement results may be communicated to the classical computing system 203 and / or the controller 206, and further the classical computing system 203 may provide directions and / or instructions to the controller 206 and the measurement system 208 to guide the behavior of the quantum computing system to perform a quantum computation. For example, the classical computing system 203 may provide classical data signals used for quantum state preparation within the quantum information processing system 205, in response to which the controller may prepare the states of the qubits 210 into a desired initial state for a particular quantum computation.

[0032] In some embodiments, physical qubits 210 are provided to the measurement system 208 and controller 206, where the measurement system 208 and the controller 206 function as a logical qubit encoder that perform a sequence of measurements on the physical qubits to produce a logical qubit (e.g., a logical qubit prepared in a quantum state). For example, the measurement system 208 and controller 206 may perform a sequence of measurements on the physical qubits to entangle them in such a way as to produce a logical qubit. Encoding the logical qubit will also produce syndrome graph data for the logical qubit as classical information, which is output to the FTPS controller of the classical computing system 203 via the channel 212. The FTPS controller analyzes the syndrome graph data to determine an error metric for the logical qubit. It is then determined whether the error metric satisfies a quality threshold specified by a policy. If the quality threshold is not met, the FTPS controller outputs instructions back to the quantum information processing system 205 along the channel 212 to discard the logical qubit. Alternatively, in some embodiments when the quality threshold is not met, the logical qubit may be flagged as poor quality and / or rerouted to a different aspect of the quantum computation. If the quality threshold is met, the FTPSPSIQ-558WO1 / 6224.009WO1 controller outputs instructions to the quantum information processing system 205 to keep the logical qubit. Additional examples of physical systems to implement qubits of a quantum state for processing are discussed in further detail with reference to FIGs. 7A-7C below.

[0033] As discussed, some quantum computational tasks can include one or more differential equations to be processed. In some example embodiments, ODEs of a given computational task are as follows: () = ( ) + , with (0) = Eq. 1

[0034] , where ( ) is the -dimensional solution vector, is an ×time-independent matrix and is an -dimensional vector, also heretaken to be time-independent. When = , with Hermitian, and= 0, the problem in Eq. 1 reduces to well-known quantumHamiltonian dynamics. The approach discussed herein enables generation of a quantum encoding (e.g., quantum state 110) ofdynamic equations of form = + , where construction of theunitary UAencodes the ODE matrix A, and the unitary UBencodes the forcing b, and the unitary U0encodes the initial condition (0), from which the solution state (e.g., final state) or history state can be accessed for further analysis of the quantum system on the quantum computer. For example, target observables (e.g., for a given analysis of the quantum state 110) of the quantum encoding can be measured and output data (e.g., results data 135) can be generated, stored, and / or displayed on a display screen of a user device.

[0035] Given Eq. 1, there are different approaches that could be employed to solve this system of equations on a quantum computer. In some example embodiments, the quantum information processing architecture 201 here is configured to embed the dynamical problem into a linear system of equations. In some example embodiments, the quantum information processing systems 205 can generate one or more types of states (e.g., quantum state 110) for a given dynamical problem (e.g., computational task 105) including: the solution statePSIQ-558WO1 / 6224.009WO1and the history state. On the one hand, given > 0, outputting a state-close in 1-norm to the final state (e.g., solution state) can be implemented as follows: |( ), at some time final 0 Eq. 2

[0036] , if, for instance, the steady-state properties are to be processed. Further, for the history state, to process the solution over the interval[0, ], the system is configured to output a quantum state -close to ahistory state encoding the solution at the discrete times… , }

[0037] , where | ( ) is a quantum state proportional to the solution ofEq. 1 at time , and {| } are clock states that provide classicallabels for the discrete times = , with a timestep.

[0038] In Eq. 1, and are given. The system discretizes time insteps of size 1 / . Setting:= / = / ==

[0039] , the problem is rescaled such that 1 and the discretetimes are {0, … , = } (and is increased slightly if needed toensure integer separations). In some example embodiments, for simplicity the bar is dropped and it is assumed it is assumed rescaling has been implemented.

[0040] In some example embodiments, the quantum information processing system 205 prepares unitaries, O1, O2, and O3. The unitary O1 is a block-encoding of , meaning a unitary with:

[0041] , where |0 : = |0 , | is an arbitrary vector, is a rescalingfactor required to encode into a unitary (assume 1 ), and | isa vector orthogonal to the subspace spanned by |0 . The values ( , )determine the quality of the block-encoding and depend on the specificPSIQ-558WO1 / 6224.009WO1 form of . For constructions for sparse and certain dense matriceswith = (1).

[0042] O2 is a unitary such that |0 = | , encoding the initialcondition as a quantum state when applied to a reference state |0 .

[0043] O3 (If) is a unitary such that |0 = | , encodingup to normalization.

[0044] To generate results data, the quantum processing system 205outputs a quantum state close in 1-norm to Eq. 2, in the case of thefinal state, or Eq. 3, for the history state over the time interval [0, ].The cost of obtaining | or | is evaluated via the query count Q,which is the number of times the above unitaries are to be applied ina given scheduling plan. In some example embodiments, the quantuminformation processing system 205 implements Q calls to the block-encoding of UA and n*Q calls the state preparation unitaries (e.g.,n=2, n=4). In some example embodiments, Q depends on the ODEparameters summarized in Table I.

[0045] FIG. 3 shows an example scheduling plan 257 for execution ona classical computer and quantum information processing system, in accordance with some example embodiments. The scheduling plan 257 comprises both an algorithm (ordering of steps) and scheduling information, such as the number of calls on oracles, which may change based on a determined QQLSA calls determined from aPSIQ-558WO1 / 6224.009WO1 quantum linear solver algorithm (QLSA) subroutine, further discussed in FIG. 4.

[0046] In the example of FIG. 3, the parallelogram boxes (e.g., blocks 301-303) correspond to operations performed on a classical information processing system (e.g., classical precomputations), and the rectangle boxes correspond to operations performed on the quantum information processing system (e.g., calls to unitaries via logical qubit operations, measurements), as is understood by someone of ordinary skill in the art. A descriptive summary of the operations follows:

[0047] Blocks 301 includes determining ODE parameters, such as those found in Table 1.

[0048] Block 302 includes computing the discretized time step h such that h is normalized.

[0049] Block 303 includes computing a number of times the unitaries need to be called (Q) in the QLSA subroutine.

[0050] Block 304 uses unitary O1 to perform block-encoding of A, the generator of the dynamics, on the quantum information processing system.

[0051] Block 305 uses unitary O2 to perform state preparation (e.g., on the quantum information processing system) of the initial condition

[0052] Block 306 uses unitary O3 to perform state preparation (e.g., on the quantum information processing system) of the forcing term b.

[0053] Algorithm Summary: The solution of Eq. 1 can be:

[0054] By discretizing time, the classical information processing system recursively approximates the solution at time from that at time 1 via a Taylor series truncated at degree :PSIQ-558WO1 / 6224.009WO1One canupper bound the required such that the time discretization relative error is at most , meaning () ( ) , = 0, … ,

[0056] In some example embodiments, the system then determines a value for the truncation degree k, that suffices as follows:log

[0057] In some example embodiments, the recursive equations are embedded by the system into a linear system L whose blocks only contain matrices proportional to A or the identity matrix. This linear system takes the form Ly = c, where y encodes the dynamical solution at all times [0, T]. In some example embodiments, an idling parameter p in the linear system manages evolution generated by A = 0 for p time-steps at the end, which increases the amplitude of the solution at time T, which can then be used to output |xF >.

[0058] In some example embodiments, the linear system of equations is solved via a QLSA subroutine at block 309 thereby generating a quantum encoding (e.g., quantum state 110, history state, final (solution) state). In some example embodiments, amplitude amplification is further applied as an option to generate the quantum encoding of the dynamical system.

[0059] In FIG. 3, the unitaries O1, O2, O3 are used to construct a block-encoding of the linear system matrix L (at block 307) and the vector of constants c (at block 308). The number of times these need to be called (Q) in the QLSA subroutine is generated in a scheduling plan using a classical processor as discussed above and seen in block 303. In some example embodiments, the output solution is further post processed via amplitude amplification (block 310) as needed,PSIQ-558WO1 / 6224.009WO1 which can then be used to generate output data, such as (a) a quantum state encoding of an approximation of the final state, or (b) a quantum state encoding of an approximation of the history state of the system being processed per a given computational task.

[0060] FIG. 4 shows a flow diagram of a method 400 for determining query counts for results of a differential equation of a quantum computational task, according to some example embodiments. The query counts guarantee convergence to a target output (e.g., performing the identified queries counts ensures the computation outputs a target encoding). In some example embodiments, the target output comprises a quantum encoding (e.g., encoded quantum state) of the solution to a set of dynamical equations (e.g., differential equations). In the following example, as an option, in some embodiments amplitude amplification is implemented using a fixed point approach as is known in the art (e.g., Yoder, Low, Chaung - APS, Vol. 113, Issue 21, Nov. 2014). In some alternative example embodiments, as discussed in further detail with reference to FIG. 5B, sampling until success is performed. For example, as the lower boundof , is computed, ( , ) is then generated (e.g., resampling:1 / , , precomputed on classical system).

[0061] Definition 1: Explicit query counts for ODE: Given a linearODE of the form = + , (0) = , assume access to , in( 1), ( 2), as wellThe final (Eq. 2) and history(Eq. 3) states can be prepared by the system 205 performing a number of queries to , and nQ queries to (e.g., n=2 for 2Qqueries; n=4 for 4Q queries) and, if 0, nQ queries to (. . , = 2, = 4). Q can be analytically upper bounded as shownby the blocks of FIG. 4:

[0062] Block 401 sets the time discretization error = / 8.

[0063] Block 402 computes the Taylor truncation:PSIQ-558WO1 / 6224.009WO1

[0064] At block 403, for the history state, set an idling parameter to =0. For the final (e.g., solution) state, at block 403, set = / ( +1) ( + 1) in the semistable case, and = / / ( + 1) ( + 1) for thestable case.

[0065] Block 404 computes

[0066] Block 405 computes the upper bound on the condition number

[0068] Block 406 fixes = (3 ) for the inhomogeneous case ( 0)and = 1 for the homogeneous case ( = 0). For outputting thehistory state compute the success probability= Eq. 4, and for outputting the final state compute

[0069] Block 407 computes

[0070] Block 408 computes = ( ˜ , , ), the best availablequery counts for QLSA given the parameters ˜ , , . Here theclosed formulabeing used.PSIQ-558WO1 / 6224.009WO1

[0071] Block 409 computes Pr , , the query counts for amplitudeamplification from success probability Pr , or Pr , = 1 / Pr ,where sampled till success.

[0072] Block 410 generates =where the total logical qubitcount for the algorithm is a + 12 + log [(( + 1)( + 1) + ) ] , inaccordance with some example embodiments.

[0073] As discussed, the Quantum Linear Solver Algorithm (QLSA) can be implemented as a subroutine in the above processing.

[0074] In accordance with some example embodiments, QLSA can be implemented as follows. Set a system of linear equations, L y = c, with an N x N matrix, and preconfigure L and c are scaled so that the singular values of L lie in [1 / k, 1]. Denote |c > as the normalizedstate that is proportional to1 c > as the normalizedstate proportional to A-1|c >.

[0075] Further, and in accordance with some example embodiments, prepare quantum states (e.g., initialize qubits, logical qubits) on a quantum information processing system as follows: (1) A unitary ULthat encodes the matrix L / , for some constant > 1 , using aquantity of qubits equal to , and (2) configure quantum state as Uc (e.g., oracle), preparing |c >.

[0076] With the above configuration, the quantum computational task (e.g., quantum algorithm to be processed) can output a quantum state -close in 1-norm to |A-1c >, using an expected calls to UL orand n* callswhere:

[0077] , for any and 12. In some example embodiments, thesuccess probability is lower bounded by 0.39 – 0.201 . In somePSIQ-558WO1 / 6224.009WO1 example embodiments, where the block encoding of the N x N dimensional matrix L requires a-auxiliary qubits, then a full implementation of QLSA requires a + 6 + [log2N] qubits (e.g., photonic qubits, ion trap qubits, charge / flux qubits, etc.; logical qubits formed thereof).

[0078] FIG. 5A is a flowchart of an example method 500, according to some example embodiments. In some implementations, one or more method operations of FIG. 5A may be performed by a device, such as classical computing system 203.

[0079] As shown in FIG. 5A, method 500 may include identifying, on a classical processor, a computational task for processing on a quantum information processing system, the computational task having a differential equation (operation 505). For example, device may identify, on a classical processor, a computational task for processing on a quantum information processing system, the computational task having a differential equation, as described above.

[0080] As also shown in FIG. 5A, method 500 may include generating, on the classical processor, a scheduling plan for execution of the differential equation on the quantum information processing system, the scheduling plan having a query count specifying a quantity of query calls to be performed against a block-encoding of the differential equation to generate results data for the computational task, the query count generated by combining a quantum linear solver query count generated for the differential equation with a probabilistic success query count generated for the differential equation (operation 510). For example, device may generate, on the classical processor, a scheduling plan for execution of the differential equation on the quantum information processing system, the scheduling plan having a query count specifying a quantity of query calls to be performed against a block-encoding of the differential equation to generate results data for the computational task, the query count generated by combining a quantum linear solver queryPSIQ-558WO1 / 6224.009WO1 count generated for the differential equation with an probabilistic success query count generated for the differential equation, as described above (e.g., via amplitude amplification, or sampling).

[0081] As further shown in FIG. 5A, method 500 may include transmitting the scheduling plan to the quantum information processing system for processing of the computational task according to the quantity of query calls to the block-encoding of the differential equation (operation 515). For example, device may transmit the scheduling plan to the quantum information processing system for processing of the computational task according to the quantity of query calls to the block-encoding of the differential equation, as described above.

[0082] As further shown in FIG. 5A, the method may include executing the scheduling plan on a quantum information processing system (operation 520). For example, at operation 520, the quantum information processing system operates qubits to instantiate an initial state and performs calls and processing as discussed above with reference to FIG. 3 (solid box operations).

[0083] As further shown in FIG. 5A, the method may include output results data, such as solutions to differential equations, or a description of the linear system at a different state or position (operation 525). Further processing may include generating visualizations of the results data (e.g., text, numbers, graphs, animations) for display on a display screen (e.g., monitor of a user device).

[0084] Although FIG. 5A shows example operations of method 500, in some implementations, method 500 may include additional operations, fewer operations, different operations, or differently arranged operations than those depicted in FIG. 5A. Additionally, or alternatively, two or more of the operations of method 500 may be performed in parallel.PSIQ-558WO1 / 6224.009WO1

[0085] FIG. 5B is a flowchart of an example method 550, according to some example embodiments. In some implementations, one or more process blocks of FIG. 5B may be performed by a device, such as the classical computing system 203. In example example embodiments, the method 550 is implemented as a sub-routine of operation 510 of method 500 (FIG. 5A, plan generation).

[0086] As shown in FIG. 5B, method 550 may include generating, on the classical processor, one or more encodings (e.g., block-encoding of a differential equation matrix of the differential equation) at operation 555. For example, device may generate, on the classical processor, one or more encodings as follows.

[0087] ENCODINGS: Configure 3 registers | , , . The register is aclock degree of freedom, labeling the solution at each time-step within[ 0, ], plus potentially further idling steps to increase the amplitudeof the solution at the final state, if nec-essary. It takes values =0, … , + / ( + 1), where is a multiple of + 1 to be fixed. Theregister is used to compute Taylor sums, = 0, … , . Finally, labelsthe components of the vector , = 1, … , . The data is encoded toinclude data about the solution to the ODE problem into the solution := of a linear system= Eq. 5where = + + , and

[0088] , with the Heaviside function ( ) = 0 for = 0 and ( ) = 1 for> 0. The term generates Taylor components, which are summedat each by to generate a truncated Taylor series thatapproximates the dynamics up to = . If + 1, from = toPSIQ-558WO1 / 6224.009WO1 =+ / ( + 1) 1 and for all = 0, … , , so steps overall, theterm generates 'idling', which allows for an increase of the amplitude of the final state component. In Eq. 5 set

[0089] It can be shown that the solution to this system can be split up=( , 0, … ,0, , 0, … ,0, , … , )

[0090] and is an orthogonal 'junk' component, that is unwanted, and can be removed by post-selection. When the history state isoutput, is set as follows: = 0; further, (as a normalizedquantum state ) provides a discrete approximation to the exact history state encoded in Eq. (3) :

[0091] Instead, to output the final state p is set to be: > 0, and thecomponents are targeted with amplitude amplification.

[0092] As also shown in FIG. 5B, method 550 may include generating, on the classical processor, the quantum linear solver query count by applying a quantum linear solver query scheme to the differential equation (operation 560). For example, device may generate, on the classical processor, the quantum linear solver query count by applying a quantum linear solver query scheme to the differential equation, as follows.

[0093] LINEAR SOLVER: Eq. 5 is processed using the asymptotic and non-asymptotic query complexity upper bound for QLSA. The running cost depends on three parameters: (a) The cost of constructing a block-encoding of from that of . (b) An upper bound on the condition number of , i.e., on : =. (c) The QLSA does not output the ideal solution | , but aPSIQ-558WO1 / 6224.009WO1 quantum state which is -close to it in 1-norm. The cost depends on .

[0094] In some example embodiments, the following are determined from the ODE parameters.

[0095] (a) Constructing the linear system block-encoding. Since := + 1 + 2, the following is defined: ˜ = / , ˜ = / , andproblem is then solved as ˜ = ˜. This has ˜ 1, as normallyrequired by QLSA. The detailed running cost of a QLSA is in terms ofcalls to a unitary block-encoding ˜ of ˜ and a state preparationunitary˜of ˜. These need to be constructed from the availableunitaries , , (see Block Encoding, below). Concerning ˜, it canbe constructed with a single call to both and . As for˜, a( ˜ , + 6,0)-block-encoding of ˜ is constructed with a single call to ,where ˜ = (1 + + 1 + ) / , as given in Definition 1. The QLSArequires further ancilla qubits, as well as log [(( + 1)( + 1) + ) ]qubits to encode ˜ .

[0096] (b) Condition number bound. Recall that the Lyapunov analysisleads to stability parameters ( , ( )). For the case when thehistory state is output, set = 0. For the case where the the finalstate is output, take = / ( + 1) ( + 1) in the semistable case, and= / / ( + 1) ( + 1) for the stable case. Then, for both final andhistory state, = / in the stable case and = ( ) in thesemi-stable case.

[0097] In some example embodiments, varying amplifications at each discrete time-step enables broader idling as specified by the integers{ : = 0,1, … ,This allows us to increase the amplitude of selectedterms | ( ) in the history state (3), which can be understood asapplying a filter on the dynamical data.

[0098] (c) Required precision. To determine , the output processing is analyzed, in accordance with some example embodiments.

[0099] Linear System Matrix - Block Encoding : In some example embodiments, the running cost of QLSA in terms of calls to a unitaryPSIQ-558WO1 / 6224.009WO1block-encoding ( ˜ ) and a state preparation unitary ( ˜) isimplemented as follows:

[0100] For quantum linear solver algorithms it is typically assumed that the defining matrix has operator norm bounded above by 1 . As shown in the previous section, for a Taylor truncation order =,×, the following applies: + 1 + 2, and therefore thematrix is rescaled such that it is in a canonical form. Therefore the following is defined:

[0101] , and the rescaled problem ˜ = ˜ is solved, which has˜ 1, as normally required. Note that the coherent solution | tothis system coincides with the coherent solution of the originalsystem. Moreover, the condition number ˜ of the matrix ˜ equalssince the expression is invariant under rescaling.

[0102] In some example embodiments, for the linear-solveralgorithm, both a unitary ˜ that prepares the normalized state | ˜ aswell as ˜ , a block-encoding of ˜ is implemented. For the former it isnoted that | ˜ = | , since the rescaling is removed under thenormalization of the state. It is readily seen that˜can be constructed via a single call to and a single call to .

[0103] For ˜ , recall that an ( , , ) block-encoding of amatrix is defined as a unitary such that

[0104] Here, the state |0 is denoted by |0 , for any integer .Further, assume oracle access to via an ( , , 0) block encoding of .From this a block-encoding ˜ of the matrix ˜ is then constructed. Thefollowing result shows this can be readily done.

[0105] Definition 2. Let ˜ : = , where is defined in Eq. 6. Take 1. Given a Taylor truncation scale = ,×, construct an (PSIQ-558WO1 / 6224.009WO1˜ , + 6,0)-block-encoding ˜ of ˜ via a single call to an ( , , 0) block-encoding of the matrix , with

[0106] being the rescaling factor of the block-encoding for ˜ .

[0107] A proof analysis for the above: Construct a block-encoding for = + + , which in turn provides a block-encodingfor the rescaled matrix ˜ . Consider the problem separately for ,and and then combine them with linear combination of unitaries.

[0108] First, : this is just the identity matrix, hence the block-encoding is trivial: there is a (1,0,0) block-encoding of .

[0109] Second, :

[0110] , which is rewritten as = wherecan be (1, , 0)-block-encoded by , wherecan be (1,1,0)-block-encoded asPSIQ-558WO1 / 6224.009WO1

[0111] , where it is assumed that 1. This gives a ( 1,1,0 )- block-encoding of . Finally, can be blockencoded as = , where acts on the space where acts augmented by an extra qubit ancilla:

[0112] Using = |0 0| and = , one can check thatis unitary. Hence, a (1,1,0) block-encoding of isobtained.

[0113] Overall, set : = , where each one of thethree block-encoding unitaries has implicit identity operators on theauxiliary qubits of the other block-encodings. This gives an ( , +2,0 )block-encoding of involving a single query to .

[0114] Third, : Rewrite|0 = |+

[0115] and |+Here is a uniform statepreparation circuit (if is a power of 2 , this is just the tensor productof log Hadamards). Furthermore, the projector onto |0 0| can bedecomposed as

[0116] and so one can obtain a ( 1,1,0 )-block-encoding of|0 0 via Linear Combination of Unitaries (LCU). Finally,construct a ( 1,1,0 )-block-encoding of ˜ in the same way as thePSIQ-558WO1 / 6224.009WO1encoding for above was constructed. Overall, obtain a ( + 1, 2,0)-block-encoding of . Finally, consider

[0117] This can be realized by the LCUusing 2 extra qubits. Hence constructed a (1 + + 1 + , + 6,0)-block-encoding of using a single call to . This implies that

[0118] , as claimed. Therefore, all the ingredients needed for the linear-solver component are obtained. The optimal asymptotic query complexity of the quantum linear solver algorithm invokes the oracles ˜ and˜for a target error a number of times ˜˜ log (1 / ) = + / log (1 / )

[0119] Since these oracles are in turn constructed from singlecalls to , and the above expression also gives the querycomplexity to the defining oracles of the linear ODE problem.

[0120] As further shown in FIG. 5B, method 550 may include generating, on the classical processor, the probabilistic success query count by applying an success query scheme to the differential equation (operation 565). For example, device may perform sampling, on the classical processor, to form the probabilistic success query count. In some example embodiments, the scheduling plan includes an instruction to perform one or more iterations of amplitude amplification on the quantum information processing system until a likelihood of success if 100% or near 100%.PSIQ-558WO1 / 6224.009WO1

[0121] As also shown in FIG. 5B, method 550 may include generating, on the classical processor, the query count by multiplying the the quantum linear solver query count and the success indicator query count (operation 570). For example, device may generate, on the classical processor, the query count by multiplying the the quantum linear solver query count and the probabilistic success query count, as follows:

[0122] (3) OUTPUT PROCESSING: An analysis with assumedaccess to the an idealized linear system solution | is first performed;followed by consideration of the effect of finite precision.

[0123] Idealized analysis. The ideal solution of the linear system encodes ODE data in the sense that a state close torecovered by performing a projective measurement , , , onthe auxiliary / clock registers. Specifically, ifTr [ | | ] = Pr | |where the trace discards clock and ancilla qubits and2 .

[0125] The values Pr , are success probabilities of thecorresponding measurements. For outputting the history state the success probability in Eq. 4 is obtained, so the overhead to repeat the measurement till success or via to use amplitude amplification is (1). For outputting the final state the expressions given in Definition 1 are referenced, which implyPSIQ-558WO1 / 6224.009WO1 for stablesemistableUsing amplitude amplification there is an overhead ofrepetitions.

[0126] Error analysis. The effect of the finite precision of the QLSA is here considered, and generate an answer the question of what linear solver error suffices, as well what discretization error is to be targeted, given a total error budget .

[0127] In some example embodiments, implementing the protocol presented (including the output processing) returns aquantum state ˜ , that satisfies:

[0128] with a success probability at least Pr , . If the twoterms in Eq. 7 are chosen to have equal magnitude, then it suffices to take errors as given in Definition 1.

[0129] Although FIG. 5B shows example operations of method 550, in some implementations, method 550 may include additional operations, fewer operations, different operations, or differently arranged operations than those depicted in FIG. 5B. Additionally, or alternatively, two or more of the operations of method 550 may be performed in parallel.

[0130] Logical Qubits and Surface Codes: FIGs. 6A-6H show examples of qubits and error correction approaches (e.g., via logical qubits), in accordance with some example embodiments. One of ordinary skill in the art appreciates that qubits can correspond to different physical implementations for processing on corresponding different types of quantum information processing systems. Examples of different physical quantum information processing systems is discussed in further detail with reference to FIGs. 7A-7C below.

[0131] Qubits: Generally, quantum computing relies on the dynamics of quantum objects, e.g., photons, electrons, atoms, ions, molecules, nanostructures, and the like, which follow the rules ofPSIQ-558WO1 / 6224.009WO1 quantum theory. 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 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.

[0132] 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).

[0133] As used herein, a distinction is made between a “physical qubit” which is a physical quantum system such as a molecule, atom, photon, etc. that exists in a 2-level quantum state, and a “logical qubit” which includes a plurality of physical qubits encoded (e.g., entangled) together according to a quantum error correcting code (such as a surface code) to encode logical quantum information. These terms are described in greater detail below.

[0134] Physical implementations: Qubits (and operations on qubits) may be implemented using a variety of physical systems. InPSIQ-558WO1 / 6224.009WO1 some examples described herein, 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.

[0135] 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 vacuum mode) may correspond to the |0 stateof a photonic qubit. Alternatively, a state with a photon in the second waveguide and no photon in the first waveguide may correspond tothe |1 state of the photonic qubit. To prepare a photonic qubit in aknown 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 throughPSIQ-558WO1 / 6224.009WO1 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.

[0136] 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 withprobability 1 (e.g., a state | 10 in Fock notation), mode coupling mayresult in a state in which both modes have a nonzero probability ofbeing occupied, e.g., a statesome 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 a1and a2depend on the reflectivity (or transmissivity) of the beam splitters and on any phase shifts that are introduced.

[0137] A single physical qubit (e.g., such as the 2-level physicalqubit illustrated in FIG. 6A with a quantum state | = | 0 +| 1 ) may in principle be used for quantum computation. 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 such as those illustrated in FIG. 6B arePSIQ-558WO1 / 6224.009WO1 entangled together according to a specific error correcting code to produce a single logical qubit that is less susceptible to noise and decoherence. 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. Embodiments herein address these and other issues by implementing fault-tolerant post-selection (FTPS) of fault-tolerant codes and channels to improve the accuracy and fidelity of encoded logical qubits. At a high level, various quality metrics may be employed to assess the fidelity of an encoded logical qubit, or more generally of any type of encoded logical block, to determine whether to discard the logical block or keep it for use in the quantum computation.

[0138] In some embodiments, FTPS may utilize information metrics based on visible syndrome and erasure information. In some embodiments, different metrics may be employed for ranking the quality of logical blocks based on their respective configurations of syndromes and erasures. For example, some embodiments utilize a logical gap (and variants thereof) which determines an unsigned weight difference between inequivalent logical corrections as a metric for predicting logical error rates of logical blocks (also known as fault- tolerant channels) based on error-correcting codes. Advantagously, this metric is highly adaptable to various types of noise and decoders. In some embodiments, FTPS may be deployed to prepare low-error surface code magic states with low overheads under an i.i.d. model of Pauli error and erasure error rates. Post-selection strategies based on the logical gap may suppress the encoding error rate of a magic state preparation channel to that of the physical error rate with a relative overhead factor of 2, when operating at 60% of the bulk threshold of the corresponding surface code. The FTPS framework may also be utilized for mitigating errors in more general fault-tolerant logical channels.PSIQ-558WO1 / 6224.009WO1

[0139] In some quantum computing methodologies, such as fusion-based quantum computing and circuit-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, e.g., 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.

[0140] In some quantum computing implementations, the classical information takes the form of syndrome graph data, where the syndrome graph is a geometric representation of the outcomes of the measurement sequence. Because the input physical qubits are prepared in an initial state and measured according to a predetermined measurement sequence, it may be determined how the syndrome should appear in the absence of any errors involving the physical qubits during the measurement sequence (e.g., Pauli or erasure errors).

[0141] Accordingly, any deviation of the syndrome graph data from the expected result may be indicative of one or more errors within the logical qubit. In general, these deviations may not indicate precisely which measurement(s) had an error, or which type of error has occurred, as there may be more than one type of error or combination of errors that is consistent with a given observed deviation from the anticipated error-free syndrome graph. ForPSIQ-558WO1 / 6224.009WO1 example, a syndrome graph may be determined as a grid of parity checks for adjacent nodes of the grid, whereby a parity error may indicate that one or more of the adjacent nodes had an error, but the parity error may not indicate precisely which adjacent node had an error, or which error occurred.

[0142] As used herein, the term “syndrome graph data” refers to a set of classical information (e.g., data represented by digital values such as ones and zeros) that specifies the location of one or more syndromes and / or one or more erasure errors within the syndrome graph of a logical block. A series of measurements (e.g., stabilizer measurements) are applied to the physical qubits of the error correcting code containing the encoded logical information, producing measurement outcomes as classical information. As described in further detail below, based on the knowledge of the particular geometry of the cluster state / error correcting code, these measurement outcomes may be used to determine classical data referred to herein as the “syndrome graph data.” The syndrome graph data may further include correction operators for the syndrome graph output by a decoder.

[0143] Errors that occur during operations on an encoded logical qubit may have varying degrees of severity. For example, errors in a fault-tolerant logical qubit may cause logical failure if they link up in a way that spans the syndrome graph of the logical qubit. Conversely, localized errors that do not span the syndrome graph may be identifiable and correctable via quantum error correction. Embodiments herein perform FTPS by determining an error metric based on the syndrome graph data, and comparing the error metric to a quality threshold to determine whether to keep or discard a logical qubit. In some embodiments, multiplexing may be employed where multiple copies of each logical qubit are produced and the higher fidelity logical qubits are kept and used in a quantum computation,PSIQ-558WO1 / 6224.009WO1 whereas the lower fidelity logical qubits are discarded, increasing the fidelity of the computation.

[0144] FIG. 6B shows an arrangement of physical qubits that can be used to encode a fault-tolerant logical qubit using a surface code according to one or more embodiments. In FIG. 6B, the solid grid lines are guides to the eye and form an array of squares, also referred to herein as a “surface code”, with physical “data qubits” disposed on the four vertices of each square and physical “measure qubits” disposed on the face of each square. As used herein, measure qubits are the physical qubits which are measured to perform measurement checks on adjacent data qubits without directly measuring the data qubits and collapsing the quantum information. In this example the surface code has a length (or more precisely, a code distance) d of 12, but any length can be employed. The surface code arrangement of qubits also includes four lines of boundary measure qubits disposed adjacent to the outermost lines of data qubits. Each square is referred to herein as a plaquette. Within the bulk of the surface code (e.g., the plaquettes which don’t form the outer boundary of the code) each data qubit may be coupled, via 4 two-qubit gates, to its 4 nearest neighbor measure qubits (each on four different plaquettes) and likewise, each measure qubit may be coupled, via 4 two-qubit gates to its 4 nearest neighbor data qubits. On the boundaries of the code, each boundary measure qubit may be coupled, via two two-qubit gates to its nearest adjacent data qubits. According to one or more embodiments, the two- qubit gates can be CNOT gates, CZ gates, and the like.

[0145] In order to operate the collection of data and measure qubits as a logical qubit that is protected against errors, the following set of measurements may be repetitively performed on the system. For each plaquette within the bulk of the surface code, 4-qubit stabilizers are measured. For example, as shown in FIG. 6D, if the data qubits of a given plaquette are labeled 1, 2, 3 ,4 (e.g., data qubits 605-611) and the measure qubit is labeled a (e.g., measure qubit 603),PSIQ-558WO1 / 6224.009WO1 the stabilizer to be measured can be X1Z2Z3X4. The “quantum circuit” (which is a term that refers to the sequence of gates and measurement operations to be performed on physical qubits) used to implement this stabilizer measurement is also shown in FIG. 6D and includes first initializing the measure qubit a in the |+> state, then performing the following gates: a CNOT gate between the measure qubit a and data qubit 1, respective CZ gates between the measure qubit a and qubit 2 and qubit 3, and a CNOT gate between the measure qubit a and qubit 4; followed by an x-basis measurement Mx of measure qubit a. The resulting measurement outcome (which takes the form of a classical bit, e.g., 0 or 1 or -1 or 1, depending on the choice of conventions) is equal to the outcome of the measurement of the parity check stabilizer X1Z2Z3X4 and becomes part of the syndrome graph. For the plaquettes found at the boundary of the surface code, and shown in FIG. 6E, a two-qubit stabilizer of the form Z1X2 is measured. The quantum circuit used to implement this two- qubit stabilizer measurement is also shown in FIG. 6E and includes first initializing the boundary measure qubit a in the |+> state then performing the following gates: a CZ gate between the measure qubit a and qubit 1 and CNOT gate between the measure qubit a and qubit 2; followed by an x-basis measurement Mx of measure qubit a.

[0146] In the example shown in FIG. 6C, there are two different types of boundaries depending on whether the boundary includes shaded plaquettes or unshaded plaquettes. A boundary surface that includes shaded plaquettes is referred to as a “dual boundary surface” and measurements including measure qubits within the dual boundary surface contribute to the “dual syndrome graph.” Similarly, a boundary surface that includes unshaded plaquettes is referred to herein as a “primal boundary surface,” and measurements including measure qubits within a primal boundary surface contribute to the “primal syndrome graph.”PSIQ-558WO1 / 6224.009WO1

[0147] In order to implement the surface code scheme shown in FIGs. 6C-6E, the plaquette measurements may be broken into two groups of measurements: a first group of measurements that measures the stabilizers associated with the shaded plaquettes during a first duration of time and a second group of measurements that measures the stabilizers with the unshaded plaquettes during a second duration of time. These two sets of measurements are performed in different times to ensure that each qubit only participates in one quantum gate at a time. One of ordinary skill in the art will appreciate that any gates that can commute with one another may be performed in the same time step, or even simultaneously, if desired. The classical data generated by each one of these measurements, referred to herein as “syndrome graph data”, is then passed to a decoder for quantum error correction according to known methods, e.g., using union find decoding, minimum weight perfect matching or any other decoding process.

[0148] One of ordinary skill will appreciate that the example shown in FIG. 6C is using a particular choice of local basis for the surface code and that other choices for the basis may be employed. For example, in some contexts, taking certain assumptions on the likely form of the errors that may occur on the underlying data and measure qubits, one may apply a single qubit gate to each data qubit to obtain a modified surface code. One may modify the basis for each check to obtain a scheme for the modified code. One example includes the CSS (Calderbank, Shor, Steane) version, where stabilizer measurements are either x-type or z-type. To obtain this version ofthe surface code, the stabilizers are conjugated by a Hadamard H:Z, Z X on half the data qubits in a bipartition, thereby resulting inthe CSS surface code. Note that the measurement schedule described above remains the same, but the new stabilizers are given by that summarized in FIG. 6F.PSIQ-558WO1 / 6224.009WO1

[0149] 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.

[0150] FIG. 6G illustrates a 3-dimensional graphical depiction of such a fault-tolerant logical identity gate. The surface labeled 614 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 618 identifies the output qubits after the identity gate I has been applied to it. The input and output surfaces, which may be associated with the physical 2D arrangement of data and measure qubits described above, are connected to each other via an intervening volume that represents the unique set of measurements to be applied over time as described above in reference to FIGs. 6C-6E. Accordingly, in FIG. 6G, 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 as described above in reference to FIGs. 6C-6E. FIG. 6H represents the same concept but written in a more familiar quantum circuit notation illustrating the analogy between the more familiar quantum circuit. While FIG. 6G 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.PSIQ-558WO1 / 6224.009WO1

[0151] The sequence of measurements performed over the flow of time illustrated in FIG. 6G (e.g., a sequence of measurements including the circuit measurements shown in FIGs. 6C-6E) may include a subset of measurements that incur a logical error (e.g., a Pauli error) or an erasure error. To identify errors in the measurement outcomes, syndrome graph data may be generated from the collection of measurement outcomes resulting from the measurements of the physical qubits. For example, the bit values associated with a plurality of edge qubits may be combined to create a syndrome value associated with an adjacent vertex that results from the intersection of the respective edges, e.g., the result of the measurements shown in FIGs. 6D and 6E. A set of syndrome values (or “syndromes”), also referred to herein as parity checks, may be associated with each vertex of the syndrome graph. The parity check values may be found by computing the parity of the bit values associated with each edge of the syndrome graph incident to the vertex. In some embodiments, a parity computation entails determining whether the sum of the edge values is an even or odd integer, with the parity result being the result of the sum modulo 2. If no errors have occurred in the quantum state or in the qubit measurements, then all syndrome values should be even (or 0). On the contrary, if an error occurs, it may result in some odd (or 1) syndrome values.

[0152] In some embodiments, half of the bit values from the qubit measurements are associated with the primal boundary surfaces, and this syndrome graph is referred to herein as the “primal graph”. The syndrome graph resulting from measurements on the dual boundary surfaces is referred to as the “dual graph”. There is generally an equivalent decoding problem on the syndrome values of the primal and dual graphs.

[0153] Syndromes may be identified and appropriately removed via quantum error correction, via a process known as decoding.PSIQ-558WO1 / 6224.009WO1 Decoding produces a recovery that is consistent with the syndrome, attempting to correct for the error. Decoding succeeds when the combined effect of the error and recovery does not give rise to a logical error. However, this process does not always succeed, and certain combinations of error and recovery may result in an error chain that spans the surface code and damages the logical information. In some example embodiments, the logical gap magnitudes for correcting syndromes may be extracted from syndrome graph data to determine an error metric associated with a logical encoded qubit (e.g., for the output surface code shown in FIG. 6G). This error metric may then be used for fault-tolerant post-selection to determine whether to keep or discard the logical qubit, in some embodiments.

[0154] FIGs. 7A-7C show example quantum information processing systems (e.g., quantum information processing system 205, FIG. 2), according to some example embodiments. The quantum information processing systems illustrated in FIGs. 7A-7C can implement different physical embodiments that correspond to quantum computing (e.g., photonic qubits, ion-based qubits, charge qubits, flux qubits). Each of the example quantum information processing systems can implement different physical components to prepare different quantum states and implement unitary evolution of the qubit systems (e.g., collectively a quantum state) to perform different quantum circuits, such as a quantum circuit comprising one or more differential equations.

[0155] FIG. 7A shows a photonic quantum computing system 700, in accordance with some example embodiments. In the example illustrated in FIG. 7A, qubits are implemented as photonic qubits 710, which can be based on degrees of freedom of photonic systems, such as photon location (e.g., which way systems) and photon polarization. The photonic quantum computing system 700 further includes a photonic controller system 702 to prepare and control the photonic qubits 710. For example, the photonic controller 702 can include aPSIQ-558WO1 / 6224.009WO1 photonic state system 702 to prepare the photonic qubits (e.g., weak coherent laser, single photon sources, quantum dots), and further include photonic controls 707 to perform unitary evolution of the quantum state corresponding to the photonic qubits 710. The photonic controls 707 can include optical components to process the photonic qubits 707, such as phase shifters, polarizers, and beam splitters to impart photonic qubit operations (e.g., gates, unitary operations). The optical measurement system 707 further comprises an optical measurement system 707 to perform measurements on photons of the photonic qubits (e.g., 710). In some example embodiments, the measurements performed by the system 707 can include readout measurements to generate output data or measurements as gates (e.g., projective measurements, fusions) to further process the photonic qubits 710.

[0156] FIG. 7B shows an ion trap-based quantum computing system 720, in accordance with some example embodiments. In the example illustrated in FIG. 7B, qubits are implemented as ion-based qubits 729, which can be implemented based on degrees of freedom of ionic systems, such as hyperfine states. The ion trap-based quantum computing system 720 further includes a ion controller system 722 to prepare and control the ion-based qubits 730. For example, the ion controller system 722 can include a ion state system 724 to prepare the ion-based qubits (e.g., quadrupole trap, optical pumping system, cooling system), and further include ion controls 726 to perform unitary evolution of the quantum state corresponding to the ion-based qubits 729. The ion controller 707 can include control components, such as a laser pulse system, to process the ion based 730 to impart qubit operations (e.g., gates, unitary operations) according to a given quantum task (e.g., qubit gate, circuit comprising a plurality of gates in an order). The ion trap-based quantum computing system 720 further comprises an ion measurement system 727 to perform measurements on the ion qubits, such as a photomultiplier tube,PSIQ-558WO1 / 6224.009WO1 avalanche photodiode or CCD based imaging device to generate readout data or perform measurements for qubit processing.

[0157] FIG. 7C shows a current-based quantum computing system 730, in accordance with some example embodiments. In the example illustrated in FIG. 7C, qubits are implemented as current qubits 740 (charge-based, flux-based, transmons), which can be implemented based on degrees of freedom of charge or current state (e.g., in superconducting circuits). The system 730 further includes a current controller system 732 to prepare and control the current- based qubits 730, and further includes current controls 736 to perform unitary evolutions of the quantum state corresponding to the current- based qubits 740. The current controller 736 can include control components, such as flux couplers, microwave photons, to process the current-based qubits 740 to impart qubit operations (e.g., gates, unitary operations) according to a given quantum task (e.g., qubit gate, circuit comprising a plurality of gates in an order). The system 730 further comprises an current measurement system 737 to perform measurements on the current-based qubits 740, such a readout cavity (e.g., readout resonator) to generate readout data or perform measurements for qubit processing.

[0158] In some example embodiments, given a target quantum state for preparation (e.g., such as a Unitary, O1, O2, O3), any of the systems of FIGs. 7A-7C can implement respective qubits via gates (e.g., gates in circuit based quantum computing) and / or measurements (in measurement based approaches, such as Measurement Based Quantum Computing MBQC, and Fusion Based Quantum Computing) to implement the target quantum state.

[0159] FIG. 8 illustrates a diagrammatic representation of a machine 800 in the form of a computer system (e.g., classical information processing system 203, FIG. 2) within which a set of instructions may be executed for causing the machine 800 to perform any one or more of the methodologies discussed herein, according toPSIQ-558WO1 / 6224.009WO1 an example embodiment. Specifically, FIG. 8 shows a diagrammatic representation of the machine 800 in the example form of a computer system, within which instructions 816 (e.g., software, a program, an application, an applet, an app, or other executable code) for causing the machine 800 to perform any one or more of the methodologies discussed herein may be executed. For example, the instructions 816 may cause the machine 800 to execute any one or more operations of method 1000. As another example, the instructions 816 may cause the machine 800 to implement portions of the data flows illustrated in at least FIG. 10. In this way, the instructions 816 transform a general, non-programmed machine into a particular machine 800 that is specially configured to carry out any one of the described and illustrated functions in the manner described herein.

[0160] In alternative embodiments, the machine 800 operates as a standalone device or may be coupled (e.g., networked) to other machines. In a networked deployment, the machine 800 may operate in the capacity of a server machine or a client machine in a server- client network environment, or as a peer machine in a peer-to-peer (or distributed) network environment. The machine 800 may comprise, but not be limited to, a server computer, a client computer, a personal computer (PC), a tablet computer, a laptop computer, a netbook, a smart phone, a mobile device, a network router, a network switch, a network bridge, or any machine capable of executing the instructions 816, sequentially or otherwise, that specify actions to be taken by the machine 800. Further, while only a single machine 800 is illustrated, the term “machine” shall also be taken to include a collection of machines 800 that individually or jointly execute the instructions 816 to perform any one or more of the methodologies discussed herein.

[0161] The machine 800 includes processors 810, memory 830, and input / output (I / O) components 850 configured to communicate with each other such as via a bus 802. In an example embodiment, the processors 810 (e.g., a central processing unit (CPU), a reducedPSIQ-558WO1 / 6224.009WO1 instruction set computing (RISC) processor, a complex instruction set computing (CISC) processor, a graphics processing unit (GPU), a digital signal processor (DSP), an application-specific integrated circuit (ASIC), a radio-frequency integrated circuit (RFIC), another processor, or any suitable combination thereof) may include, for example, a processor 812 and a processor 814 that may execute the instructions 816. The term “processor” is intended to include multi- core processors 810 that may comprise two or more independent processors (sometimes referred to as “cores”) that may execute instructions 816 contemporaneously. Although FIG. 8 shows multiple processors 810, the machine 800 may include a single processor with a single core, a single processor with multiple cores (e.g., a multi-core processor), multiple processors with a single core, multiple processors with multiple cores, or any combination thereof.

[0162] The memory 830 may include a main memory 832, a static memory 834, and a storage unit 836, all accessible to the processors 810 such as via the bus 802. The main memory 832, the static memory 834, and the storage unit 836 store the instructions 816 embodying any one or more of the methodologies or functions described herein. The instructions 816 may also reside, completely or partially, within the main memory 832, within the static memory 834, within machine storage medium 838 of the storage unit 836, within at least one of the processors 810 (e.g., within the processor’s cache memory), or any suitable combination thereof, during execution thereof by the machine 800.

[0163] The I / O components 850 include components to receive input, provide output, produce output, transmit information, exchange information, capture measurements, and so on. The specific I / O components 850 that are included in a particular machine 800 will depend on the type of machine. For example, portable machines such as mobile phones will likely include a touch input device or other such input mechanisms, while a headless server machine will likely notPSIQ-558WO1 / 6224.009WO1 include such a touch input device. It will be appreciated that the I / O components 850 may include many other components that are not shown in FIG. 8. The I / O components 850 are grouped according to functionality merely for simplifying the following discussion and the grouping is in no way limiting. In various example embodiments, the I / O components 850 may include output components 852 and input components 854. The output components 852 may include visual components (e.g., a display such as a plasma display panel (PDP), a light emitting diode (LED) display, a liquid crystal display (LCD), a projector, or a cathode ray tube (CRT)), acoustic components (e.g., speakers), other signal generators, and so forth. The input components 854 may include alphanumeric input components (e.g., a keyboard, a touch screen configured to receive alphanumeric input, a photo-optical keyboard, or other alphanumeric input components), point-based input components (e.g., a mouse, a touchpad, a trackball, a joystick, a motion sensor, or another pointing instrument), tactile input components (e.g., a physical button, a touch screen that provides location and / or force of touches or touch gestures, or other tactile input components), audio input components (e.g., a microphone), and the like.

[0164] Communication may be implemented using a wide variety of technologies. The I / O components 850 may include communication components 864 operable to couple the machine 800 to a network 880 or devices 870 via a coupling 882 and a coupling 872, respectively. For example, the communication components 864 may include a network interface component or another suitable device to interface with the network 880. In further examples, the communication components 864 may include wired communication components, wireless communication components, cellular communication components, and other communication components to provide communication via other modalities. The devices 870 may be another machine or any of a wide variety of peripheral devices (e.g., aPSIQ-558WO1 / 6224.009WO1 peripheral device coupled via a universal serial bus (USB)). For example, as noted above, the machine 800 may correspond to any one of the compute service manager 108 or the execution platform 80, and the devices 870 may include the client device 84 or any other computing device described herein as being in communication with the network-based database system 102 or the cloud storage platform 104.

[0165] The various memories (e.g., 830, 832, 834, and / or memory of the processor(s) 810 and / or the storage unit 836) may store one or more sets of instructions 816 and data structures (e.g., software) embodying or utilized by any one or more of the methodologies or functions described herein. These instructions 816, when executed by the processor(s) 810, cause various operations to implement the disclosed embodiments.

[0166] As used herein, the terms “machine-storage medium,” “device-storage medium,” and “computer-storage medium” mean the same thing and may be used interchangeably in this disclosure. The terms refer to a single or multiple tangible storage devices and / or tangible media (e.g., a centralized or distributed database, and / or associated caches and servers) that store executable instructions and / or data. The terms shall accordingly be taken to include, but not be limited to, solid-state memories, and optical and magnetic media, including memory internal or external to processors. Specific examples of machine-storage media, computer-storage media, and / or device-storage media include non-volatile memory, including by way of example semiconductor memory devices, e.g., erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), field-programmable gate arrays (FPGAs), and flash memory devices; magnetic disks such as internal hard disks and removable disks; magneto-optical disks; and CD-ROM and DVD-ROM disks. The terms “machine-storage media,” “computer-storage media,” and “device-storage media”PSIQ-558WO1 / 6224.009WO1 specifically exclude carrier waves, modulated data signals, and other such media, at least some of which are covered under the term “signal medium” discussed below.

[0167] In various example embodiments, one or more portions of the network 880 may be an ad hoc network, an intranet, an extranet, a virtual private network (VPN), a local-area network (LAN), a wireless LAN (WLAN), a wide-area network (WAN), a wireless WAN (WWAN), a metropolitan-area network (MAN), the Internet, a portion of the Internet, a portion of the public switched telephone network (PSTN), a plain old telephone service (POTS) network, a cellular telephone network, a wireless network, a Wi-Fi® network, another type of network, or a combination of two or more such networks. For example, the network 880 or a portion of the network 880 may include a wireless or cellular network, and the coupling 882 may be a Code Division Multiple Access (CDMA) connection, a Global System for Mobile communications (GSM) connection, or another type of cellular or wireless coupling. In this example, the coupling 882 may implement any of a variety of types of data transfer technology, such as Single Carrier Radio Transmission Technology (1xRTT), Evolution- Data Optimized (EVDO) technology, General Packet Radio Service (GPRS) technology, Enhanced Data rates for GSM Evolution (EDGE) technology, third Generation Partnership Project (3GPP) including 3G, fourth generation wireless (4G) networks, Universal Mobile Telecommunications System (UMTS), High-Speed Packet Access (HSPA), Worldwide Interoperability for Microwave Access (WiMAX), Long Term Evolution (LTE) standard, others defined by various standard-setting organizations, other long-range protocols, or other data transfer technology.

[0168] The instructions 816 may be transmitted or received over the network 880 using a transmission medium via a network interface device (e.g., a network interface component included in the communication components 864) and utilizing any one of a number ofPSIQ-558WO1 / 6224.009WO1 well-known transfer protocols (e.g., hypertext transfer protocol (HTTP)). Similarly, the instructions 816 may be transmitted or received using a transmission medium via the coupling 872 (e.g., a peer-to-peer coupling) to the devices 870. The terms “transmission medium” and “signal medium” mean the same thing and may be used interchangeably in this disclosure. The terms “transmission medium” and “signal medium” shall be taken to include any intangible medium that is capable of storing, encoding, or carrying the instructions 816 for execution by the machine 800, and include digital or analog communications signals or other intangible media to facilitate communication of such software. Hence, the terms “transmission medium” and “signal medium” shall be taken to include any form of modulated data signal, carrier wave, and so forth. The term “modulated data signal” means a signal that has one or more of its characteristics set or changed in such a manner as to encode information in the signal.

[0169] Described implementations of the subject matter can include one or more features, alone or in combination as illustrated below by way of example. The following are example embodiments.

[0170] Example 1 is a method comprising: identifying, on a classical processor, a computational task for processing on a quantum information processing system, the computational task comprising a differential equation; generating, on the classical processor, a scheduling plan for generation of a quantum state that corresponds to the differential equation on the quantum information processing system, the scheduling plan comprising a query count specifying a quantity of queries to be performed against a block-encoding of the differential equation to generate the quantum state; and transmitting the scheduling plan to the quantum information processing system for processing of the computational task according to the quantity of queries to the block-encoding of the differential equation.PSIQ-558WO1 / 6224.009WO1

[0171] In Example 2, the subject matter of Example 1 includes, generating, on the classical processor, block-encoding data for the block-encoding of the differential equation; and generating, on the classical processor, the query count by applying a quantum linear solver query scheme to the differential equation.

[0172] In Example 3, the subject matter of Examples 1–2 includes, wherein the block-encoding comprises one or more quantum oracles.

[0173] In Example 4, the subject matter of Examples 1–3 includes, wherein the quantity of queries comprises a first quantity of queries to a first quantum oracle and a second quantity of queries to a second quantum oracle, and wherein the second quantity of queries is larger than the first quantity of queries.

[0174] In Example 5, the subject matter of Examples 1–4 includes, wherein the second quantity of queries is at least four times larger than the first quantity of queries.

[0175] In Example 6, the subject matter of Examples 1–5 includes, wherein the quantum state comprises a solution state.

[0176] In Example 7, the subject matter of Example 6 includes, wherein the quantum information processing system executes the scheduling plan and performs the quantity of queries on the block- encoding of the differential equation to generate a quantum encoding of the solution state.

[0177] In Example 8, the subject matter of Examples 1–7 includes, wherein the quantum state comprises a history state.

[0178] In Example 9, the subject matter of Example 8 includes, wherein the quantum information processing system executes the scheduling plan and performs the quantity of queries on the block- encoding of the differential equation to generate a quantum encoding of the history state.

[0179] In Example 10, the subject matter of Examples 1–9 includes, wherein the differential equation is an ordinary differentialPSIQ-558WO1 / 6224.009WO1 equation comprising a N x N time-independent matrix and a N- dimensional vector, and wherein generating the block-encoding of the differential equation comprises discretizing time into a linear system that comprises matrices proportional to the N x N time-independent matrix or an Identity matrix.

[0180] In Example 11, the subject matter of Example 10 includes, wherein applying the quantum linear solver query scheme to the differential equation comprises generating a quantum state representation using a first quantity of calls to a first set of unitary matrices that correspond to the linear system, and a second quantity of calls to a second set of unitary matrices that correspond to a prepared state unitary.

[0181] In Example 12, the subject matter of Example 11 includes, wherein the prepared state unitary is generated from calls to an initial state unitary and a normalization unitary that encodes a normalization of the N-dimensional vector of the differential equation.

[0182] Example 13 is a system comprising: a classical processor configured to: identify a computational task for processing on a quantum information processing system, the computational task comprising a differential equation; generate a scheduling plan for generation of a quantum state that corresponds to the differential equation on the quantum information processing system, the scheduling plan comprising a query count specifying a quantity of queries to be performed against a block-encoding of the differential equation to generate the quantum state; and transmit the scheduling plan to the quantum information processing system for processing of the computational task according to the quantity of queries to the block-encoding of the differential equation.

[0183] In Example 14, the subject matter of Example 13 includes, wherein the classical processor is further configured to: generate block-encoding data for the block-encoding of the differentialPSIQ-558WO1 / 6224.009WO1 equation; and generate the query count by applying a quantum linear solver query scheme to the differential equation.

[0184] In Example 15, the subject matter of Examples 13–14 includes, wherein the block-encoding comprises one or more quantum oracles, and wherein the quantity of queries comprises a first quantity of queries to a first quantum oracle and a second quantity of queries to a second quantum oracle, wherein the second quantity of queries is larger than the first quantity of queries.

[0185] In Example 16, the subject matter of Examples 13–15 includes, wherein the quantum state comprises one of a solution state and a history state, and wherein the quantum information processing system executes scheduling plan and performs the quantity of queries to generate a quantum encoding of the one of the solution state and the history state.

[0186] In Example 17, the subject matter of Examples 1–16 includes, wherein the differential equation is an ordinary differential equation comprising a N x N time-independent matrix and a N- dimensional vector, wherein generating the block-encoding of the differential equation comprises discretizing time into a linear system that comprises matrices proportional to the N x N time-independent matrix or an Identity matrix, wherein applying the quantum linear solver query scheme to the differential equation comprises generating a quantum state representation using a first quantity of calls to a first set of unitary matrices that correspond to the linear system, and a second quantity of calls to a second set of unitary matrices that correspond to a prepared state unitary, and wherein the prepared state unitary is generated from calls to an initial state unitary and a normalization unitary that encodes a normalization of the N- dimensional vector of the differential equation.

[0187] Example 18 is a tangible computer-readable medium storing a set of instructions, the set of instructions comprising: identifying, on a classical processor, a computational task forPSIQ-558WO1 / 6224.009WO1 processing on a quantum information processing system, the computational task comprising a differential equation; generating, on the classical processor, a scheduling plan for generation of a quantum state that corresponds to the differential equation on the quantum information processing system, the scheduling plan comprising a query count specifying a quantity of queries to be performed against a block-encoding of the differential equation to generate the quantum state; and transmitting the scheduling plan to the quantum information processing system for processing of the computational task according to the quantity of queries to the block-encoding of the differential equation.

[0188] In Example 19, the subject matter of Example 18 includes, the set of instructions further comprising: generating, on the classical processor, block-encoding data for the block-encoding of the differential equation; and generating, on the classical processor, the query count by applying a quantum linear solver query scheme to the differential equation.

[0189] In Example 20, the subject matter of Examples 18–19 includes, wherein the block-encoding comprises one or more quantum oracles.

[0190] Example 21 is at least one machine-readable medium including instructions that, when executed by processing circuitry, cause the processing circuitry to perform operations to implement of any of Examples 1–20.

[0191] Example 22 is an apparatus comprising means to implement of any of Examples 1–20.

[0192] Example 23 is a system to implement of any of Examples 1–20.The terms “machine-readable medium,” “computer-readable medium,” and “device-readable medium” mean the same thing and may be used interchangeably in this disclosure. The terms are defined to include both machine-storage media and transmission media.PSIQ-558WO1 / 6224.009WO1 Thus, the terms include both storage devices / media and carrier waves / modulated data signals.

[0193] The various operations of example methods described herein may be performed, at least partially, by one or more processors that are temporarily configured (e.g., by software) or permanently configured to perform the relevant operations. Similarly, the methods described herein may be at least partially processor-implemented. For example, at least some of the operations of the methods 500 and 550 of FIG. 5A and 5B may be performed by one or more processors. The performance of certain of the operations may be distributed among the one or more processors, not only residing within a single machine, but also deployed across a number of machines. In some example embodiments, the processor or processors may be located in a single location (e.g., within a home environment, an office environment, or a server farm), while in other embodiments the processors may be distributed across a number of locations.

[0194] Although the embodiments of the present disclosure have been described with reference to specific example embodiments, it will be evident that various modifications and changes may be made to these embodiments without departing from the broader scope of the inventive subject matter. Accordingly, the specification and drawings are to be regarded in an illustrative rather than a restrictive sense. The accompanying drawings that form a part hereof show, by way of illustration, and not of limitation, specific embodiments in which the subject matter may be practiced. The embodiments illustrated are described in sufficient detail to enable those skilled in the art to practice the teachings disclosed herein. Other embodiments may be used and derived therefrom, such that structural and logical substitutions and changes may be made without departing from the scope of this disclosure. This Detailed Description, therefore, is not to be taken in a limiting sense, and the scope of various embodiments is defined only by the appended claims,PSIQ-558WO1 / 6224.009WO1 along with the full range of equivalents to which such claims are entitled.

[0195] Such embodiments of the inventive subject matter may be referred to herein, individually and / or collectively, by the term “invention” merely for convenience and without intending to voluntarily limit the scope of this application to any single invention or inventive concept if more than one is in fact disclosed. Thus, although specific embodiments have been illustrated and described herein, it should be appreciated that any arrangement calculated to achieve the same purpose may be substituted for the specific embodiments shown. This disclosure is intended to cover any and all adaptations or variations of various embodiments. Combinations of the above embodiments, and other embodiments not specifically described herein, will be apparent, to those of skill in the art, upon reviewing the above description.

[0196] In this document, the terms “a” or “an” are used, as is common in patent documents, to include one or more than one, independent of any other instances or usages of “at least one” or “one or more.” In this document, the term “or” is used to refer to a nonexclusive or, such that “A or B” includes “A but not B,” “B but not A,” and “A and B,” unless otherwise indicated. In the appended claims, the terms “including” and “in which” are used as the plain- English equivalents of the respective terms “comprising” and “wherein.” Also, in the following claims, the terms “including” and “comprising” are open-ended; that is, a system, device, article, or process that includes elements in addition to those listed after such a term in a claim is still deemed to fall within the scope of that claim.

Claims

PSIQ-558WO1 / 6224.009WO1 Claims What is claimed is:

1. A method comprising: identifying, on a classical processor, a computational task for processing on a quantum information processing system, the computational task comprising a differential equation; generating, on the classical processor, a scheduling plan for generation of a quantum state that corresponds to the differential equation on the quantum information processing system, the scheduling plan comprising a query count specifying a quantity of queries to be performed against a block-encoding of the differential equation to generate the quantum state; and transmitting the scheduling plan to the quantum information processing system for processing of the computational task according to the quantity of queries to the block-encoding of the differential equation.

2. The method of claim 1, further comprising: generating, on the classical processor, block-encoding data for the block-encoding of the differential equation; and generating, on the classical processor, the query count by applying a quantum linear solver query scheme to the differential equation.

3. The method of claim 1, wherein the block-encoding comprises one or more quantum oracles.PSIQ-558WO1 / 6224.009WO1 4. The method of claim 1, wherein the quantity of queries comprises a first quantity of queries to a first quantum oracle and a second quantity of queries to a second quantum oracle, and wherein the second quantity of queries is larger than the first quantity of queries.

5. The method of claim 1, wherein the second quantity of queries is at least four times larger than the first quantity of queries.

6. The method of claim 1, wherein the quantum state comprises a solution state.

7. The method of claim 6, wherein the quantum information processing system executes the scheduling plan and performs the quantity of queries on the block-encoding of the differential equation to generate a quantum encoding of the solution state.

8. The method of claim 1, wherein the quantum state comprises a history state.

9. The method of claim 8, wherein the quantum information processing system executes the scheduling plan and performs the quantity of queries on the block-encoding of the differential equation to generate a quantum encoding of the history state.

10. The method of claim 1, wherein the differential equation is an ordinary differential equation comprising a N x N time-independent matrix and a N-dimensional vector, and wherein generating the block-encoding of the differential equation comprises discretizing time into a linear system that comprises matrices proportional to the N x N time-independent matrix or an Identity matrix.PSIQ-558WO1 / 6224.009WO1 11. The method of claim 10, wherein applying the quantum linear solver query scheme to the differential equation comprises generating a quantum state representation using a first quantity of calls to a first set of unitary matrices that correspond to the linear system, and a second quantity of calls to a second set of unitary matrices that correspond to a prepared state unitary.

12. The method of claim 11, wherein the prepared state unitary is generated from calls to an initial state unitary and a normalization unitary that encodes a normalization of the N-dimensional vector of the differential equation.

13. A system comprising: a classical processor configured to: identify a computational task for processing on a quantum information processing system, the computational task comprising a differential equation; generate a scheduling plan for generation of a quantum state that corresponds to the differential equation on the quantum information processing system, the scheduling plan comprising a query count specifying a quantity of queries to be performed against a block-encoding of the differential equation to generate the quantum state; and transmit the scheduling plan to the quantum information processing system for processing of the computational task according to the quantity of queries to the block-encoding of the differential equation.

14. The system of claim 13, wherein the classical processor is further configured to: generate block-encoding data for the block-encoding of the differential equation; andPSIQ-558WO1 / 6224.009WO1 generate the query count by applying a quantum linear solver query scheme to the differential equation.

15. The system of claim 13, wherein the block-encoding comprises one or more quantum oracles, and wherein the quantity of queries comprises a first quantity of queries to a first quantum oracle and a second quantity of queries to a second quantum oracle, wherein the second quantity of queries is larger than the first quantity of queries.

16. The system of claim 13, wherein the quantum state comprises one of a solution state and a history state, and wherein the quantum information processing system executes scheduling plan and performs the quantity of queries to generate a quantum encoding of the one of the solution state and the history state.

17. The method of claim 1, wherein the differential equation is an ordinary differential equation comprising a N x N time-independent matrix and a N-dimensional vector, wherein generating the block- encoding of the differential equation comprises discretizing time into a linear system that comprises matrices proportional to the N x N time-independent matrix or an Identity matrix, wherein applying the quantum linear solver query scheme to the differential equation comprises generating a quantum state representation using a first quantity of calls to a first set of unitary matrices that correspond to the linear system, and a second quantity of calls to a second set of unitary matrices that correspond to a prepared state unitary, and wherein the prepared state unitary is generated from calls to an initial state unitary and a normalization unitary that encodes a normalization of the N-dimensional vector of the differential equation.PSIQ-558WO1 / 6224.009WO1 18. A tangible computer-readable medium storing a set of instructions, the set of instructions comprising: identifying, on a classical processor, a computational task for processing on a quantum information processing system, the computational task comprising a differential equation; generating, on the classical processor, a scheduling plan for generation of a quantum state that corresponds to the differential equation on the quantum information processing system, the scheduling plan comprising a query count specifying a quantity of queries to be performed against a block-encoding of the differential equation to generate the quantum state; and transmitting the scheduling plan to the quantum information processing system for processing of the computational task according to the quantity of queries to the block-encoding of the differential equation.

19. The tangible computer-readable medium of claim 18, the set of instructions further comprising: generating, on the classical processor, block-encoding data for the block-encoding of the differential equation; and generating, on the classical processor, the query count by applying a quantum linear solver query scheme to the differential equation.

20. The tangible computer-readable medium of claim 18, wherein the block-encoding comprises one or more quantum oracles.

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