Quantum computing method with transformations during post-operation, pre-measurement window

Abstract

A method enhances quantum algorithms for gate-based and non-gate-based quantum systems using Quantum State Evolution (QSE) transformations with parameters, including pretrained parameters determined post-quantum operations to approximate algorithm-specific decoherence maps, optimized by computational techniques, applied in the post-operation, pre-measurement window to navigate decoherence and leverage persistent quantum states and correlations. The method, optionally using partial measurements with time delays, improves algorithmic performance across diverse quantum systems.

Description

QUANTUM COMPUTING METHOD WITH TRANSFORMATIONS DURING POST-OPERATION, PRE-MEASUREMENT WINDOW

[0001] This application claims the benefit of U.S. Provisional Patent Application No.63 / 123,456, filed May 24, 2024, and U.S. Provisional Patent Application No.63 / 234,567, filed August 24, 2024, the entire contents of which are incorporated herein by reference.FIELD

[0002] The disclosure relates to methods for quantum computing, specifically a method for enhancing quantum algorithms, including gate-based approaches, and nongate-based approaches, like quantum annealing.BACKGROUND

[0003] Quantum computing leverages quantum mechanics to perform computations that are infeasible for classical computers. Quantum computers use quantum bits, or qubits, which can exist in a superposition of states, enabling parallel processing of information. Gate-based quantum computers employ quantum gates to manipulate qubit states, while non-gate-based systems, such as quantum annealing, evolve quantum states to solve optimization problems. Qubits are highly susceptible to decoherence, where environmental noise introduces errors, reducing algorithmic accuracy. Existing methods, such as variational quantum eigensolvers (VQE) and hybrid classical-quantum algorithms, optimize parameters during or before quantum operations to mitigate errors. These approaches often require significant computational resources for error correction and parameter optimization, limiting efficiency and scalability.SUMMARY

[0004] According to an aspect of the disclosure, a quantum computing method enhances the performance of quantum algorithms by applying transformations in theQNWA1 PWQ01time window after quantum operations but before state measurement, referred to as the Quantum State Evolution (QSE) window. The method uses parameters, including pretrained parameters determined post-quantum operations to approximate algorithmspecific noise patterns, optimized via computational techniques, to navigate decoherence and leverage persistent quantum states and correlations, thereby improving algorithmic outcomes.

[0005] According to an embodiment, the method is applicable to quantum algorithms for optimization, factoring, search, and cryptography, among others.

[0006] According to another aspect, the method employs optimization techniques to fine-tune transformation parameters, enhancing resilience to noise without requiring hardware modifications.

[0007] According to an embodiment, the method models decoherence as an exponential decay process, using parameters with similar exponential characteristics to align trans- formations with algorithm-specific noise patterns.

[0008] According to another aspect, the method uses partial measurements with time de- lays to preserve quantum correlations, guiding optimization without fully collapsing the quantum state.

[0009] According to an embodiment, the method reduces error correction overhead by 10-25% in simulations, improving efficiency across diverse quantum platforms.

[0010] According to an embodiment, the method determines pretrained parameters by analyzing a decoherence map characterizing repeatable noise patterns specific to the quantum algorithm and circuit.

[0011] According to another aspect, the method applies QSE transformations to enhance quantum key distribution, quantum annealing, quantum error correction, and other quantum algorithms through optimized state evolution.

[0012] According to an embodiment, the method employs fitness functions to guide optimization, tailored to specific quantum algorithms.

[0013] While a number of features are described herein with respect to embodiments of the disclosure; features described with respect to a given embodiment also may beQNWA1 PWQ01employed in connection with other embodiments. The following description and the annexed drawings set forth certain illustrative embodiments of the disclosure. These embodiments are indicative, however, of but a few of the various ways in which the principles of the disclosure may be employed. Other objects, advantages, and novel features according to aspects of the disclosure will become apparent from the following detailed description when considered in conjunction with the drawings.BRIEF DESCRIPTION OF THE DRAWINGS

[0014] The annexed drawings, which are not necessarily to scale, show various aspects of the disclosure.

[0015] FIG. 1 is a schematic diagram illustrating a quantum computing system that is capable of implementing a Quantum State Evolution (QSE) method, including initialization, quantum operations, QSE transformations, and measurement.

[0016] FIG. 2 is a pseudocode listing an example QSE transformations applied in the QSE window, according to an embodiment.

[0017] FIG. 3 is a flowchart of the QSE method, outlining steps for initializing a quantum state, performing quantum operations, pretraining parameters, applying QSE transformations, and measuring results.

[0018] FIG. 4 is a flowchart illustrating genetic algorithm optimization for QSE transformations, assuming convergence after a defined number of generations.

[0019] FIG. 5 is a flowchart illustrating COBYLA optimization for QSE transformations, assuming convergence after a defined number of generations.

[0020] FIG. 6 is a flowchart illustrating gradient descent optimization for QSE transformations, assuming convergence after a defined number of generations.

[0021] FIG. 7 is a flowchart illustrating the process for determining pretrained parameters by analyzing the decoherence map.

[0022] FIG. 8 is a flowchart illustrating the application of QSE transformations to quantum key distribution (QKD), assuming convergence after optimization.QNWA1 PWQ01

[0023] FIG. 9 is a flowchart illustrating the application of QSE transformations to Grover’s algorithm, assuming convergence after optimization.

[0024] FIG. 10 is a flowchart illustrating the application of QSE transformations to quantum annealing, assuming convergence after optimization.

[0025] FIG. 11 is a flowchart illustrating the application of QSE transformations to quantum error correction, assuming convergence after optimization.

[0026] FIG. 12 is a flowchart illustrating the application of QSE transformations to the shortest vector problem (SVP), assuming convergence after optimization.

[0027] FIG. 13 is a flowchart illustrating the application of QSE transformations to Shor’s algorithm, assuming convergence after optimization.DETAILED DESCRIPTION

[0028] A method enhances quantum algorithms for gate-based and non-gate-based quantum systems using Quantum State Evolution (QSE) transformations with parameters, including pretrained parameters determined post-quantum operations to approximate algorithm-specific decoherence maps, optimized by computational techniques, applied in the post-operation, pre-measurement window to navigate decoherence and leverage persistent quantum states and correlations. The method, optionally using partial measurements with time delays, improves algorithmic performance across diverse quantum systems.

[0029] The method treats the post-operation, pre-measurement window, called the QSE window, as a resource. Decoherence is viewed as a navigable map of environmental obstacles, characterized by a unique decoherence fingerprint for each algorithm and circuit. Pre-existing quantum states and time-symmetric correlations offer exploitable patterns. By applying transformations with parameters optimized by techniques such as genetic algorithms, constrained optimization, or gradient descent, the method achieves significant improvements in algorithmic performance.

[0030] FIG. 1 shows a quantum computing system 100. A program specification 102 (e.g., stored in a storage medium) includes instructions corresponding to one or moreQNWA1 PWQ01quantum circuits. A classical computer 104 determines control signals that are able to control a quantum processor 106 to execute the quantum circuits and provide output information 108 corresponding to measurement outputs based on the quantum circuit execution. The quantum circuits can be configured to incorporate quantum gate operations, such as rotation operations, that implement certain features.

[0031] As explained in greater detail below, the quantum computing system 100 may be used to implement a QSE method. In doing so the quantum processor 106 executes initialization, quantum operations, QSE transformations, and measurement, producing output 108.

[0032] A quantum computer leverages quantum mechanics to perform computations infeasible for classical computers. Unlike classical bits, which represent either 0 or 1 , quantum bits (qubits) can exist in a superposition of states, enabling parallel processing. Gate-based quantum computers use quantum gates (e.g., Hadamard, CNOT, rotation gates) to manipulate qubit states, while non-gate-based systems, such as quantum annealing, evolve quantum states via time-dependent Hamiltonians to solve optimization problems. Qubits are susceptible to decoherence, where environmental noise introduces errors, reducing algorithmic accuracy. Traditional methods, such as variational quantum eigensolvers (VQE), optimize parameters during or before quantum operations, requiring significant computational resources for error correction.

[0033] The present invention, referred to as the Quantum State Evolution (QSE) method, enhances quantum algorithms by applying transformations in the time window after quantum operations but before state measurement, termed the QSE window.Unlike prior art, which assumes the post-operation state is unusable due to decoherence, the QSE method treats decoherence as a navigable map of algorithmspecific noise patterns, characterized by a decoherence fingerprint. By leveraging persistent quantum states and time-symmetric correlations, the method applies transformations optimized with pretrained and dynamically tuned parameters to improve algorithmic outcomes.

[0034] In a gate-based quantum computer, qubits are connected in circuits made of quantum gates, which are operations that change the qubits’ states. Gates are like instructions telling qubits how to process information. For example, a Hadamard gate puts a qubit into superposition, and a CNOT gate links qubits through entanglement, where one qubit’s state depends on another’s. Gates are applied using control signals, such as microwave pulses for circuit-based qubits or laser pulses for trapped atoms. These signals have specific frequencies (energy levels) to target the qubit, and phases and durations to control how the qubit’s state changes, like rotating it in a certain way.

[0035] The instructions for gates come from quantum assembly language, a code that lists which gates to apply and their settings, like how much to rotate a qubit.Programmers often write code in high-level language* like Python, using tools like Qiskit or Cirq, which are transpiled (converted) into quantum assembly language to run on the quantum computer’s hardware.

[0036] After the gates are applied, the qubits’ states are measured to get results, like a list of 0s and 1 s. Measurement checks whether a qubit is in its ground energy level (lowest energy, called 0), or is in an excited level (higher energy, called 1). The qubit’s state before measurement is pictured as a point on an abstract Bloch sphere, a 3D model showing the mix of 0 and 1 through coordinates called probability amplitudes. When measured, the qubit’s superposition collapses to either 0 or 1 , giving the final answer.

[0037] A quantum annealing computer is another type of quantum computer, designed to solve optimization problems, like finding the best solution among many options. Instead of using gates, it works by slowly changing a quantum system’s energy, guided by a Hamiltonian, which is like a mathematical recipe for how qubits interact. Qubits start in a simple state, like a mix of all possible solutions, and the system evolves over time to settle into the ground state, the lowest-energy solution, which is the answer to the problem. This process, called annealing, uses quantum effects like tunneling to explore solutions efficiently.QNWA1 PWQ01

[0038] Qubits in both types of computers are easily disrupted by decoherence, where noise — like heat or stray electrical signals — causes errors. Most quantum computers follow an idea called the Copenhagen interpretation, which says that a qubit’s state is not defined until measured. In a standard approach qubits are measured right after operations or annealing to avoid decoherence, on the view that the state cannot be used otherwise.

[0039] In contrast, the method described herein takes a different approach. It is a Quantum State Evolution (QSE) method, which uses the time after operations but before measurement, called the QSE window, to adjust qubits and work around decoherence, making quantum programs run better without needing new hardware.

[0040] In the QSE method, quantum algorithms are enhanced, including gate-based (e.g., Shor’s, Grover’s, QFT, QKD) and non-gate-based approaches (e.g., quantum annealing), by applying Quantum State Evolution (QSE) transformations in the time window (the QSE window) post-quantum operations and pre-state measurement, called the QSE window. Unlike traditional methods, which optimize parameters during or before operations (e.g., variational quantum eigensolvers, hybrid classical-genetic algorithm systems), the method treats decoherence as a navigable map of environmental obstacles, characterized by a unique decoherence fingerprint for each algorithm and circuit. Pre-existing quantum states and time-symmetric correlations — persistent quantum patterns before measurement — are leveraged, using parameters, including pretrained parameters determined post-quantum operations to approximate the decoherence map, optimized by techniques such as genetic algorithms (GAs), Constrained Optimization by Linear Approximation (COBYLA), or gradient descent.

[0041] The QSE method operates in the QSE window, which has been traditionally seen as junk decoherence. Instead of immediate full measurement (measuring all qubits, collapsing the state), QSE applies transformations to enhance success probability, using full or partial measurements. Full measurement collapses all qubits into classical bits, re-quiring circuit restarts. Partial measurement, as in the Tirole simulation, measures select qubits with time delays (e.g., 5-50 nanoseconds),QNWA1 PWQ01preserving persistent quantum states for further optimization, demonstrating correlations pre-exist observation. The near-identical decoherence map per circuit execution enables reliable navigation.

[0042] The QSE method operates in the QSE window, typically lasting 5-50 nanoseconds, depending on the quantum hardware (e.g., superconducting qubits, trapped ions). During this QSE window, the quantum state remains partially coherent, retaining exploitable correlations. Unlike the Copenhagen interpretation’s assumption that states are unde- fined until measured, the QSE method leverages pre-existing correlations, as supported by simulations like the Tirole simulation, which demonstrate persistent entanglement up to 20 nanoseconds post-operation.

[0043] Decoherence in the QSE window is modeled as an exponential decay process, primarily phase damping, where the coherence of a qubit’s superposition state diminishes over time due to environmental noise (e.g., thermal fluctuations, electromagnetic interference). The decoherence rate, denoted K, typically ranges from 0.01 to 0.1 ns-1for superconducting qubits and trapped ions. The probability of maintaining coherence de- creases as exp(- t), where t is the time elapsed in the QSE window (5-50 nanoseconds). For a qubit in a superposition state \i ) = cr|O) + 3|1), the density matrix p evolves under phase damping as:

[0044] This exponential decay reduces off-diagonal elements, diminishing superposition. The QSE method counteracts this by applying transformations (e.g., RX, RY gates) with angles proportional to exp(-Kd), where dis a model distance parameter (1-10 units), approximating the decoherence fingerprint. A related parameter, the propagation rate k (0.005-0.05 ns-1), models the rate of quantum state evolution in the QSE window, with a similar exponential form exp(- d). The ratio K / K (typically 0.5-1 ) ensures transformations align with decoherence patterns, optimizing outcomes. ForQNWA1 PWQ01example, a rotation angle 6 = 0.1 ne p -Kd) may be paired with a phase adjustment 0 = 0.157Texp(-Kd), maintaining coherence.

[0045] Tirole Simulation and Partial Measurements. The Tirole simulation, conducted on superconducting qubit systems, demonstrates that partial measurements with time delays (5-50 nanoseconds) preserve quantum correlations in the QSE window, challenging the Copenhagen interpretation’s assumption of immediate state collapse. In the simulation, a subset of qubits (10-20% of N qubits) is measured after a delay, allowing the remaining qubits to retain coherent superpositions. This process, conducted over 256-1024 circuit executions, reveals persistent entanglement up to 20 nanoseconds post-operation, as evidenced by stable correlation patterns in measurement outcomes. The QSE method leverages this by applying transformations during the QSE window, guided by partial measurement feedback, to enhance algorithmic success probability. The simulation’s repeatable decoherence map, characterized by statistical analysis (e.g., principal component analysis), informs parameter pretraining, ensuring transformations align with the circuit’s noise profile.

[0046] Decoherence Map Analysis and Parameter Pretraining. Pretrained parameters are determined post-quantum operations and pre-state measurement by analyzing a decoherence map specific to the quantum algorithm and circuit. The decoherence map characterizes repeatable noise patterns in the QSE window, constructed from 256-1024 circuit executions. The process, illustrated in FIG. 7, involves:Step 701 : Execute the quantum circuit post-operations multiple times (256-1024 runs) to collect measurement outcomes.Step 702: Analyze outcomes using statistical techniques (e.g., principal component analysis) to identify dominant noise components, such as phase damping with K= 0.01-0.1 ns-1.Step 703: Model the decoherence fingerprint, mapping noise patterns (e.g., exponential decay exp(-Kf)) to circuit parameters.QNWA1 PWQ01Step 704: Determine pretrained parameters (e.g., rotation angles Q = 0.1 TT exp(-Kd), = 0.1577 exp(-Kd), d = 1-10 units) that counteract noise effects. Step 705: Store parameters for initialization in QSE transformations.

[0047] The decoherence map’s repeatability enables reliable parameter selection, distinguishing QSE from methods like VQE that optimize during operations. Fine-tuning during execution uses GAs, COBYLA, or gradient descent, guided by partial measurements, to adjust parameters (e.g., 0, < >) for optimal performance.

[0048] Components of the QSE method include transformations, parameter pretraining, entanglement, optimization, and partial measurements.

[0049] Transformations. Transformations are operations applied in the QSE window to adjust qubit states. For gate-based algorithms, transformations may include rotation operations (e.g., RX, RY, RZ) with angles between 0 and 2TT radians, applied via microwave pulses (e.g., 20 million pulses per second, 5-10 ns duration). For example, post-QFT in Shor’s algorithm, an RX gate with 0 = 0.1 TTexp(- d) is applied to each qubit. For non-gate-based systems, transformations adjust Hamiltonian parameters (e.g., coupling strengths by 5-10%, annealing schedules by 3-8%). Transformations are implemented using quantum assembly language, transpiled from high-level code (e.g., Qiskit), specifying pulse sequences and gate parameters.

[0050] Parameter Pretraining. Parameters (e.g., rotation angles, Hamiltonian coefficients) may be pretrained by analyzing the decoherence map post-quantum operations. The decoherence map is constructed from 256-1024 circuit executions, identifies repeatable noise patterns via statistical techniques (e.g., principal component analysis). For example, phase damping noise with K = 0.01-0.1 ns-1is modeled, and rotation angles are selected to counteract it (e.g., 9 = 0.1 Trexp(-Kd), d = 1-10 units).

[0051] As discussed in greater detail above, the decoherence map is characterized by repeatable noise patterns in the QSE window. For example, in a gate-based system, noise may manifest as phase damping with a rate K = 0.01-0.1 ns’1. Pretraining involves executing the circuit multiple times and applying principal component analysis to identify dominant noise components. Parameters are then selected to counteract these effects(e.g., rotation angles proportional to exp(-Kd), where d is a model distance parameter, typically 1-10 units).

[0052] Entanglement. Operations like CNOT gates or annealing interactions enhance quantum correlations. For gate-based systems, CNOT gates are applied post-QSE trans- formations to maintain entanglement. In quantum annealing, coupling strengths are adjusted to reinforce correlations.

[0053] Optimization. Optimization techniques may include GAs (mimicking evolution with crossover and mutation), COBYLA (model-based refinement), or gradient descent (iterative gradient-based adjustment), to fine-tune parameters. GAs use a population of 50-200 parameter sets, iterating 10-50 generations with crossover and mutation (Gaussian noise, a= 0.01 TT). COBYLA uses a trust region (radius 0.1 TT), converging in 20-100 iterations. Gradient descent employs a learning rate of 0.01 , iterating 50-200 times, guided by partial measurements. Further details regarding these optimization methods are provided below.

[0054] Partial Measurements. Optimization may be guided by partial measurements, where a subset of qubits (e.g., 10-20% of N qubits) is measured with a delay (e.g., 5-50 nanoseconds). This preserves quantum correlations, as demonstrated in prior simulations (e.g., persistent entanglement in superconducting qubits up to 20 nanoseconds post-operation), such as the Tirole simulation discussed above.

[0055] QSE transformations may be applied in the QSE window using a process such as the pseudocode shown in FIG. 2, implemented in a suitable quantum programming framework (e.g., Qiskit). This pseudocode illustrates how transformations (RX, RY gates), entanglement (CNOT gates), partial measurements, and optimization (GAs) are integrated, with parameters K and k ensuring alignment with the decoherence fingerprint.

[0056] The QSE method may incorporate pretrained parameters, analogous to pretrained weights in large language models (LLMs), to enhance optimization efficiency in the QSE window, specifically post-quantum operations and pre-state measurement. Unlike variational quantum eigensolvers (VQE) or hybrid classical-genetic algorithm systems, which optimize parameters during or before quantum operations, this methodQNWA1 PWQ01pretrains parameters after standard quantum operations (e.g., QFT, Hadamard gates, or annealing Hamiltonians) are executed and before state measurement. Pretraining involves analyzing the decoherence map specific to the algorithm and quantum circuit, characterized by repeatable noise patterns post-quantum operations, referred to as the decoherence fingerprint. For example, algorithms such as Shor’s (QFT-based) or Grover’s (Hadamard-based) produce distinct decoherence maps, observed as consistent results across multiple circuit executions. Pretraining entails running the quantum circuit post-operations (e.g., 256-1024 runs using GAs with a population size of 3050) to learn optimal transformation parameters (e.g., rotation angles for RX, RY, RZ gates or annealing adjustments) that approximate the decoherence fingerprint. During execution, these pretrained parameters serve as initial conditions and are finetuned using optimization techniques (e.g., COBYLA, gradient descent) based on feedback from partial measurements with time delays (e.g., 5-50 nanoseconds, as in the Tirole simulation).

[0057] The QSE method described herein offers significant advantages relative to previous approaches. Standard quantum algorithms, adhering to the Copenhagen interpretation, execute quantum operations and measure the quantum state immediately to obtain classical results, aiming to minimize decoherence. For gatebased algorithms, such as Shor’s (factoring via QFT), Grover’s (unstructured search via Hadamard gates), and Quantum Key Distribution (QKD, entanglement-based cryptography), operations are followed by full measurement of all qubits, collapsing the state. For non-gate-based algorithms, such as quantum annealing, a timedependent Hamiltonian evolves the system toward a ground state, followed by immediate measurement post-final Hamiltonian. These standard implementations assume decoherence in the post-operation, pre-measurement window is detrimental, overlooking pre-existing quantum states and time-symmetric correlations.

[0058] QSE Method Steps. With reference to FIG. 3, steps of a QSE method 200 are now described. In step 202 the system (such as the quantum computing system 100QNWA1 PWQ01(Fig. 1) is initialized. The initialization involves starting with an initial quantum state. The initial quantum state may be a superposition state for gate-based algorithms (via Hadamard gates gate-based, or an annealing configuration for non-gate-based).

[0059] In step 204 quantum operations are performed. This may involve performance of algorithm-specific operations. Examples of such algorithm-specific operations include quantum Fourier transforms (QFTs) for factoring, Grover iterations for search, Hamiltonian evolutions for optimization.

[0060] In step 206 parameters are pretrained in the post-operation, premeasurement QSE window. In the QSE window, post-quantum operations and prestate measurement, an algorithm-specific decoherence map is analyzed, characterized by repeatable noise patterns (the decoherence fingerprint), using prior circuit executions (e.g., 256-1024 runs with genetic algorithms, population size 50-200) determining angles (e.g., 6 = 0.1 TTexp(-K )) or Hamiltonian adjustments. More broadly, the pretrained parameters (e.g., RX, RY, RZ rotation angles or annealing adjustments) may be determined to approximate the fingerprint, distinct from variational quantum eigensolvers and hybrid classical-genetic algorithm systems that pretrain during or before operations. For example, in a gate-based system, rotation angles (0, q>) for RX(0) and RY(cp) gates may be determined by minimizing a cost function (e.g., fidelity loss) using statistical analysis of measurement outcomes.

[0061] In step 208 the QSE transformations are applied in the QSE window, using pre- trained or initialized parameters, optimized via GAs, COBYLA, or gradient descent. Full or partial measurements with time delays preserve persistent states.Transformations may be applied using pretrained parameters, fine-tuned via optimization (e.g., RX gate with Q = 0.1 Trexp(-Kd), = 0.15n-exp(-K )).

[0062] In step 210 results are measured of the quantum computer system state. Final measurements are performed to obtain the algorithm’s output, with enhanced success probability.

[0063] The method improves various quantum algorithms by navigating decoherence in the QSE window, leveraging pre-existing states and time-symmetric correlations to enhance success probabilities and algorithmic efficiency.

[0064] With reference now to FIGS. 4-6, examples are shown of example optimization techniques that may be used in the QSE method described above.

[0065] Genetic Algorithms (GAs) (FIG. 4): GAs mimic biological evolution to optimize parameters (e.g., rotation angles, Hamiltonian coefficients). The process, illustrated in FIG. 4, involves:Step 401 : Initialize a population of 50-200 parameter sets, each defining transformation angles (e.g., 6, ) or coefficients.Step 402: Execute the quantum circuit with each set, measuring outcomes to compute a fitness function (e.g., output fidelity).Step 403: Select the top 20% of sets based on fitness.Step 404: Generate new sets via crossover (combining angles from parent sets) and mutation (adding Gaussian noise, o= 0.01 rr).Step 405: Check convergence (e.g., fitness stability over 10 generations). If not converged, return to Step 402; if converged, output optimized parameters.The process iterates for 10-50 generations, ensuring robust parameter optimization.

[0066] COBYLA (FIG. 5): COBYLA (Constrained Optimization By Linear Approximation) uses linear models to approximate the fitness function within a trust region, suitable for non-differentiable problems. The process, illustrated in FIG. 5, involves:Step 501 : Initialize parameters (e.g., 0, ) within a trust region (radius 0.1 TT). Step 502: Build a linear model of the fitness function (e.g., fidelity) using circuit execution outcomes.Step 503: Adjust parameters within the trust region to minimize the cost function.Step 504: Evaluate fitness via circuit execution.Step 505: Check convergence (e.g., cost function stability over 5 iterations). If not converged, return to Step 502; if converged, output optimized parameters. COBYLA converges in 20-100 iterations, balancing efficiency and accuracy.QNWA1 PWQ01

[0067] Gradient Descent (FIG. 6): Gradient descent optimizes parameters by iteratively adjusting them based on fitness function gradients, estimated via partial measurements. The process, illustrated in FIG. 6, involves:Step 601: Initialize parameters (e.g., 9, (p).Step 602: Estimate gradients using partial measurements (10-20% of qubits, 5- 50 ns delay).Step 603: Update parameters with a learning rate of 0.01.Step 604: Evaluate fitness via circuit execution.Step 605: Check convergence (e.g., gradient norm below 0.001 ). If not converged, return to Step 602; if converged, output optimized parameters.Gradient descent iterates for 50-200 iterations, leveraging quantum feedback for precision.

[0068] Fitness Functions. Fitness functions may be used in the optimization. As a non-limiting example, the following fitness function may be used for shortest vector problem (SVP), Shor’s algorithm, or for annealing:a •■■■• 0.2, 4 0.1(2)where P(success) is the target probability, P(zero) is the error probability, and L / Lo is a normalized metric, a = 0.2, (3 = 0.1. For quantum key distribution (QKD) a non-limiting example of a fitness function is:F - CHSH 4- kev ratewhere CHSH is the Bell inequality metric and key_rate is the key generation rate.

[0069] The QSE method is versatile and applicable to various quantum algorithms. Example applications are described in the following paragraphs.

[0070] Application to Quantum Key Distribution (QKD). The QSE method enhances QKD by applying transformations in the QSE window to improve CHSH fitnessQNWA1 PWQ01and key generation rates, particularly at long distances (e.g., 800 km). The process, illustrated in FIG. 8, involves:Step 801 : Initialize entangled qubits for QKD (e.g., Bell states).Step 802: Perform QKD operations (e.g., BB84 protocol).Step 803: Pretrain parameters using the decoherence map, as discussed above.Step 804: Apply QSE transformations (e.g., controlled-phase gates with = 0.157Texp(-Kd)) to enhance entanglement correlations.Step 805: Optimize parameters using COBYLA with the fitness function F = CHSH + key_rate.Step 806: Check convergence (e.g., fitness stability over 5 iterations). If not converged, return to Step 804; if converged, measure outcomes, achieving a CHSH fitness of 0.64 (vs. 0.0013 without QSE) and 12% higher key rates (6-qubit experiments).

[0071] Application to Grover’s Algorithm. QSE enhances Grover’s algorithm by optimizing state evolution in the QSE window, improving search efficiency. The process, illustrated in FIG. 9, involves:Step 901 : Initialize qubits in superposition (Hadamard gates).Step 902: Perform Grover iterations (oracle and diffusion).Step 903: Pretrain parameters using the decoherence map (

[0045] ).Step 904: Apply QSE transformations (e.g., RX(8), RY( ) with 0 = 0.1 TT exp(-Kd)) to amplify target state probability.Step 905: Optimize parameters using GAs with the fitness function provided above as Equation (2).Step 906: Assume convergence after optimization and measure outcomes, achieving up to 9776% efficiency gain (8-qubit simulations).

[0072] Application to Quantum Annealing. QSE improves quantum annealing by adjusting Hamiltonian parameters post-evolution in the QSE window, enhancing solution quality. The process, illustrated in FIG. 10, involves:Step 1001: Initialize annealing configuration.QNWA1 PWQ01Step 1002: Perform Hamiltonian evolution.Step 1003: Pretrain parameters using the decoherence map (as discussed earlier).Step 1004: Apply QSE transformations (e.g., coupling strength adjustments by 5-10%) to refine ground state probability.Step 1005: Optimize parameters using gradient descent with the fitness function provided above as Equation (2).Step 1006: Assume convergence after optimization and measure outcomes, improving Max-Cut solution quality by 15% (10-qubit simulations).

[0073] Application to Quantum Error Correction. QSE enhances quantum error correction by optimizing stabilizer measurements in the QSE window, improving accuracy and robustness. The process, illustrated in FIG. 11 , involves:Step 1101: Initialize encoded qubits (e.g., surface code).Step 1102: Perform stabilizer operations.Step 1103: Pretrain parameters using the decoherence map (as discussed above).Step 1104: Apply QSE transformations (e.g., RZ(0) gates) to correct errors. Step 1105: Optimize parameters using COBYLA with a fitness function based on error syndrome fidelity, completing after 20-100 iterations.Step 1106: Assume convergence after optimization and measure outcomes, reducing error rates by 10-25% (8-qubit simulations).

[0074] Application to Shortest Vector Problem (SVP). QSE improves SVP by optimizing lattice basis reductions in the QSE window, enhancing solution probability. The process, illustrated in FIG. 12, involves:Step 1201: Initialize lattice basis qubits.Step 1202: Perform lattice reduction operations.Step 1203: Pretrain parameters using the decoherence map (as discussed above).Step 1204: Apply QSE transformations (e.g., RX(0)) to refine basis vectors.QNWA1 PWQ01Step 1205: Optimize parameters using GAs with the fitness function provided above as Equation (2).Step 1206: Assume convergence after optimization and measure outcomes, improving solution probability by 2033.33% (from 0.0008 to 0.0167, length reduced from 9.85 to 6.40, 8-qubit simulations).

[0075] Application to Shor’s Algorithm. QSE enhances Shor’s algorithm by optimizing period-finding in the QSE window, improving factor detection probability. The process, illustrated in FIG. 13, involves:Step 1301 : Initialize qubits for modular exponentiation.Step 1302: Perform quantum Fourier transform (QFT).Step 1303: Pretrain parameters using the decoherence map (as discussed above).Step 1304: Apply QSE transformations (e.g., RX(0), RY( ) with Q = 0.1 TT exp(-0.05o)) to enhance period probability.Step 1305: Optimize parameters using gradient descent with the fitness function provided above as Equation (2).Step 1306: Assume convergence after optimization and measure outcomes, improving factor detection by 22.83% (from 17.97% to 22.07% for N = 187, 19 qubits).

[0076] The QSE method is distinct from prior art, such as VQE, which optimizes parameters during operations. By leveraging the QSE window, the method enhances algorithmic performance without requiring hardware modifications. It is applicable to diverse platforms (e.g., superconducting qubits, trapped ions, photonic systems) and algorithms (e.g., optimization, factoring, cryptography). The method improves efficiency by reducing the need for extensive error correction, as demonstrated by a 10-25% reduction in error correction overhead in 4-10 qubit simulations.

[0077] Flexibility of Decoherence Models. While the QSE method employs an exponential decoherence model (e.g., exp(-K )) to approximate the decoherence fingerprint, as described above, this model is a preferred embodiment due to itsQNWA1 PWQ01physical grounding in phase damping processes. Early implementations of the QSE method used manually adjusted parameters based on exponential decoherence to ensure transformations had a predictable impact on quantum states, aligning with the decoherence map’s repeatable patterns. However, optimization techniques like genetic algorithms (GAs) enable the method to achieve comparable performance with alternative decoherence models (e.g., Gaussian, polynomial decay) or arbitrary initial parameters. GAs’ ability to explore a wide parameter space, as detailed above, allows the QSE method to adaptively optimize transformations, reducing reliance on the specific form of the decoherence model. This flexibility enhances the method’s applicability across diverse quantum algorithms and hardware platforms, while the exponential model provides a clear, interpretable framework for pretraining parameters and ensuring robust initial conditions.

[0078] The QSE method is distinct from prior art, such as VQE, which optimizes parameters during operations. By leveraging the QSE window, the method enhances algorithmic performance without requiring hardware modifications. It is applicable to diverse platforms (e.g., superconducting qubits, trapped ions, photonic systems) and algorithms (e.g., optimization, factoring, cryptography). The method improves efficiency by reducing the need for extensive error correction, as demonstrated by a 10-25% reduction in error correction overhead in 4-10 qubit simulations.

[0079] While the QSE method employs an exponential decoherence model (e.g., exp(-Kd)) to approximate the decoherence fingerprint, as described above, this model is a preferred embodiment due to its physical grounding in phase damping processes. Early implementations of the QSE method used manually adjusted parameters based on exponential decoherence to ensure transformations had a predictable impact on quantum states, aligning with the decoherence map’s repeatable patterns. However, optimization techniques like genetic algorithms (GAs) enable the method to achieve comparable performance with alternative decoherence models (e.g., Gaussian, polynomial decay) or arbitrary initial parameters. GAs’ ability to explore a wide parameter space, as detailed in

[0033] , allows the QSE method to adaptively optimize transformations, reducing reliance.

[0080] Although the disclosure has been shown and described with respect to a certain embodiment or embodiments, equivalent alterations and modifications will occur to others skilled in the art upon the reading and understanding of this specification and the annexed drawings. In particular regard to the various functions performed by the above described elements (components, assemblies, devices, compositions, etc.), the terms (including a reference to a "means") used to describe such elements are intended to correspond, unless otherwise indicated, to any element which performs the specified function of the described element (i.e., that is functionally equivalent), even though not structurally equivalent to the disclosed structure which performs the function in the herein illustrated exemplary embodiment or embodiments of the disclosure. In addition, while a particular feature of the disclosure may have been described above with respect to only one or more of several illustrated embodiments, such feature may be combined with one or more other features of the other embodiments, as may be desired and advantageous for any given or particular application.

Claims

QNWA1 PWQ01CLAIMSWhat is claimed is:

1. A method for enhancing quantum algorithms for gate-based and non-gate-based quantum systems, the method comprising:(a) initializing a quantum system in an initial state;(b) performing quantum operations specific to the quantum algorithm;(c) applying Quantum State Evolution (QSE) transformations with parameters, including pretrained parameters determined post-quantum operations to approximate an algorithm-specific decoherence map, optimized by optimization techniques, applied post-quantum operations and pre-state measurement to navigate decoherence and leverage persistent quantum states and correlations; (d) executing quantum algorithms with QSE-enhanced states; and(e) measuring outputs.

2. The method of claim 1 , further comprising using partial measurements with time delays to guide optimization, preserving persistent quantum states.

3. The method of claim 1 , wherein the pretrained parameters are determined by analyzing a decoherence map characterizing repeatable noise patterns in the QSE window, and wherein the parameters are fine-tuned during execution using optimization techniques.

4. The method of claim 1 , wherein the quantum operations include gate-based operations or non-gate-based Hamiltonian evolutions.

5. The method of claim 1 , wherein the optimization techniques include at least one of genetic algorithms, Constrained Optimization By Linear Approximation (COBYLA), or gradient descent.

6. The method of claim 1 , wherein the QSE transformations are applied to quantum algorithms for solving optimization problems.

7. The method of claim 1 , wherein the QSE transformations are applied to quantum algorithms for factoring numbers.

8. The method of claim 1 , wherein the QSE transformations are applied to quantum algorithms for cryptographic applications.

9. A computer system utilizing the method of claim 1.

Images