Method and system for quantum probability estimation of an event

Abstract

A method of quantum probability estimation of an event in a stochastic process is provided. The stochastic process is modelled based on a finite state-machine model. A classical predictive model for the event is modelled based on the stochastic process and includes a set of states and a set of state transition matrices associated with the finite state-machine model. The method includes: generating a circuit description of a quantum state-preparation operator based on a past observation sequence of the stochastic process and the set of state transition matrices of the classical predictive model; generating a circuit description of a quantum search operator based on the circuit description of the quantum state-preparation operator and an event indication function, whereby the event indication function is configured to indicate whether the event is to occur with respect to the future sequence of the stochastic process; generating a plurality of circuit descriptions for a plurality of measurement quantum circuits, respectively, for compiling the plurality of measurement quantum circuits on a quantum computer, whereby each circuit description of the measurement quantum circuit is generated based on the circuit description of the quantum state-preparation operator and the circuit description of the quantum search operator; for each of the plurality of circuit descriptions of the plurality of measurement quantum circuits: compiling the circuit description of the measurement quantum circuit on the quantum computer to form the measurement quantum circuit thereon; and executing the measurement quantum circuit formed on the quantum computer to obtain a measurement result associated with the measurement quantum circuit that indicates a number of times that the measurement quantum circuit produces an indication that the event is to occur; and estimating a probability of the event based on the measurement results obtained. There is also provided a corresponding system for quantum probability estimation of an event.

Description

METHOD AND SYSTEM FOR QUANTUM PROBABILITY ESTIMATION OF AN EVENT CROSS-REFERENCE TO RELATED APPLICATION

[0001] This application claims the benefit of priority of Singapore Patent Application No.10202500065Y filed on 8 January 2025, the content of which being hereby incorporated by reference in its entirety for all purposes.TECHNICAL FIELD

[0002] The present invention generally relates to a method of quantum probability estimation of an event in a stochastic process, and a system thereof.BACKGROUND

[0003] In many situations, optimal decisions in the present depend on the likelihood of particular future events in stochastic systems (systems that evolve dynamically with time). Examples include, but not limited to, supply chain design that must account for future demand, insurance pricing based on probabilities of adverse events, and portfolio optimisation based on expected future pricing of various types of assets. Thus, the more accurately the probability of future events can be estimated, the more informed the decisions will be.

[0004] Formally, a stochastic system can be modelled by a stochastic process that emits outputs xtat regular discrete time-steps t. The statistics of each xtare governed by random variables Xtand typically depend on the outputs that occur at various time-steps prior. Consider now a subset of future events, whose probability p of occurrence is relevant to making optimal decisions. In such scenarios, an industry-standard method for estimating the probability of such an event p is through stochastic sampling - potential future sequences are sampled a suitable large number of times (e.g., denoted as N) and calculate, out of the sampling results, the proportion of the samples that falls within the target subset of events. However, the accuracy of the estimates is ultimately constrained by the following statistical considerations:• Random errors from finite sampling constraints - Estimating p to some standard error e « p requires order N ≈ O(1 / ε2). Thus, halving e tends to quadruple sampling time, and estimating events to high levels of relative accuracy may require a significant number of samples;• Bias from memory constraints - more complex processes are highly non-Markovian,such that data from the distant past can affect the likelihood of current events. Tracking all this data requires immense memory, and simulating such processes becomes the most costly per sample. Thus, practical simulation often involves dimensional reduction techniques to “prune” past data (e.g., Principal Component Analysis). However, discarding too much information induces systematic bias - such that the estimates of p contain error even when N → ∞.

[0005] These limitations imply that estimating the probability p of an event is particularly challenging when (1) estimating the event is required to be within an extremely small error tolerance e « p, and (2) the underlying process is highly non -Markovian, such that exactly modelling requires tracking events in the distant past. The above-mentioned scenario (1) is typical in many settings, such as risk analysis when dealing with catastrophic events or where there are interdependent events and small margins of error for each estimate to constrain the total error. The above-mentioned scenario (2) occurs whenever underlying processes have long memories, such as high-order autoregressive models or dynamic systems near criticality. In extreme cases, such as modelling human language (such as in LLMs), model inference becomes exceptionally costly (and is a primary contributor to the energetic cost of modern Al).

[0006] While quantum algorithms for estimating the probability of events have been proposed, they are mostly only concerned with mitigating random errors. Present state-of-art involves the use of quantum amplitude estimation (QAE) (Brassard et al, “Quantum Computation and Information”, American Mathematical Society; Providence, Rl, 2002, pages 53-74) or its shallow circuit variants (Suzuki et al., “Amplitude Estimation without Phase Estimation”, Quantum Information Processing 19, 75 (2020) and Grinko et al., “Iterative Quantum Amplitude Estimation”, Quantum Information 7, 1 (2021), arXiv: 191205559), which enables up to a quadratic scaling improvement in random error reduction. To that, reducing random error to order e requires a computational cost of O(1 / ε), compared to O(1 / ε2) via classical Monte Carlo methods. While such approaches have gained notable attention (including dedicated studies by J. P. Morgan), they face a critical constraint, namely, the data loading problem. To sample the probability of a random variable Z, QAE algorithms assume a quantum machine 풜 that maps |0⟩ to the quantum state ||ϕZ⟩ = Σz√pz|z⟩, which is often referred to as a quantum sample of Z. The difficulty of quantum machine 풜 is entirely neglected. In sampling events over complex stochastic systems, executing quantum machine 풜 is extremely costly. Direct quantum loading certainly is not practical. It takes O(2L) entangling gates to synthesize a quantum state of L bits, which is an exponential overhead that overwhelmsany quantum advantage. The alternative approach is to translate a classical predictive model of the process into a quantum circuit gate-for-gate. However, the memory costs of doing so will be at least as costly, with a caveat that instead of using N » 1 bits, N quantum bits (qubits) will be needed.

[0007] Existing quantum computers (and indeed any quantum computer shortly) will likely have severe memory constraints. As such, the bias error introduced for sampling any process of meaningful complexity will likely overwhelm any meaningful advantages in random error reduction. To surmount such obstacles, it is critical to implement quantum-enhanced models that exhibit an advantage in reducing both random error and bias.

[0008] A need therefore exists to provide a method of quantum probability estimation of an event, as well as a system thereof, that seeks to overcome, or at least ameliorate, one or more deficiencies in existing methods of quantum probability estimation of an event, and more particularly, that relaxes both computational constraint (reducing random error from finite sampling constraint) and memory constraint (reducing systematic bias from memory constraint). It is against this background that the present invention has been developed.SUMMARY

[0009] According to a first aspect of the present invention, there is provided a method of quantum probability estimation of an event in a stochastic process, wherein the stochastic process is modelled based on a finite state-machine model, and a classical predictive model for the event is modelled based on the stochastic process and comprises a set of states and a set of state transition matrices associated with the finite state-machine model, the method comprising:generating a circuit description of a quantum state-preparation operator based on a past observation sequence, comprising a series of past observed symbols, of the stochastic process and the set of state transition matrices of the classical predictive model, wherein the quantum state-preparation operator comprises an initialization operator having encoded therein the past observation sequence of the stochastic process; and a series of state transition operators for a future sequence, comprising a series of future symbols, of the stochastic process, each state transition operator having encoded therein the set of state transition matrices;generating a circuit description of a quantum search operator based on the circuit description of the quantum state-preparation operator and an event indication function, wherein the event indication function is configured to indicate whether the event is to occur with respect to the future sequence of the stochastic process, and the quantum search operator is configuredto facilitate a search for a particular future sequence, comprising a particular series of future symbols, of the stochastic process that results in an indication that the event is to occur;generating a plurality of circuit descriptions for a plurality of measurement quantum circuits, respectively, for compiling the plurality of measurement quantum circuits on a quantum computer, wherein each circuit description of the measurement quantum circuit is generated based on the circuit description of the quantum state-preparation operator and the circuit description of the quantum search operator;for each of the plurality of circuit descriptions of the plurality of measurement quantum circuits:compiling the circuit description of the measurement quantum circuit on the quantum computer to form the measurement quantum circuit thereon; andexecuting the measurement quantum circuit formed on the quantum computer to obtain a measurement result associated with the measurement quantum circuit that indicates a number of times that the measurement quantum circuit produces an indication that the event is to occur; andestimating a probability of the event based on the measurement results associated with the plurality of measurement quantum circuits obtained.

[0010] According to a second aspect of the present invention, there is provided a system for quantum probability estimation of an event in a stochastic process, wherein the stochastic process is modelled based on a finite state-machine model, and a classical predictive model for the event is modelled based on the stochastic process and comprises a set of states and a set of state transition matrices associated with the finite state-machine model, the system comprises:at least one memory; andat least one processor communicatively coupled to the at least one memory and configured to:generate a circuit description of a quantum state-preparation operator based on a past observation sequence, comprising a series of past observed symbols, of the stochastic process and the set of state transition matrices of the classical predictive model, wherein the quantum state-preparation operator comprises an initialization operator having encoded therein the past observation sequence of the stochastic process; and a series of state transition operators for a future sequence, comprising a series of future symbols, of the stochastic process, each state transition operator having encoded therein the set of state transition matrices;generate a circuit description of a quantum search operator based on the circuit description of the quantum state-preparation operator and an event indication function, wherein the event indication function is configured to indicate whether the event is to occur with respect to the future sequence of the stochastic process, and the quantum search operator is configured to facilitate a search for a particular future sequence, comprising a particular series of future symbols, of the stochastic process that results in an indication that the event is to occur;generate a plurality of circuit descriptions for a plurality of measurement quantum circuits, respectively, for compiling the plurality of measurement quantum circuits on a quantum computer, wherein each circuit description of the measurement quantum circuit is generated based on the circuit description of the quantum state-preparation operator and the circuit description of the quantum search operator,for each of the plurality of circuit descriptions of the plurality of measurement quantum circuits:compile the circuit description of the measurement quantum circuit on the quantum computer to form the measurement quantum circuit thereon; andexecute the measurement quantum circuits formed on the quantum computer to obtain a measurement result associated with the measurement quantum circuit that indicates a number of times that the measurement quantum circuit produces an indication that the event is to occur; andestimate a probability of the event based on the measurement results associated with the plurality of measurement quantum circuits obtained.

[0011] According to a third aspect of the present invention, there is provided a computer program product, embodied in one or more non-transitory computer-readable storage mediums, comprising instructions executable by at least one processor to perform the method of quantum probability estimation of an event in a stochastic process according to the above-mentioned first aspect of the present invention.BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Embodiments of the present invention will be better understood and readily apparent to one of ordinary skill in the art from the following written description, by way of example only, and in conjunction with the drawings, in which:FIG. 1 depicts a schematic diagram of a method of quantum probability estimation of an event in a stochastic process, according to various embodiments of the present invention;FIG. 2 depicts a schematic block diagram of a system for quantum probability estimation of an event in a stochastic process, according to various embodiments of the present invention;FIG. 3 depicts a schematic block diagram of a system for quantum probability estimation of an event in a stochastic process, including a quantum computer, according to various embodiments of the present invention,FIG. 4A shows a schematic drawing illustrating a classical model (classical predictive model) whereby at each timestep t, a classical predictive model stores information about the past, x̃ = ... xt-1, xtinside a memory system M;FIG. 4B shows a schematic drawing illustrating a quantum model (quantum predictive model) encoding X̃ inside a quantum memory by setting it to certain state |ϕ⟩_X̃, and applies a quantum process at each subsequent time-step;FIG. 5 shows a schematic drawing of a classical finite state-machine model of an example binary autoregressive AR(2) process;FIG. 6A shows a graph that plots the random error in estimates of p using the quantum probability estimation algorithm, according to various example embodiments of the present invention, compared with its classical counterpart;FIG. 6B shows a graph that plots the standard error of the estimates on a log-linear plot to demonstrate the scaling advantage, according to various example embodiments of the present invention;FIG. 7 shows a graph for demonstrating quantum bias reduction in estimates of p using the quantum probability estimation algorithm according to various example embodiments of the present invention, compared with its classical counterpart;FIG. 8 depicts an example circuit schematic of the quantum probability estimator (quantum probability estimation (QPE) machine), according to various example embodiments of the present invention;FIG. 9A shows the hidden Markov model of the dynamics of the undiscretized process; FIG. 9B shows the hidden Markov model of FIG. 9A where each state is rewritten with the causal state notation si, for i = 0, 1, 2, 3, and mapping the output values from { — 1, l} to {0, 1 } for ease of coding; andFIG. 10 shows a comparison of the absolute error of the probability estimate between the true probability and the probabilities predicted by the approximate quantum and classical models over all futures of length four given a past of length 3.DETAILED DESCRIPTION

[0013] Various embodiments of the present invention provide a method of quantum probability estimation of an event in a stochastic process, and a system thereof.

[0014] As discussed in the background, often in time-series analysis, the probability of an event is desired or required to be estimated. Yet, as the stochastic systems underlying such events become ever more complex such that the future can depend on observations from the discontent pasts, the accuracy of such estimates is ultimately limited by both computational and memory constraints, and more particularly, random errors from finite sampling constraints and bias from memory constraints. In this regard, various embodiments of the present invention provide a method of quantum probability estimation of an event in a stochastic process, as well as a system thereof, that seeks to overcome, or at least ameliorate, one or more deficiencies in existing methods of quantum probability estimation of an event, and more particularly, that relaxes both computational constraint (reducing random error from finite sampling constraint) and memory constraint (reducing systematic bias from memory constraint).

[0015] FIG. 1 depicts a schematic diagram of a method 100 of quantum probability estimation of an event in a stochastic process, according to various embodiments of the present invention. According to the method 100, the stochastic process is modelled based on a finite state-machine model. Furthermore, a classical predictive model for the event is modelled based on the stochastic process and comprises a set of states and a set of state transition matrices associated with the finite state-machine model. The method 100 comprises generating (at 106) a circuit description of a quantum state-preparation operator based on a past observation sequence, comprising a series of past observed symbols, of the stochastic process and the set of state transition matrices of the classical predictive model. In this regard, the quantum statepreparation operator comprises: an initialization operator having encoded therein the past observation sequence of the stochastic process; and a series of state transition operators for a future sequence, comprising a series of future symbols, of the stochastic process, each state transition operator having encoded therein the set of state transition matrices. The method 100 further comprises generating (at 108) a circuit description of a quantum search operator based on the circuit description of the quantum state-preparation operator and an event indication function. The event indication function is configured to indicate whether the event is to occur with respect to the future sequence of the stochastic process. The quantum search operator (e.g., a Grover operator) is configured to facilitate a search for a particular future sequence,comprising a particular series of future symbols, of the stochastic process that results in an indication that the event is to occur. The method 100 further comprises generating (at 110) a plurality of circuit descriptions for a plurality of measurement quantum circuits, respectively, for compiling the plurality of measurement quantum circuits on a quantum computer, wherein each circuit description of the measurement quantum circuit is generated based on the circuit description of the quantum state-preparation operator and the circuit description of the quantum search operator. The method 100 further comprises (at 112), for each of the plurality of circuit descriptions of the plurality of measurement quantum circuits: compiling the circuit description of the measurement quantum circuit on the quantum computer to form the measurement quantum circuit thereon; and executing the measurement quantum circuit formed on the quantum computer to obtain a measurement result associated with the measurement quantum circuit that indicates a number of times that the measurement quantum circuit produces an indication that the event is to occur. The method 100 further comprises estimating (at 114) a probability of the event based on the measurement results associated with the plurality of measurement quantum circuits obtained.

[0016] In various embodiments, the circuit descriptions (or circuit schematics) of operators or quantum circuits described herein are generated using a classical computer. For example, circuit descriptions of operators may be generated by a classical computer and selected circuit descriptions of operators may be assembled to form a circuit description of a desired quantum circuit on classical computer using a quantum programming package such as but not limited to Qiskit in Python (a high level description of the quantum circuit, where the operators are still described in terms of mathematical transformations). Thereafter, the circuit description of the quantum circuit may be compiled and executed on a quantum computer. The compiler is classical (may run on a classical computer or a server of a quantum computer provider). For example, the compiler may obtain the high-level circuit description and the specific instruction set (gate set) of the target quantum computer, and apply decomposition on the high level quantum circuit operators into smaller gates, such that all the operations in the compiled quantum circuit are defined within the available instruction set. It will be appreciated by a person skilled in the art that various quantum circuit compiling algorithms (including open source algorithms and proprietary algorithms) exist in the art. For example, a quantum computer may comprise two key components, namely, a quantum processor and a classical control system. The quantum processor mainly holds the physical qubits. For each compiled quantum circuit (which may now be a series of low level instructions understandable by the controlsystem), the control system may apply the corresponding instruction as a physical operation onto the corresponding qubits (e.g., a rotation gate can be interpreted as a electromagnetic pulse of a certain duration on a specific qubit). It will be appreciated by a person skilled in the art that different quantum computers have different instruction sets. Therefore, a compiled physical quantum circuit may refer to an ordered set of physical qubits coupled with an ordered set of defined physical operations on the qubits.

[0017] In various embodiments, each circuit description of the plurality of circuit descriptions of the plurality of measurement quantum circuits is generated to comprise a predefined number of the circuit description of the quantum search operator. In various embodiments, the predefined number of the circuit description of the quantum search operator for each circuit description of the plurality of circuit descriptions of the plurality of measurement quantum circuits is different.

[0018] In various embodiments, the quantum computer comprises a quantum memory register and a series of output registers respectively for the series of future symbols of the future sequence of the stochastic process. In this regard, each of the plurality of measurement quantum circuits formed on the quantum computer communicatively couples to the quantum memory register and the series of output registers.

[0019] In various embodiments, each of the plurality of measurement quantum circuits comprises a quantum state-preparation operator circuit compiled according to the circuit description of the quantum state-preparation operator and one or more quantum search operator circuits compiled according to the circuit description of the quantum search operator. In various embodiments, for each of the plurality of measurement quantum circuits, the quantum statepreparation operator circuit of the measurement quantum circuit comprises an initialization operator circuit configured according to the initialization operator; and a series of state transition operator circuits configured according to the series of state transition operators. In this regard, the initialization operator circuit communicatively couples (e.g., connects) to the quantum memory register. Furthermore, the series of state transition operator circuits respectively communicatively couple to the series of output registers and communicatively couples to the quantum memory register. For example, for a certain number of time steps into the future, the same certain number of state transition operator circuits may be applied, each state transition operator circuit being applied on the same quantum memory register and a new output register. Accordingly, the state transition operator may also be referred to as a step operator since it moves one time step at a time into the future.

[0020] In various embodiments, the above-mentioned executing (at 112) the measurement quantum circuit formed on the quantum computer comprises initializing each output register of the series of output registers in a zero state.

[0021] In various embodiments, the quantum search operator comprises a reverse quantum state-preparation operator having a reverse configuration of the quantum state-preparation operator, a forward quantum state-preparation operator being the quantum state-preparation operator, a state reflection operator arranged between the forward and reverse quantum statepreparation operators, and a phase oracle arranged before the reverse quantum state-preparation operator. Furthermore, the phase oracle has encoded therein the event indication function.

[0022] In various embodiments, the quantum search operator is a Grover operator.

[0023] In various embodiments, the event indication function is a binary function configured to return a binary number ‘1 ’ or a binary number ‘0’ with respect to the future sequence of the stochastic process. In this regard, the binary numbers ‘ 1’ and ‘0’ indicate that the event is to occur and not occur, respectively, with respect to the future sequence of the stochastic process.

[0024] In various embodiments, the series of state transition operators are each generated based on a set of Kraus operators obtained based on the set of state transition matrices.

[0025] In various embodiments, each of the plurality of measurement quantum circuits is executed a predefined number of times to obtain the measurement result associated with the measurement quantum circuit.

[0026] In various embodiments, the above-mentioned estimating (at 114) the probability of the event is based on a maximum likelihood estimation on the measurement results associated with the plurality of measurement quantum circuits obtained.

[0027] FIG. 2 depicts a schematic block diagram of a system 200 for quantum probability estimation of an event in a stochastic process, according to various embodiments of the present invention, corresponding to the above-mentioned method 100 of quantum probability estimation of an event as described hereinbefore with reference to FIG. 1 according to various embodiments of the present invention. The system 200 comprises: at least one memory 202; and at least one processor 204 communicatively coupled (e g., connected) to the at least one memory 202 and configured to perform the method 100 of quantum probability estimation of an event in a stochastic process according to various embodiments of the present invention. Accordingly, the at least one processor 204 is configured to: generate a circuit description of a quantum state-preparation operator based on a past observation sequence, comprising a seriesof past observed symbols, of the stochastic process and the set of state transition matrices of the classical predictive model, wherein the quantum state-preparation operator comprises an initialization operator having encoded therein the past observation sequence of the stochastic process; and a series of state transition operators for a future sequence, comprising a series of future symbols, of the stochastic process, each state transition operator having encoded therein the set of state transition matrices; generate a circuit description of a quantum search operator based on the circuit description of the quantum state-preparation operator and an event indication function, wherein the event indication function is configured to indicate whether the event is to occur with respect to the future sequence of the stochastic process, and the quantum search operator is configured to facilitate a search for a particular future sequence, comprising a particular series of future symbols, of the stochastic process that results in an indication that the event is to occur; generate a plurality of circuit descriptions for a plurality of measurement quantum circuits, respectively, for compiling the plurality of measurement quantum circuits on a quantum computer, wherein each circuit description of the measurement quantum circuit is generated based on the circuit description of the quantum state-preparation operator and the circuit description of the quantum search operator; for each of the plurality of circuit descriptions of the plurality of measurement quantum circuits: compile the circuit description of the measurement quantum circuit on the quantum computer to form the measurement quantum circuit thereon; and execute the measurement quantum circuits formed on the quantum computer to obtain a measurement result associated with the measurement quantum circuit that indicates a number of times that the measurement quantum circuit produces an indication that the event is to occur; and estimate a probability of the event based on the measurement results associated with the plurality of measurement quantum circuits obtained.

[0028] It will be appreciated by a person skilled in the art that the at least one processor 204 may be configured to perform various functions or operations through set(s) of instructions (e.g., software modules) executable by the at least one processor 204 to perform various functions or operations. Accordingly, as shown in FIG. 2, the system 200 may comprise: a quantum state-preparation operator generating module (or a quantum state-preparation operator generating circuit) 206 configured to perform the above-mentioned generating a circuit description of a quantum state-preparation operator based on a past observation sequence, comprising a series of past observed symbols, of the stochastic process and the set of state transition matrices of the classical predictive model; a quantum search operator generating module (or a quantum search operator generating circuit) 208 configured to perform the above-mentioned generating a circuit description of a quantum search operator based on the circuit description of the quantum state-preparation operator and an event indication function; a measurement quantum circuit generating module 210 configured to perform the above-mentioned generating a plurality of circuit descriptions for a plurality of measurement quantum circuits, respectively, for compiling the plurality of measurement quantum circuits on a quantum computer; a measurement quantum circuit compiling and executing module (or a measurement quantum circuit compiling and executing circuit) 212 configured to perform, for each of the plurality of circuit descriptions of the plurality of measurement quantum circuits, the above-mentioned compiling the circuit description of the measurement quantum circuit on the quantum computer; and the above-mentioned executing the measurement quantum circuits formed on the quantum computer; and an event probability estimation module (or an event probability estimation circuit) 214 configured to perform the above-mentioned estimate a probability of the event based on the measurement results associated with the plurality of measurement quantum circuits obtained.

[0029] It will be appreciated by a person skilled in the art that the above-mentioned modules of the system 200 are not necessarily separate modules, and two or more modules may be realized by or implemented as one functional module (e g., a circuit or a software program) as desired or as appropriate without deviating from the scope of the present invention. For example, two or more of the quantum state-preparation operator generating module 206, the quantum search operator generating module 208, the measurement quantum circuit generating module 210, the measurement quantum circuit compiling and executing module 212 and the event probability estimation module 214 may be realized (e.g., compiled together) as one or more executable software programs (e g., software applications), which for example may be stored in the at least one memory 202 and executable by the at least one processor 204 to perform the corresponding functions or operations as described herein according to various embodiments of the present invention.

[0030] In various embodiments, the system 200 for quantum probability estimation of an event corresponds to the method 100 of quantum probability estimation of an event as described hereinbefore with reference to FIG. 1, therefore, various operations, functions or steps configured to be performed by the at least one processor 204 may correspond to various operations, functions or steps of the method 100 of quantum probability estimation of an event described hereinbefore according to various embodiments, and thus need not be repeated with respect to the system 200 for quantum probability estimation of an event for clarity andconciseness. In other words, various embodiments described herein in context of methods (e.g., the method 100 of quantum probability estimation of an event) are analogously valid for the corresponding systems or devices (e g., the system 200 for quantum probability estimation of an event), and vice versa.

[0031] A computing system, a controller, a microcontroller or any other system providing a processing capability may be provided according to various embodiments in the present invention. Such a system may be taken to include one or more processors and one or more computer-readable storage mediums. For example, the system 200 for quantum probability estimation of an event described hereinbefore includes at least one processor 204 and at least one computer-readable storage medium (or memory) 202 which are for example used in various processing carried out therein as described herein. A memory or computer-readable storage medium used in various embodiments for a classical computer may be a volatile memory, e.g., a DRAM (Dynamic Random Access Memory) or a non-volatile memory, e.g., a PROM (Programmable Read Only Memory), an EPROM (Erasable PROM), EEPROM (Electrically Erasable PROM), or a flash memory, e.g., a floating gate memory, a charge trapping memory, an MRAM (Magnetoresistive Random Access Memory) or a PCRAM (Phase Change Random Access Memory).

[0032] In various embodiments, with respect to classical computers, a “circuit” may be understood as any kind of a logic implementing entity, which may be special purpose circuitry or a processor executing software stored in a memory, firmware, or any combination thereof. Thus, in an embodiment for a classical computer, a “circuit” may be a hard-wired logic circuit or a programmable logic circuit such as a programmable processor, e.g., a microprocessor (e.g., a Complex Instruction Set Computer (CISC) processor or a Reduced Instruction Set Computer (RISC) processor). A “circuit” with respect to a classical computer may also be a processor executing software, e.g., any kind of computer program, e.g., a computer program using a virtual machine code, e g., Java. Any other kind of implementation of various functions or operations may also be understood as a “circuit” in accordance with various other embodiments. Similarly, a “module” may be a portion of a system according to various embodiments in the present invention and may encompass a “circuit” as above, or may be understood to be any kind of a logic-implementing entity therefrom.

[0033] In various embodiments, with respect to quantum computers, a “quantum circuit” may be referred to as a collection of interconnected quantum gates (interconnected by quantum wires), which are used to carry out unitary transformations on qubits. For example, a quantumcircuit may be configured to perform a deterministic sequence of operations on a plurality of qubits arranged in a quantum register. A qubit register is a physical substrate (e.g., superconducting loops, trapped ions, neutral atoms or combinations thereof) configured to maintain a plurality of qubits in a coherent state. The quantum circuit may be defined by a spatial and temporal arrangement of quantum logic gates, each represented by a unitary matrix that operates on one or more qubits to induce a controlled evolution of the probability amplitudes of the qubit state within a multi-dimensional Hilbert space. These operations are implemented via a control interface that translates the mathematical logic of the matrices into specific physical control signals, such as electromagnetic or optical pulses, to induce state transformations. The quantum circuit is also configured to interface with a readout module to collapse the quantum state into a classical output.

[0034] Some portions of the present disclosure may be explicitly or implicitly presented in terms of algorithms and functional or symbolic representations of operations on data within a computer memory. These algorithmic descriptions and functional or symbolic representations are the means used by those skilled in the data processing arts to convey most effectively the substance of their work to others skilled in the art. An algorithm may be, and generally, conceived to be a self-consi stent sequence of steps leading to a desired result.

[0035] The present specification also discloses a system (e.g., which may also be embodied as one or more devices or apparatuses), such as the system 200 for quantum probability estimation of an event, for performing various operations, functions or steps of various methods described herein. Such a system may each be specially constructed for the required purposes or may comprise a general purpose computer system selectively activated or reconfigured by a computer program stored in the computer system. In general, various algorithms that may be presented herein are not limited to being implemented or executed by any particular computer system. Alternatively, the construction of more specialized computer system to perform various operations, functions or steps of various methods described herein may be provided as desired or as appropriate without going beyond the scope of the present invention.

[0036] In addition, the present specification also at least implicitly discloses computer program(s) or software / functional module(s), in that it would be apparent to a person skilled in the art that various operations, functions or steps of various methods described herein may be put into effect by computer code The computer program(s) is not intended to be limited to any particular programming language and implementation thereof, and it will be appreciated by a person skilled in the art that a variety of programming languages and coding thereof may beused to implement the computer program(s). Moreover, the computer program(s) is not intended to be limited to any particular control flow as there are a variety of programming languages which can use different control flows. It will be appreciated by a person skilled in the art that a computer program may be stored on any computer-readable storage medium (non-transitory computer-readable storage medium), such as but not limited to, a magnetic disk, an optical disk or a memory chip. For example, a computer program stored on a computer-readable storage medium may be loaded and executed on a computer system to implement various operations, functions or steps of various methods described herein according to various embodiments of the present invention.

[0037] Accordingly, in various embodiments, there is provided a computer program product, embodied in one or more computer-readable storage mediums (non-transitory computer-readable storage medium), comprising instructions (e g., the quantum statepreparation operator generating module 206, the quantum search operator generating module 208, the measurement quantum circuit generating module 210, the measurement quantum circuit compiling and executing module 212 and / or the event probability estimation module 214) executable by one or more computer processors to perform the method 100 of quantum probability estimation of an event as described hereinbefore with reference to FIG. 1 according to various embodiments of the present invention. Accordingly, various computer programs or software modules described herein may be stored in a computer program product receivable by a system therein, such as the system 200 for quantum probability estimation of an event as shown in FIG. 2, for execution by at least one processor 204 of the system 200 to perform various operations, functions or steps of various methods described herein according to various embodiments of the present invention.

[0038] It will be appreciated by a person skilled in the art that various modules of systems described herein (e.g., the quantum state-preparation operator generating module 206, the quantum search operator generating module 208, the measurement quantum circuit generating module 210, the measurement quantum circuit compiling and executing module 212 and / or the event probability estimation module 214) may be software module(s) realized by computer program(s) or set(s) of instructions executable by a computer processor to perform various functions or operations. Various modules described herein (e g., the quantum statepreparation operator generating module 206, the quantum search operator generating module 208, the measurement quantum circuit generating module 210, the measurement quantum circuit compiling and executing module 212 and / or the event probability estimation module214) may also be implemented as hardware module(s) being functional hardware unit(s) designed to perform various functions or operations. More particularly, in the hardware sense, a module is a functional hardware unit designed for use with other components or modules. For example, a module may be implemented using discrete electronic components, or it can form a portion of an entire electronic circuit such as an Application Specific Integrated Circuit (ASIC) or a Field Programmable Gate Array (FPGA). Numerous other possibilities exist. It will also be appreciated by a person skilled in the art that a combination of hardware and software modules may be implemented. Furthermore, various operations, functions or steps of various methods described herein may be performed in parallel rather than sequentially as desired or as appropriate (e g., as long as it does not render the method(s) inoperable or unsatisfactory for its intended purpose).

[0039] FIG. 3 depicts a schematic block diagram of a system 300 for quantum probability estimation of an event in a stochastic process, including a quantum computer, according to various embodiments of the present invention. The system 300 is the same or similar as the system 200 for quantum probability estimation of an event as described hereinbefore with reference to FIG. 2 according to various embodiments of the present invention, except that the system 200 further comprises the quantum computer 340 described hereinbefore with respect to the system 200 (or the method 100) for quantum probability estimation of an event. That is, the system 300 comprises: a classical computer 320 comprising the at least one memory 202 and the at least one processor 204, along with the quantum state-preparation operator generating module 206, the quantum search operator generating module 208, the measurement quantum circuit generating module 210, the measurement quantum circuit compiling and executing module 212 and / or the event probability estimation module 214, configured to perform the method 100 of quantum probability estimation of an event according to various embodiments of the present invention; and the quantum computer 340 described hereinbefore with respect to the system 200 (or the method 100). Accordingly, the quantum computer 340 comprises: the quantum memory register; and the series of output registers respectively for the series of future symbols of the future sequence of the stochastic process. In this regard, each of the plurality of measurement quantum circuits 350 formed on the quantum computer 340 communicatively couples (e.g., connects) to the quantum memory register and the series of output registers. In various embodiments, as described hereinbefore, for each of the plurality of measurement quantum circuits, the quantum state-preparation operator circuit of the measurement quantum circuit comprises an initialization operator circuit configured according to the initializationoperator; and a series of state transition operator circuits configured according to the series of state transition operators. In this regard, the initialization operator circuit communicatively couples to the quantum memory register, and the series of state transition operator circuits respectively communicatively couples to the series of output registers and communicatively couples to the quantum memory register

[0040] It will be appreciated by a person skilled in the art that the terminology used herein is for the purpose of describing various embodiments only and is not intended to be limiting of the present invention. As used herein, the singular forms “a”, “an” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the tenns “comprises” and / or “comprising,” when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and / or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and / or groups thereof.

[0041] Any reference to an element or a feature herein using a designation such as “first”, “second” and so forth does not limit the quantity or order of such elements or features, unless stated or the context requires otherwise. For example, such designations may be used herein as a convenient way of distinguishing between two or more elements or instances of an element. Thus, a reference to first and second elements does not necessarily mean that only two elements can be employed, or that the first element must precede the second element, unless stated or the context requires otherwise. In addition, a phrase referring to “at least one of’ a list of items refers to any single item therein or any combination of two or more items therein.

[0042] In order that the present invention may be readily understood and put into practical effect, various example embodiments of the present invention will be described hereinafter by way of examples only and not limitations. It will be appreciated by a person skilled in the art that the present invention may, however, be embodied in various different forms or configurations and should not be construed as limited to the example embodiments set forth hereinafter. Rather, these example embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the present invention to those skilled in the art.

[0043] Often in time-series analysis, the probability of an event is desired or required to be estimated. Yet, as the stochastic systems underlying such events become ever more complex such that the future can depend on observations from the discontent pasts, the accuracy of such estimates is ultimately limited by both computational and memory constraints. In this regard,various example embodiments of the present invention provide a quantum machine that leverages unique properties of quantum information to significantly relax both computational and memory constraints. For example, quantum machines according to various example embodiments of the present invention are advantageously able to reduce systematic bias from memory constraints by nearly an order of magnitude while simultaneously offering a quadratic reduction of random error due to finite sampling. Meanwhile, their reduced quantum memory requirements significantly advance the timeline for benefits of quantum technologies in analysis of time-series data. Accordingly, various example embodiments of the present invention advantageously provide a quantum machine for error-reduced probability estimation in forecasting of stochastic processes.

[0044] As described above, various example embodiments of the present invention advantageously provide a quantum machine for estimating the probability of events that simultaneously reduces random error and bias. Without wishing to be bound by theory but for better understanding, various example embodiments note that quantum causal models can better isolate the potential causes of natural things. In this regard, to achieve such a quantum machine, various example embodiments seek to provide a means to convert such quantum causal models into quantum circuits and integrate these circuits with quantum amplitude estimation.

[0045] More formally, various example embodiments first recognise that the task of sampling futures in stochastic systems is essentially a task of next-token prediction. At each time-step / , x =... xt-1,xtmay represent past observations. Various example embodiments note that provided one can find a way to sample X[+1faithfully, then the past can be updated as x =... xt-1(xt, xt+1at time-step t + 1 and repeat this process recursively to sample Vt+2and beyond. To do this with minimal memory, certain information contained in x that is necessary for repeating this process can be identified and stored in a memory system; such that repeated actions on M alone can generate the correct conditional future P(x|x), where X = X[+1Xt+2... are random variables that govern future outputs (see FIG. 4A). In particular, FIG. 4A shows a schematic drawing illustrating a classical model (classical predictive model) whereby at each timestep / , a classical predictive model stores information about the past, x —... xt-pxtinside a memory system M. Repeated application of some systematic computational process at each time step / then allows the classical predictive model to generate potential future trajectory x = x1,x2,x2,... according to the underlying stochastic process’ future probability distribution P(%|x). From computational mechanics, it can be understood that every stochastic process P has an intrinsic measure of complexity dc= log2(Z)c), where Dcis the memorydimension of M (the number of distinct configurations has). Referred to as the topological state complexity of P, dclower bounds the number of bits must contain. Classical machines with minimal memory dimension are known as e-machines and have seen diverse use cases from financial analytics and atmospherics to measuring consciousness from neurological data. Complex processes (e.g., human language) possess immense Dcas they are highly non-Markovian, where the future may depend on events in the distant past. Devoting less memory than Dcwill make it impossible to sample P(x|x) with perfect statistical fidelity. This then induces bias.

[0046] The canonical method of quantum data loading could simply be to translate a classical model by decomposing it into logical gates. Switching each gate with a quantum mechanical counterpart can then realize a quantum machine capable of generating a quantum sample of P(x|%). Even assuming the provably memory-minimal classical machines are known, the resulting quantum machine would have memory dimension DQ — DC. The only benefit of quantum algorithms in this context is reducing the number of future samples, but the memory cost of creating each sample does not change. The downside is that qubits (while theoretically a fair comparison to a bit) are far more challenging to engineer. As of 2024, state-of-art quantum computers are limited to about 103physical qubits, with many works estimating that only 10 to 20 are fully useful. As such, memory limitations result in massive distortion.

[0047] The approach according to various example embodiments leverages quantum models that can better isolate past helpful information for future prediction, and thus reduce memory-constrained distortion. Specifically, quantum mechanics give the freedom to store different pasts into mutually non-orthogonal states |s^) of a quantum memory system Q (see FIG. 4B). In particular, FIG. 4B shows a schematic drawing illustrating a quantum model (quantum predictive model) encoding X inside a quantum memory by setting it to certain state and applies a quantum process at each subsequent time-step such that when measured in the computational basis, they replicate expected conditional statistics. The amount of information a model needs to track can be quantified by the dimensions of the quantum memory system Q. When the memory dimension is constrained to values below Dc, quantum techniques allow quantum devices to be constructed that can sample from P (X |x ) with drastically reduced distortion.

[0048] Accordingly, technical contributions according to various example embodiments of the present invention include developing systematic adjustments to such quantum models,enabling their integration with shallow circuit quantum phase estimation techniques. This involves taking into account that existing quantum models do not prepare a pure quantum sample, as said outputs must remain entangled with model memory JVC. Therefore, according to various example embodiments of the present invention, unique end-to-end quantum circuit designs for event-probability estimation are developed that (i) alleviated the data loading problem, providing an efficient means to generate quantum samples of X, and are (ii) optimized for memory-limited quantum computers by use of dimensionality reduction in quantum models. The resulting quantum machines then possess dual advantages in reducing both sampling error and bias in resource-limited regimes.

[0049] Given a stochastic process, an event can be defined by a binary function f configured to map from a set of possible futures x to a binary number {0, 1 }. That is, the binary function / is configured to return a binary number ‘1’ or a binary number ‘0’ with respect to a future sequence, comprising a series of future symbols, of the stochastic process, whereby the binary numbers ‘ 1’ and ‘0’ indicate that the event is to occur and not occur, respectively, with respect to the future sequence of the stochastic process. An event E is said to be true if f(x) = 1.Accordingly, the binary function / is an event indication function configured to indicate whether the event is to occur with respect to a future sequence of the stochastic process. Let P then be the probability that E is true. In this regard, a quantum probability estimation method according to various example embodiments of the present invention allows an estimation of the probability PE given any observed past x. The resulting quantum machine according to various example embodiments of the present invention has two key advantages over all classical counterparts:• Quadratic speed-up - it exhibits a quadratic reduction in the number of simulations N required to estimate PE to some desired random error e for any E. That is, all classical machines require that A scale with O(l / c2), while the quantum machines according to various example embodiments of the present invention enables N to scale with 0(1 / e).• Reduce Bias - when contained to using the same memory dimension d, the quantum machines according to various example embodiments of the present invention can generate estimates with significantly reduced systematic bias. The exact reduction factor can reach a full order of magnitude, even in simple proof-of-concept trials. There exist theoretical examples while the reduction factor is unbounded.

[0050] For better understanding, these advantages are illustrated via a proof-of-concept example according to various example embodiments of the present invention. Consider anexample stochastic process generated by the dynamics of an example finite-state machine in FIG. 5. In particular, FIG. 5 shows a schematic drawing of a classical finite state-machine model of an example binary autoregressive AR(2) process. Accordingly, the finite state-machine model comprises 4 internal memory states {si}3i=0The edge labels x|r from s;to Sj indicates that an instance of the finite-state machine in state skwill output x and transition to state s, with probability r = p±±. The process arises when discretizing an autoregressive (AR) model of order 2 (discretization of AR(2) process will be described in detail later below according to various example embodiments of the present invention). In machine learning literature, discretization is more often referred to in classical literature as quantization and involves representing a real-time number with a low-precision approximation to save memory. Without limitation, this process may be referred to herein as discretization to prevent confusion with quantization in the context of converting a classical model to a quantum models. Such autoregressive (AR) models describe continuous stochastic variables Ytthat evolve under the equation:= K1Yt-1+ K2Yt-2+ &(Equation 1) where Ytdenotes the current value of the series at time t, K1, K2G R denote real-valued constants and denotes a random variable governing white noise at time t. Such AR models are non-Markovian (since Ytdepends on more than just Xt-1), and appear in the context of describing security prices in finance. Discretization then involves representing each Ytwith a binary variable Xt, such that Xt= sign(Yt) is the sign of Ytthat is 1 when Yt> 0 and —1 otherwise. This results in a process in which any classical model must track four distinct states, with transition probabilities p±±= P(ξt> ±κ1± κ2).

[0051] An example method or algorithm of quantum probability estimation of an event in a stochastic process according to various example embodiments of the present invention is provided or utilized to estimate the probability of an event in the above-described example stochastic process (AR(2) process), with parametersκ1= 0.5 and κ2= 0.4, and assume the observed past of x-1x0= 11. Consider the probability that we subsequently observed 1001 (future sequence of the stochastic process), which occurs with approximate probability p = 0.00842. First, both quantum and classical models are allowed unlimited memory so that they can store all 4 finite-machine states. FIG 6A shows a graph that plots the random error in estimates of p using the quantum probability estimation algorithm according to various example embodiments of the present invention, compared with its classical counterpart. In particular,FIG. 6A shows a graph that plots the random errors in estimates of p by both the quantum and classical models to demonstrate significantly improved capability of the quantum model according to various example embodiments of the present invention in mitigating random error without memory constraints. FIG. 6A showed the lower and upper quartile for the estimates of p for both classical and quantum models for 1000 separate trials, as a function of the number of samples per trial N. Clearly, the quantum model exhibits a much quicker convergence to the true value of / ?, whereby a clear quadratic advantage can be observed At N = 4 × 104samples, the standard error of the classical model hovers at around 4.5 × 10-4, whereas the standard error for the quantum model is at around 9.1 × 10-5. The scaling advantage can be verified by plotting the standard error of the estimates on a log-linear plot as shown in FIG. 6B

[0052] To illustrate quantum-reduced bias under memory constraints, the scenario where both classical and quantum models can track only 3 fully distinguishable states (instead of 4 necessary for perfect simulation) during the sampling process is considered. In the example, since the minimal number of memory distinct memory states needed to model this process is 4, this constraint naturally introduces bias.

[0053] In classical models, a classical lower bound can be determined as bias by identifying all possible ways to merging two of the 4 states in the AR(2) model, taking the one with minimal distortion (as measured by the KL divergence). The distortion caused by this is represented by the fainter dashed line in FIG. 7. FIG. 7 shows a graph for demonstrating quantum bias reduction in estimates of p using the quantum probability estimation algorithm according to various example embodiments of the present invention, compared with its classical counterpart. In particular, in FIG 7, the total error estimation of error (random plus bias) for estimating P(0.4= 1001|x_3._1= 000) with memory dimensional of at most 3 for both quantum models (solid fainter line) and classical counterparts (solid darker line) are compared as a function of number of samples N. In classical models, this introduces a bias of about 6 × 10-3(darker dashed line) that remains even as the number of samples goes to oo (the dimension reduction comparison with Causal-State Splitting Reconstruction (CSSR) will be discussed in further detail later below). In contrast, quantum models of equivalent memory dimension can reduce this bias an order of magnitude to about 3 × 10-4(dashed fainter line in FIG. 7). In fact, when the total standard error of the estimates is plotted using a dimension 3 quantum model, it surpasses ultimate classical bounds (for infinite N) with as little as 300 samples. The classical bounds refer to the lowest possible error a classical model under memory constrain can achieve, found through an exhaustive search over all possible clustering of the classical states to reachthe desired memory constrain. The metric that is minimized is the Kullback-Liebler Divergence between the probability distribution generated by the memory-constrained model and the original model. It can be seen that this bias is approximately 6 × 10-6. In contrast, the standard error of the quantum models according to various example embodiments of the present invention dips below this using as little as 300 samples and converges to a bias that is more than 1 order of magnitude lower.

[0054] According to various example embodiments of the present invention, a quantum probability estimation (QPE) machine may be systematically configured using the following input information:1. A classical predictive model of a stochastic process P(X̆, X⃗) (i.e., joint distribution of the past X and future ). The classical predictive model is completed and specified by (a) a set of n internal states 5 = {5)}, j — 0, 1,...,n — 1 and (b) a set of transition matrices T — {Txk}, representing the probability that the machine in state Sj transitions to Skon the output of x.2. An efficiently computable binary function f that maps each possible future sequence of length up to L (i.e., x1, x2, ..., xL) into {0, 1}, that is, a binary number ‘1’ or a binary number ‘0’ with respect to the future sequence of the stochastic process, whereby the binary numbers ‘1’ and ‘0’ indicate that the event is to occur and not occur, respectively, with respect to the future sequence of the stochastic process.3. The past q observations (past observation sequence, comprising a series of past observed symbols) of the stochastic process that have been observed x-q:0= x-q+1... x-2x-1, x0.4. A set of circuit parameters {mfc} (which defines the number of quantum search operators (e g., Grover operators) to apply for each measurement quantum circuit Ck) and the number of measurement shots Ns(which defines the number of measurements for each measurement quantum circuit Cfc). For example, the values of {mk} may be predefined (e.g., user-defined). In this regard, {mfc} may be defined in various ways such as based on the desired convergence rate of the searching algorithm, as well as based on the specific constraints of the quantum hardware Accordingly, it will be understood by a person skilled in the art that the present invention is not limited to any particular values of (mk) which may be defined as desired or as appropriate By way of examples only and without limitation, {mk} may be set as mk— 2kor simply according to various number patterns such as {0, 1, 2, 4, 8,...} or {0, 1, 2, 3, 4,...}. Accordingly, each circuit description of the plurality of circuit descriptions of the plurality ofmeasurement quantum circuits {Ck} is generated to comprise a predefined number mkof the circuit description of the quantum search operator.

[0055] The method or algorithm of quantum probability estimation of an event according to various example embodiments of the present invention, on completion, outputs p, an estimate of the probability p̂ = P(f = 1|x-p:0) that the future will exhibit the binary property f (i.e., the event is said to be true or (x) = 1). Note that this encompasses scenarios where one seeks the likelihood of a target string x0. Lby simply setting f = 1 if and only if x0:L= x̃0:L.

[0056] FIG. 8 depicts an example circuit schematic of the quantum probability estimator (quantum probability estimation (QPE) machine) 800 according to various example embodiments of the present invention, and a corresponding example quantum probability estimation algorithm (“Algorithm 1”) is provided later below. As shown in section (a) of FIG.8, in each round k, the QPE machine 800 executes the corresponding quantum circuit (measurement quantum circuit) CkNstimes, where Nsis a predefined positive integer constant, concluding with a measurement to sample if f is true. For example, the measurement at the end of executing each quantum circuit Ckgives a result x, which gives either (x) = 1 (meaning / is “true” on x) or f x) — 0 (meaning / is “false” on x). Each quantum circuit Ckcomprises mkquantum search operator circuits Q (e g., Grover operator circuits Q configured to perform Grover-type iterations), along with one application of the recurrent model JI (quantum statepreparation operator). Section (b) of FIG. 8 illustrates the recurrent quantum model JI which involves a single quantum memory (top wire) under repeated interaction with a string of L quantum systems (state transition operators I / tep) initialized in |0), enabling quantum models with reduced bias compared to classical models of identical memory dimension. Meanwhile, each Grover operator Q = AS0A†Sxinvolves forward and reverse applications of quantum state-preparation operator JI, together with the oracle Sxand a phase flip about the zero state (So). For example, for L time steps, the same Ustepis applied on the same memory register with L different output registers. Thus, with the recurrent model JI (quantum state-preparation operator), depending on the length of the future sequence, the memory register that holds the hidden state is repeatedly fed back in as input for the next step, creating a feedback loop. The number of loops depends on the future time steps L. This recurrent nature allows for easy extension to further time horizons, which is advantageously utilized according to various example embodiments of the present invention to produce the quantum state-preparation operator as shown in section (b) of FIG. 8. Returning to section (a), each round k produces Nsmeasurement outcomes. From this, the result hkis collated, which corresponds to the numberof measurement outcomes in hkwhich f was found to be true (i.e., f = 1). The set of all measurement outcomes (of all quantum circuit Cfe) is then collected. This information is subsequently collated and processed by a classical computer through maximum likelihood estimation to produce an estimate p of the probability of the event.

[0057] In various example embodiments, the QPE machine 800, or the corresponding QPE method or algorithm, may comprise three stages. At a first stage 820 (corresponding to section (b) of FIG. 8 and subroutine (b) (PrepareStateOperator - prepare quantum state-preparation operator I) of the QPE algorithm), the classical predictive model (comprising a set of causal states S and a set of indexed transition matrices {Tx}) for the event and the past x-q:0(past observation sequence, comprising a series of past observed symbols, of the stochastic process) serve as inputs, for producing the state-preparation operator 풜 as output. The past observation sequence is encoded into an initialization operator Uini(an example method of generating the initialization operator Uiniwill be described later below according to various example embodiments of the present invention), and the classical predictive model, or more specifically, the set of indexed transition matrices {Tx}, is encoded into a series of state transition operators Ustep. In various example embodiments, the series of state transition operators Ustepare each generated based on a set of Kraus operators KxKx...Kxobtained based on the set of state transition matrices {Tx} (an example method of generating the state transition operator I / Stepwill be described later below according to various example embodiments of the present invention). For example, the whole set of {Tx} is encoded into one Ustepoperator. Each Txis mapped to a specific Kx, and the set of { Kx} is mapped into a single Ustep. The size of the set of { Kx} is the same size as the set of {Tx}. Therefore, the series of I / Steprefers to applying the same Ustepoperator a certain number of time. Accordingly, the first stage comprises generating a circuit description of a quantum state-preparation operator 풜 based on a past observation sequence, comprising a series of past observed symbols x-q:0, of the stochastic process and the set of state transition matrices {Tx} of the classical predictive model, whereby the quantum state-preparation operator JI comprises an initialization operator Uinihaving encoded therein the past observation sequence x-q:0of the stochastic process; and a series of state transition operators Ustepfor a future sequence, comprising a series of future symbols, of the stochastic process, each state transition operator having encoded therein the set of state transition matrices {Tx}. For each symbol of the future target sequence, as illustrated in section (b) of FIG. 8, the unitary state transition operator Ustepis applied to the memory register and a new outputregister initially in the zero state (|0)). This sequence of operations thus defines the quantum state-preparation operator JI as:풜 = Ustep(ψ,L-1)... Ustep(ψ,1)Ustep(ψ,0)Uini(Equation 2) where Ustep(ψ,i)denotes the application of Ustepon the memory register and the / -th output register. The first stage (or subroutine (b)) thus generates a full circuit description of the quantum state-preparation operator JI, which can be compiled and executed on a quantum computer (e.g., universal gate-based quantum computer).

[0058] At a second stage 840 (corresponding to section (c) of FIG. 8 and subroutine (c) (PrepareGroverOperator - prepare Grover operator Q) of the QPE algorithm), the binary function is taken as input, and encoded into the operator Sx(phase oracle). For the searching predicate, this encoding may be done through a multi-control single-qubit gate. Along with the JI operator from the first stage (or the subroutine (b)), the second stage (or subroutine (c)) thus generates a full circuit description of the quantum search operator (e.g., the Grover operator) Q as output. Accordingly, the second stage comprises generating a circuit description of a quantum search operator (e g., the Grover operator) Q based on the circuit description of the quantum state-preparation operator JI and an event indication function (e g., the binary function f), whereby the event indication function is configured to indicate whether the event is to occur with respect to the future sequence of the stochastic process, and the quantum search operator Q is configured to facilitate a search for a particular future sequence, comprising a particular series of future symbols, of the stochastic process that results in an indication that the event is to occur. In various example embodiments, as shown in section (c) of FIG. 8, the quantum search operator (e.g., the Grover operator) Q comprises a reverse quantum statepreparation operator Jl^ having a reverse configuration of the quantum state-preparation operator JI, a forward quantum state-preparation operator JI being the quantum statepreparation operator JI, a state reflection operator Soarranged between the forward and reverse quantum state-preparation operators, and a phase oracle Sxarranged before the reverse quantum state-preparation operator, and more specifically, JlS0Jl',tSx.

[0059] Subroutines (b) and (c) thus generate the full circuit descriptions (or circuit components) of the quantum state-preparation operator JI and the quantum search operator (e.g., the Grover operator) Q for subroutine (a), which can be fully generated on a classical computer.

[0060] At a third stage 860 (corresponding to section (a) of FIG. 8 and subroutine (a) (QAE - performs the quantum amplitude estimation) of the QPE algorithm), subroutine (a) takes as input the operators 풜 and Q, along with the predefined {mfc} and Ns, to configure or compile a sequence of measurement quantum circuits {Ck} on the quantum computer, where each measurement quantum circuit use mkapplications of the quantum search operator Q. The QPE machine 800 compiles these quantum circuits {Ck} onto the quantum computer (e g., a gatebased quantum computer) and executes them, performing Nsmeasurements per quantum circuit Ck. For example, operators 풜 and Q are configured as circuit descriptions or instructions on which mathematical operation to apply on which qubits, or if compiled into the basis gates of the quantum computer, the specific sequence of gates to apply on the corresponding qubits to achieve the mathematical transformation. To assemble operators into a circuit description for a quantum circuit, the circuit descriptions or instructions of the operators may be appended in a specific order, and acting on the corresponding qubits, such as illustrated in FIG. 8. The quantum circuit compilation step may obtain the high level circuit description (mathematical description) of a quantum circuit, and decompose them into low level gate / instructions that can be interpreted as a physical operation that can be performed on the quantum processor. The specific decomposition / compilation algorithm may defer across different quantum computer providers, but it is desired to achieve a set of basis instructions / gates to achieve the pre-defined mathematical transformation within a certain error tolerance. Accordingly, each quantum circuit Ckis executed a predefined number of times (Ns) to obtain the measurement result associated with the quantum circuit Ck. This yields a set of measurement results {hk}, where hkdenotes the number of times the target sequence is measured in quantum circuit Ck. These measurement results are then post-processed on a classical computer using maximum likelihood estimation to produce the final estimate p of the probability of the event.

[0061] Accordingly, the third stage 860 comprises generating a plurality of circuit descriptions for a plurality (or set) of measurement quantum circuits {Ck}, respectively, for compiling the plurality of measurement quantum circuits on a quantum computer, wherein each circuit description of the measurement quantum circuit is generated based on the circuit description of the quantum state-preparation operator A and the circuit description of the quantum search operator Q. The quantum computer comprises a quantum memory register and a series of output registers respectively for the series of future symbols of the future sequence of the stochastic process, and each of the plurality of measurement quantum circuits Ckformedon the quantum computer communicatively couples to the quantum memory register and the series of output registers. The third stage 860 further comprises, for each of the plurality of circuit descriptions of the plurality of measurement quantum circuits {Ck}, compiling the circuit description of the measurement quantum circuit Ckon the quantum computer to form the measurement quantum circuit Ckthereon; and executing the measurement quantum circuits Ckformed on the quantum computer to obtain a measurement result hkassociated with the measurement quantum circuit Ckthat indicates a number of times that the measurement quantum circuit Ckproduces an indication that the event is to occur. The third stage 860 further comprises estimating a probability of the event based on the measurement results { hk} associated with the plurality of measurement quantum circuits {Ck} obtained.

[0062] In various example embodiments, as shown in FIG. 8, for each of the plurality of measurement quantum circuits {Ck}, the quantum state-preparation operator circuit JI of the measurement quantum circuit Ckcomprises an initialization operator circuit Uiniconfigured according to the initialization operator Uini; and a series of state transition operator circuits I / Stepconfigured according to the series of state transition operators I / Step. In this regard, as also shown in FIG 8, the initialization operator circuit Uinicommunicatively couples to the quantum memory register, and the series of state transition operator circuits I / Steprespectively communicatively couples to the series of output registers and communicatively couples to the quantum memory register. In various example embodiments, for the above-mentioned executing the measurement quantum circuit Ckformed on the quantum computer comprises initializing each output register of the series of output registers in a zero state |0).

[0063] Accordingly, in various example embodiments, each quantum circuit Ckmay involve the coherent concatenation of mkGrover operator (dimensionality-reduced Grover iterators) Q, and each Grover operator Q then contains a forward and reverse application of a quantum circuit JI (see section (b) of FIG. 8). Each quantum circuit Ckis then executed Nstimes, thus making use of:MN = Ns(2mk+ 1)k=0(Equation 3) applications of quantum state-preparation operator 풜

[0064] In various example embodiments, as shown section (a) 860 of FIG 8, each execution of a quantum circuit Ckproduces a measurement outcome of whether / is true, resulting in (M +1) · Nsoutputs. The measurement outputs may then be fed into a classical computer for processing, which may use the measurement data to provide a maximum likelihood estimation of p of the probability of the event. It will be appreciated by a person skilled in the art that the method and system for quantum probability estimation of an event according to various example embodiments of the present invention may be applied to estimate the probability of any event as desired or as appropriate, such as but not limited to, estimating the probability of a rise / fall in the stock market given the past market performance, where the event can be a few days of rising / falling market prices, or estimating the probability of increasing energy demand, where the event can be increasing power usages over the next few time steps. Therefore, it will be appreciated by a person skilled in the art that the method and system for quantum probability estimation of an event according to various example embodiments of the present invention is not limited to estimating the probability of any particular or specific event.

[0065] By way of an illustrative example, the above-mentioned example QPE algorithm (Algorithm 1) according to various example embodiments of the present invention is provided below.Algorithm 1 - Quantum Probability Estimation (QPE) machine Inputs:ℳc= (풮, 풯): Classical model consisting of the set of states S and the set of indexed state transition operators T ={TX}.f Binary function that marks the target subspace.x-q:0: Past sequence of process states of length q.{mk} number of Grover operators to applyNsnumber of measurements per circuitOutput: Probability estimate pprocedure QPEMachine(ℳc, f, x-q:0, {mk}, Ns)풜 ← PrepareStateOperator(풯, x-q:0)Q ← PrepareGroverOperator(풜, f)p̂ ← QAE({mk}, Ns, 풜, Q)return pt> Subroutine (a) QAE): performs the quantum amplitude estimation -procedure QAE({mk}, Ns, 풜, Q)M ← largest index of k in {mk}Initialise {hk}, the list of target state countsfor k E {0, doCk← circuit with 풜 and mkGrover operatorsExecute Ckon quantum device with Nsmeasurementshk← number of times the target state is measuredUpdate k-th entry of {hk} with hkp̂ ← MaximumLikelihoodEstimate({hk})return p▷ Subroutine (b) (PrepareStateOperator): prepare operator 풜 – procedure PrepareStateOperator(풯, x-q:0)Ustep← QuantumModel(풯)Uini← EncodePast(x-q:0, 풯)t> Details on QuantumModel and EncodePast described in Appendix A2 and A3 풜 ← construct operator from using Uiniand Ustepequation (2)return JI▷ Subroutine (c) (PrepareGroverOperator): prepare Grover operator Q – procedure PrepareGroverOperator(풜, f)Construct phase oracle Sxusing f:e,x) =(-H =1x( |x), otherwiseConstruct state reflection operator, S0= 2 |0...0) (0...0| - IQ ← 풜S0풜†Sxreturn Q

[0066] Accordingly, the QPE machine 800 or algorithm advantageously possesses a significant technical advance / effect by utilizing recurrent quantum circuits (RQCs) design JI that replaces canonical quantum state preparation. For example, the RQC design allows for straightforward scaling up of the operator JI to cover longer time horizon, by appending moreoutput registers and applying ( / stepcorrespondingly. This results in a linear increase in the number of gate / operations with the number of simulation time steps. This is in contrast to canonical quantum state preparation, which tries to construct a matrix operation to create a quantum state based on a desired probability distribution. The resulting circuit's gate depth increases exponentially with increasing time step, making it impractical. An alternative method uses parameterized quantum gates, such as qGAN, to train a circuit to encode the distribution. This approach resolves the issue of the circuit depth, but for every increase in the time step, a new circuit needs to be trained from scratch, making it resource intensive and not scalable. The RQCs begin with a quantum memory register of dimension dq, where dqis always bounded by above by n. A output registers are then prepared which encode the quantum sample, initialised in state | O)o10^... |, where |0)j is the output register to store the / '-the index of the sequence x0: L, initialised in the computational zero state. The dynamics of the ( / -machine is then primarily defined by two operators: (1) ( / ini, which initialises the memory qubit into certain appropriate quantum state |< >s) that depends on what pasts have been observed (past observation sequence of the stochastic process); and (2) [ / step which can be constructed directly the classical transition probabilities {Tx} supplied (example methods to identify these unitaries will be described in detail later below).

[0067] The ATini operator is a state preparation unitary operating on the memory register, which would require in general O(2llog2l0<?l I) CNOT Gates. The ( / step operator operates on both the memory register and the output register, with a total of [ log2|f)q | + [log2|DX| ] qubits, where Dxis the size of the output alphabet. This results in ( / step requiring O(4llog2 Dd+riog2 *A1) To simulate L time steps into the future, the ( / step unitary is implemented L times. Therefore, the CNOT gate complexity of implementing < A is of order O(LDqDx) In comparison, direct realisation of operator c / Z using systematic quantum-state vector preparation involves the synthesis of a gate on L ■ [log21 Dx|] qubits. The number of CNOT gates is then C(4LDy). Therefore, the construction of operator < A according to various example embodiments of the present invention gives an exponential reduction in the number of CNOT gates required with respect to the number of simulation time steps L. As such, this advantageously ensures an efficient bypass of the data loading problem.

[0068] The use of such quantum models in memory-limited regimes can then immediately reduce model bias. Specifically, forcing Dq< n can result in significantly less bias than classical counterparts with the same memory dimension In certain instances, quantum modelsof reduced dimension may exhibit no bias at all, and the memory needed to estimate any future event 0 biased between quantum and classical models may grow without bound. Since the gate complexity for implementing operator < A scales quadratically with Dq, such reduced memory costs can also translate to significantly reduced gate costs. As such, the quantum probability of an event estimator according to various example embodiments of the present invention provides a significant advance towards practical quantum-enhanced forecasting in stochastic systems.

[0069] It will be appreciated by a persons skilled in the art that the method of quantum probability estimation of an event, and the corresponding system, according to various example embodiments of the present invention may be applied or implemented in various practical applications, such as but not limited to, fields ranging from logistics to finance, where reliable estimates are essential for activities ranging from demand forecasting to derivative pricing and risk analysis. In particular, many financial institutions are required to maintain capital reserves that mitigate against the ri sk of financial losses. Thus, the greater the confidence in the estimates of risk, the more the amount of capital required for this purpose and, accordingly, apply more capital to revenue generating activities. Indeed, quantum-enhanced methods of risk analysis have already been identified by J. P. Morgan to hold significant future promise (Herman et al., “ Survey of Quantum Computing for Finance”, arXiv preprint, arXiv: 2201.02773 (2022)). The technical advancements associated with the method of quantum probability estimation of an event according to various example embodiments of the present invention greatly decrease the technological barriers to realising such technologies, thus enabling practical applications thereof. Meanwhile in supply chains, better estimates of the likelihood of logistic disruptions are critical to maintaining connectivity of the supply chain. In energy markets, accurate forecasting of high demand market conditions, is crucial for balancing supply and demand (e.g., see the 2021 Texas power crises (Levin et al., “Extreme weather and electricity markets: key lessons from the February 2021 Texas crisis”, Joule, 6(1) (2022), pages 1-7). The method of quantum probability estimation of an event, and the corresponding system, according to various example embodiments of the present invention enable greatly reduced bias and random errors when estimating such probabilities, thus resulting in significant monetary savings and service reliability.

[0070] An example Maximum Likelihood Amplitude Estimation (MLAE) will now be described according to various example embodiments of the present invention The MLAE uses multiple circuits defined by a fixed schedule mk, Ns}, followed by maximum likelihood estimation on the measurement results to obtain the probability estimate p of the desiredsequence. Given the probability of obtaining the desired sequence p = sin20a, for some unknown parameter 6a6 [0, TT / 2], for each of the circuit and its measurement results {hk, Ns— hk, a likelihood function of the following form may be obtained:Lfc(hk; 0) oc [sin2(0k)]hk[cos2(0k)]Ws-?lk, 0 < k < M(Equation 4) where 9k= (2mk+ l)0a. The unified likelihood function may then be defined by:ML(h;0) = ]~[Lfe(hfc;0)fc = 0(Equation 5)

[0071] A maximum likelihood estimation is then conducted to estimate 6a.§a= argmaxlnL(h; 0)eg[o,7r / 2](Equation 6)

[0072] This gives the probability estimate pp = sin26a(Equation 7)

[0073] It will now be shown that in the noiseless case, this estimator p is an unbiased estimator for the amplitude to be estimated. Suppose that an amplitude estimation algorithm is utilized for estimating the amplitude ^fp in:W - + TT^i'Pi)(Equation 8)where = 8^-. In the ideal setting where Nkis asymptotically large, the experimental data collected from the quantum processor will follow the distribution: / ifc / Ns= sin2((2mfc+ l)0a).(Equation 9)

[0074] Looking at the form of thi s maximum likelihood estimator, it can be seen that setting yk= sin2(0k) and tk= hkIn yk+ (Ns- hk) ln(l - yk), the following can be obtained:MlnL(h; 6) = hkln(sin2(0k)) + Ns— hk)ln(cos2(0k))k=0M= ^ h.klnyk+ (Ns- hk)ln(l - yk)k=0M=2 k=0c«(Equation 10) It follows that max0In L(h; 0) < k=omaxffc

[0075] Now consider:d > dO dd0tkdykdyktkdg h.k(l - yk) - (Ns- hk)yk~ dykykCL - Vk)_ dO (hk- Nsyk)dykyk(i ~ yk)(Equation 11)

[0076] Looking at each tkindividually, dstk— 0 at yk= sin2((2mk+ l)0a) — hk / Ns. Meanwhile, the second derivative dyktk= —hk / y — (Ns- hk) / (l - yk)2< 0. Thus, each tkhas either a turning-point, maximum or minimum at yk— sin2((2m + l)ga) = hk / Ns. Considering tk(yk) as a function of yk, it can be seen that each tkhas a uniformally negative second derivative, meanwhile t (y ) > 0 on the interval (0, 1) and cuts the axis only at yk= 0, 1 (precluding points of inflection inside this interval). This implies tkyk) is uniformally concave on the interval ykG [0, 1] with a single (global) maximum at yk= sin2((2mk+ l)0a) = hk / Ns= sin2((2mk+ 1)0O).

[0077] It follows that every tkindependently obtains a global maximum at 0 = 0a+ nn for n G Z.

[0078] Furthermore, since max In L(h; O') = max£kL0tk S^=omaxand simultaneously lnL(h; g = 0a) >k=0max tk, various example embodiments find that L(h; 0a) corresponds to the global maximum of L(h; 0) and thus 0a— argmaxlnL(h; 0). This indicates that p — sin20ais an unbiased estimator for a (in the absence of noise).

[0079] An example method of generating or constructing the Ustepoperator will now be described according to various example embodiments of the present invention. In the example method, the approach to constructing the unitary Ustepinvolves using Kraus operators. These operators, denoted as Kx, x G. act only on the quantum memory and satisfy the completeness relation:(KX^KX= Ix& X(Equation 12)

[0080] For a set of output alphabets x, a set of Kraus operators {Kx} can be defined, where x e x that describe the evolution of the quantum memory following each interaction with the ancillary qubit through the unitary matrix:Kx:= < |i / step|0>(Equation 13) Specifically,Kxj = (i| <x|C7step|j> |0)(Equation 14)

[0081] These Kraus operators can be obtained from the transition matrices {T } from the ( -machine, through the following relation:(Equation 15) where the exact relation can be computed using the methods in Yang et al., “Matrix Product States for Quantum Stochastic Modeling”, Phys. Rev. Lett. 121, 260602 (2018). With the definition of the Kraus operators of the process, we can define several columns of the unitary operator l / step, with the remaining matrix elements being populated according to the definition in Equation (14), for example, through a Gram-Schmidt process. These steps define the QuantumModel(T) function / operation in the QPE algorithm (Algorithm 1) described hereinbefore

[0082] An example method of generating or constructing the set of initialiser matrices {( / nwiHnowbedescribed according to various example embodiments of the present invention. The initialiser matrices, {( / nJ, i — {0,..., d — 1} where d is the number of quantum causal states, are used to initialise the memory register to the correct initial quantum causal state at the start of the simulation. Similar to the construction of the l / stepoperator, two example ways to construct the initialiser matrices are described below.

[0083] A first method utilizes causal states. In particular, the first method to create the initialiser matrices involves defining the set of causal states in the computational basis, as described in Binder et al., “A practical, unitary simulator for non-Markovian complex processes”, Physical Review Letters 120 (2018), 10.1103 / PhysRevLett.l20.240502,arXiv: 1709.02375. To create the operators necessary for the quantum circuit, a set of square matricesis established, i = 1,..., d of size n X n, n = 2^lo8zwith the following structure:000(Equation 16) where the first column is defined by the quantum causal state. Polar decomposition can be used to obtain the initialising unitary matrix U-niof the same size as follows:At = UiniP(Equation 17) where P is some positive semidefinite Hermitian matrix. Through this method, a set of initialiser matrices U-niis obtained, i = 0,where each causal state |s is mapped to a unique initialiser matrix Uni.

[0084] A second method utilizes Kraus operators. The second method for generating the initialiser matrices involves the use of Kraus operators associated with the process, which can be obtained from the transition matrices as in Equation (15). This method is used to define the EncodePast fun ction / op eration used in in the QPE algorithm (Algorithm 1) described hereinbefore. The unifilar property of the c-machine allows for the unique determination of the initial quantum causal state given a past sequence that is equal to or longer than the Markov order of the process. If a Markov order N of the process and a specific past x_W:0= x_N...x_2x_1is given, this past sequence will, by definition, be an element of a particular quantum causal state |s. This quantum causal state |Sj) can be produced through the repeated application of the respective Kraus operators:|Si) = KX-^KX-^KX~^\4>)(Equation 18) where |< >) is in any arbitrary state. A suitable set of pasts {%} will allow the construction of the full set of causal states, which can be converted into initialiser matrices using the aforementioned method.

[0085] An example discretization of AR(2) process will now be described according to various example embodiments of the present invention.

[0086] The / 17?(2) process over real values is defined as:+ <f)2Xt-2+ Q(Equation 19)where <p1, <p2G IR and Et~ lV(0,a2)(Equation 20) for some cr| > 0. The output space is discretized into two different output symbols:(D)_ t-l, ifXt< 0cI 1, ifXt> 0(Equation 21)

[0087] Since the process has a Markov order of 2, the full set of states is given as {(— 1)(— 1), (— 1)(1), (1)(— 1), (1)(1)} The output symbols from each of these states can be computed exactly:(-1)(-1): Xt= 01- -l + 02- -l + et=et ~ 01—02 (Equation 22) (-1)(-1): Xt= 01- l + 02- -l + et=Ct + 01 “ 02(Equation 23) (l)C-l): Xt= 01- -l + 02- l + Q = G “ 01 + 02(Equation 24) (1)(1): Xt= et+ ^ + < / >2(Equation 25)

[0088] Various example embodiments define:Pi = P(.et > 0i + 02)(Equation 26) p2= P(et> -0i - 02)(Equation 27) p3- P(et> 0i - 02)(Equation 28) P4= P^t > -0i - 02)(Equation 29)|0089] With this, the definition of the discretization process defined in Equation (21) and the dynamics of the undiscretized process defined by Equations (22) to (25) gives the hidden Markov model shown in FIG. 9A.

[0090] Each state may then be rewritten with the causal state notation st, for i = 0, 1, 2, 3, and map the output values from { — 1, 1 } to {0, 1 } for ease of coding, as shown in FIG. 9B.

[0091] Thus, the following example transition matrices may be obtained.0 1 — p40 0To _0 00 1 - p30 l - p20 0 ’.0 0 0 1 — px.p40 0 0"T! _ 0 0 p30p20 0 0.0 0 p40.(Equation 30) where for i,j G {0, 1, 2, 3}, k G {0, 1}, T-j corresponds to P(xt— k,st— S ls^ = s;).

[0092] A dimensional reduction comparison with the Causal-State Splitting Reconstruction (CSSR) algorithm will now be described according to various example embodiments of the present invention. In particular, another comparison of dimensionally reduced quantum model with another equivalent classical model is provided. As mentioned, the quantum model is truncated using the methods in [8], where the memory dimension is set to two. In this comparison, the Causal-State Splitting Reconstruction (CSSR) algorithm (by Shalizi et al., “Blind construction of optimal nonlinear recursive predictors for discrete sequences”, in Proceedings of the 20th Conference on Uncertainty in Artificial Intelligence, UAI ’04 (AUAI Press, Arlington, Virginia, USA, 2004), pages 504 - 511) is utilized to create the truncated classical model To discover the two-state e-machine for the approximate process, the CSSR algorithm was fed a time series of length ten million, generated from the full AR(2) model, and its look-back length A was set at 1. Both the classical and quantum models were used to estimate a future x0:4of length four with a past of x_3:0of length three. The results are shown in FIG.10, where the absolute error in the quantum estimates is demonstrated to be lower than that of the classical estimates. In particular, FIG. 10 shows a comparison of the absolute error of the probability estimate between the true probability and the probabilities predicted by the approximate quantum and classical models over all futures of length four given a past of length 3.

[0093] While embodiments of the invention have been particularly shown and described with reference to specific embodiments, it should be understood by those skilled in the art that various changes in form and detail may be made therein without departing from the scope of the invention as defined by the appended claims. The scope of the invention is thus indicatedby the appended claims and all changes which come within the meaning and range of equivalency of the claims are therefore intended to be embraced.

Claims

CLAIMS1. A method of quantum probability estimation of an event in a stochastic process, wherein the stochastic process is modelled based on a finite state-machine model, and a classical predictive model for the event is modelled based on the stochastic process and comprises a set of states and a set of state transition matrices associated with the finite state-machine model, the method comprising:generating a circuit description of a quantum state-preparation operator based on a past observation sequence, comprising a series of past observed symbols, of the stochastic process and the set of state transition matrices of the classical predictive model, wherein the quantum state-preparation operator comprises an initialization operator having encoded therein the past observation sequence of the stochastic process; and a series of state transition operators for a future sequence, comprising a series of future symbols, of the stochastic process, each state transition operator having encoded therein the set of state transition matrices;generating a circuit description of a quantum search operator based on the circuit description of the quantum state-preparation operator and an event indication function, wherein the event indication function is configured to indicate whether the event is to occur with respect to the future sequence of the stochastic process, and the quantum search operator is configured to facilitate a search for a particular future sequence, comprising a particular series of future symbols, of the stochastic process that results in an indication that the event is to occur;generating a plurality of circuit descriptions for a plurality of measurement quantum circuits, respectively, for compiling the plurality of measurement quantum circuits on a quantum computer, wherein each circuit description of the measurement quantum circuit is generated based on the circuit description of the quantum state-preparation operator and the circuit description of the quantum search operator,for each of the plurality of circuit descriptions of the plurality of measurement quantum circuits:compiling the circuit description of the measurement quantum circuit on the quantum computer to form the measurement quantum circuit thereon; andexecuting the measurement quantum circuit formed on the quantum computer to obtain a measurement result associated with the measurement quantum circuit that indicates a number of times that the measurement quantum circuit produces an indication that the event is to occur; andestimating a probability of the event based on the measurement results associated with the plurality of measurement quantum circuits obtained.

2. The method according to claim 1, whereineach circuit description of the plurality of circuit descriptions of the plurality of measurement quantum circuits is generated to comprise a predefined number of the circuit description of the quantum search operator, andthe predefined number of the circuit description of the quantum search operator for each circuit description of the plurality of circuit descriptions of the plurality of measurement quantum circuits is different.

3. The method according to claim 1 or 2, whereinthe quantum computer comprises a quantum memory register and a series of output registers respectively for the series of future symbols of the future sequence of the stochastic process, andeach of the plurality of measurement quantum circuits formed on the quantum computer communicatively couples to the quantum memory register and the series of output registers.

4. The method according to claim 3, whereineach of the plurality of measurement quantum circuits comprises a quantum statepreparation operator circuit compiled according to the circuit description of the quantum statepreparation operator and one or more quantum search operator circuits compiled according to the circuit description of the quantum search operator, andfor each of the plurality of measurement quantum circuits, the quantum state-preparation operator circuit of the measurement quantum circuit comprises an initialization operator circuit configured according to the initialization operator; and a series of state transition operator circuits configured according to the series of state transition operators, wherein the initialization operator circuit communicatively couples to the quantum memory register, and the series of state transition operator circuits respectively communicatively couple to the series of output registers and communicatively couples to the quantum memory register.

5. The method according to claim 3 or 4, wherein said executing the measurement quantum circuit formed on the quantum computer comprises initializing each output register of the series of output registers in a zero state.

6. The method according to any one of claims 1 to 5, whereinthe quantum search operator comprises a reverse quantum state-preparation operator having a reverse configuration of the quantum state-preparation operator, a forward quantum state-preparation operator being the quantum state-preparation operator, a state reflection operator arranged between the forward and reverse quantum state-preparation operators, and a phase oracle arranged before the reverse quantum state-preparation operator, andthe phase oracle has encoded therein the event indication function.

7. The method according to any one of claims 1 to 6, wherein the quantum search operator is a Grover operator.

8. The method according to any one of claims 1 to 7, wherein the event indication function is a binary function configured to return a binary number ‘ 1 ’ or a binary number ‘0’ with respect to the future sequence of the stochastic process, wherein the binary numbers ‘ 1 ’ and ‘0’ indicate that the event is to occur and not occur, respectively, with respect to the future sequence of the stochastic process.

9. The method according to any one of claims 1 to 8, wherein the series of state transition operators are each generated based on a set of Kraus operators obtained based on the set of state transition matrices.

10. The method according to any one of claims 1 to 9, wherein each of the plurality of measurement quantum circuits is executed a predefined number of times to obtain the measurement result associated with the measurement quantum circuit.

11. The method according to any one of claims 1 to 10, wherein said estimating the probability of the event is based on a maximum likelihood estimation on the measurement results associated with the plurality of measurement quantum circuits obtained.

12. A system for quantum probability estimation of an event in a stochastic process, wherein the stochastic process is modelled based on a finite state-machine model, and a classical predictive model for the event is modelled based on the stochastic process and comprises a set of states and a set of state transition matrices associated with the finite state-machine model, the system comprises:at least one memory; andat least one processor communicatively coupled to the at least one memory and configured to:generate a circuit description of a quantum state-preparation operator based on a past observation sequence, comprising a series of past observed symbols, of the stochastic process and the set of state transition matrices of the classical predictive model, wherein the quantum state-preparation operator comprises an initialization operator having encoded therein the past observation sequence of the stochastic process; and a series of state transition operators for a future sequence, comprising a series of future symbols, of the stochastic process, each state transition operator having encoded therein the set of state transition matrices;generate a circuit description of a quantum search operator based on the circuit description of the quantum state-preparation operator and an event indication function, wherein the event indication function is configured to indicate whether the event is to occur with respect to the future sequence of the stochastic process, and the quantum search operator is configured to facilitate a search for a particular future sequence, comprising a particular series of future symbols, of the stochastic process that results in an indication that the event is to occur;generate a plurality of circuit descriptions for a plurality of measurement quantum circuits, respectively, for compiling the plurality of measurement quantum circuits on a quantum computer, wherein each circuit description of the measurement quantum circuit is generated based on the circuit description of the quantum state-preparation operator and the circuit description of the quantum search operator;for each of the plurality of circuit descriptions of the plurality of measurement quantum circuits:compile the circuit description of the measurement quantum circuit on the quantum computer to form the measurement quantum circuit thereon; andexecute the measurement quantum circuits formed on the quantum computer to obtain a measurement result associated with the measurement quantum circuit that indicates anumber of times that the measurement quantum circuit produces an indication that the event is to occur; andestimate a probability of the event based on the measurement results associated with the plurality of measurement quantum circuits obtained.

13. The system according to claim 12, whereineach circuit description of the plurality of circuit descriptions of the plurality of measurement quantum circuits is generated to comprise a predefined number of the circuit description of the quantum search operator, andthe predefined number of the circuit description of the quantum search operator for each circuit description of the plurality of circuit descriptions of the plurality of measurement quantum circuits is different.

14. The system according to claim 12 or 13, whereinthe quantum computer comprises a quantum memory register and a series of output registers respectively for the series of future symbols of the future sequence of the stochastic process, andeach of the plurality of measurement quantum circuits formed on the quantum computer communicatively couples to the quantum memory register and the series of output registers.

15. The system according to claim 14, whereineach of the plurality of measurement quantum circuits comprises a quantum statepreparation operator circuit compiled according to the circuit description of the quantum statepreparation operator and one or more quantum search operator circuits compiled according to the circuit description of the quantum search operator, andfor each of the plurality of measurement quantum circuits, the quantum state-preparation operator circuit of the measurement quantum circuit comprises an initialization operator circuit configured according to the initialization operator; and a series of state transition operator circuits configured according to the series of state transition operators, wherein the initialization operator circuit communicatively couples to the quantum memory register, and the series of state transition operator circuits respectively communicatively couple to the series of output registers and communicatively couples to the quantum memory register.

16. The system according to claim 14 or 15, wherein said execute the measurement quantum circuit formed on the quantum computer comprises initializing each output register of the series of output registers in a zero state.

17. The system according to any one of claims 12 to 16, whereinthe quantum search operator comprises a reverse quantum state-preparation operator having a reverse configuration of the quantum state-preparation operator, a forward quantum state-preparation operator being the quantum state-preparation operator, a state reflection operator arranged between the forward and reverse quantum state-preparation operators, and a phase oracle arranged before the reverse quantum state-preparation operator, andthe phase oracle has encoded therein the event indication function.

18. The system according to any one of claims 12 to 17, wherein the quantum search operator is a Grover operator.

19. The system according to any one of claims 12 to 18, wherein the event indication function is a binary function configured to return a binary number ‘1’ or a binary number ‘0’ with respect to the future sequence of the stochastic process, wherein the binary numbers ‘1’ and ‘0’ indicate that the event is to occur and not occur, respectively, with respect to the future sequence of the stochastic process.

20. The system according to any one of claims 12 to 19, wherein the series of state transition operators are each generated based on a set of Kraus operators obtained based on the set of state transition matrices.

21. The system according to any one of claims 12 to 20, wherein each of the plurality of measurement quantum circuits is executed a predefined number of times to obtain the measurement result associated with the measurement quantum circuit.

22. The system according to any one of claims 12 to 21, wherein the at least one processor is further configured to estimate the probability of the event based on a maximum likelihood estimation on the measurement results associated with the plurality of measurement quantum circuits obtained.

23. The system according to any one of claims 1 to 22, further comprising the quantum computer.

24. A computer program product, embodied in one or more non-transitory computer-readable storage mediums, comprising instructions executable by at least one processor to perform the method of quantum probability estimation of an event in a stochastic process according to any one of claims 1 to 11.

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