arXiv subjects

Peter D. Drummond

Publications and source records attributed to Peter D. Drummond.

At least 19 recordsLinked to original sources

arXiv2026

Matrix phase-space representations for quantum symmetries

We introduce a general phase-space representation that includes global quantum symmetries in the basis expansion. This method, called matrix phase-space, projects the basis onto a reduced Hilbert space, which can greatly reduce sampling errors of many-body quantum simulations and unifies several previous phase-space methods. The purpose of this paper is to provide detailed proofs of basic theorems and operator identities. We also treat several different types of symmetries. To illustrate the benefits of matrix phase-space methods, we give a detailed derivation of a recent application to the topical problem of verifying the outputs of Gaussian boson sampling (GBS) quantum computers with photon number resolving detectors. This has exponential complexity, and using parity symmetry reduces sampling errors by very large factors relative to earlier methods.

quant-ph↗

arXiv2026

Matrix phase-space representations for gaussian boson sampling

We introduce coherent matrix phase-space distributions. These use conservation laws and symmetries to improve the accuracy and speed of quantum phase-space representations. As an example, this is applied to validation of low-loss Gaussian boson sampling (GBS) quantum computational advantage experiments, where classical generation of the random photon-number counts is exponentially hard. Large improvements in sampling errors are demonstrated compared to previous methods. Matrix phase-space representations also provide a large numerical speed-up, due to their (at worst) quadratic scaling, compared to other methods for validating total count probabilities of large-scale, low-loss GBS networks.

quant-ph↗

arXiv2026

Non-periodic Fourier propagation algorithms for partial differential equations

Spectral methods for partial differential equations (PDEs) with non-periodic boundary conditions arising in computational physics often use polynomial expansions on non-uniform grids. Here, we implement a Fourier method that employs fast trigonometric expansions on a uniform grid with non-periodic boundaries using fast discrete sine transforms (DST) or/and discrete cosine transforms (DCT) to solve parabolic PDEs. We implement this method in two ways: either using a Fourier spectral derivative or a Fourier interaction picture. Both methods can treat vector fields with a combination of Dirichlet and/or Neumann boundary conditions in one or more space dimensions. As examples, we use them to solve a variety of computational physics PDEs with analytical solutions, including the Peregrine solitary wave solution. For the 1D heat equation problem, our method with an interaction picture is accurate up to machine precision. Soluble examples of stochastic partial differential equation (SPDE) with non-periodic boundaries in one and two space dimensions, with physics and interdisciplinary applications are also treated. We compare the results obtained from these algorithms with publicly available solvers that use polynomial spectral methods, and study their relative performance and error scaling. Polynomial methods with non-uniform spatial grids have lower spatial discretization errors when the solutions change slowly in space, typically with large spatial grids. For problems with rapid spatial variation, Fourier methods can outperform polynomial expansions, owing to their smaller maximum space interval, and are generally faster due to the computational efficiency of discrete Fourier transform methods. We verified this by making a complexity analysis in which we studied the total error at the optimum combination of time and space steps for a given resource use.

math.NA↗

arXiv2026

Gaussian boson sampling: Benchmarking quantum advantage

Quantum computers solve intractable problems which classically require an exponentially long time to compute. With the development of large-scale experiments that claim quantum advantage, a vital issue has now emerged. What are the errors, and how do they affect the complexity of the problem solved? Large-scale Gaussian boson sampling (GBS) experiments give an example in which random numbers are generated. Despite classical hardness, these have computable benchmarks for checking data validity. While there are other quantum computing architectures, Gaussian boson sampling is uniquely testable at all scales. Several large, pioneering quantum computing (QC) experiments have been carried out to investigate quantum advantage. Here, we introduce a highly scalable but classical algorithm that can solve GBS approximately. Our numerical simulation of the output count data is closer to the exact solution than current experiments up to 1152 modes. This algorithm outperforms all previous classical, approximate algorithms and scales efficiently to larger experiments. Our results show that effects beyond losses can cause the errors that allow classical simulability. This work will lead to more precise algorithms and is a step towards understanding how QC quantum advantage is affected by the underlying physics.

quant-ph↗

arXiv2025

Validation tests of Gaussian boson samplers with photon-number resolving detectors

An important challenge with the current generation of noisy, large-scale quantum computers is the question of validation. Does the hardware generate correct answers? If not, what are the errors? This issue is often combined with questions of computational advantage, but it is a fundamentally distinct issue. In current experiments, complete validation of the output statistics is generally not possible because it is exponentially hard to do so. Here, we apply phase-space simulation methods to partially verify recent experiments on Gaussian boson sampling (GBS) implementing photon-number resolving (PNR) detectors. The positive-P phase-space distribution is employed, as it uses probabilistic sampling to reduce complexity. It is \sim10^{18} times faster than direct classical simulation for experiments on 288 modes where quantum computational advantage is claimed. When combined with binning and marginalization to improve statistics, multiple validation tests are efficiently computable, of which some tests can be carried out on experimental data. We show that the data as a whole shows discrepancies with theoretical predictions for perfect squeezing. However, a small modification of the GBS parameters greatly improves agreement for some tests. We suggest that such validation tests could form the basis of feedback methods to improve GBS experiments.

quant-ph↗

arXiv2024

The Quantum and Stochastic Toolbox: xSPDE4.2

This is the fourth major release of the xSPDE toolbox, which solves stochastic partial and ordinary differential equations, with applications in biology, chemistry, engineering, medicine, physics and quantum technologies. It computes statistical averages, including time-step and sampling error estimation. xSPDE can provide higher order convergence, Fourier spectra and probability densities. The toolbox has graphical output and $χ^{2}$ statistics, as well as weighted, projected, or forward-backward equations. It can generate input-output quantum spectra. The equations can have independent periodic, Dirichlet, and Neumann or Robin boundary conditions in any dimension, for any vector component, and at either end of any interval. xSPDE has functions that can numerically solve both ordinary and partial differential stochastic equations of any type, obtaining correlations, probabilities and averages. The toolbox has a core treating stochastic differential equations, with averages, probability distributions and full error estimates. There are stochastic extensions treating applications to partial differential equations, projected equations, quantum stochastic equations, master equations and quantum phase-space simulations including Gaussian boson sampling experiments.

quant-ph↗

arXiv2023

Simulating Gaussian boson sampling quantum computers

A growing cohort of experimental linear photonic networks implementing Gaussian boson sampling (GBS) have now claimed quantum advantage. However, many open questions remain on how to effectively verify these experimental results, as scalable methods are needed that fully capture the rich array of quantum correlations generated by these photonic quantum computers. In this paper, we briefly review recent theoretical methods to simulate experimental GBS networks. We focus mostly on methods that use phase-space representations of quantum mechanics, as these methods are highly scalable and can be used to validate experimental outputs and claims of quantum advantage for a variety of input states, ranging from the ideal pure squeezed vacuum state to more realistic thermalized squeezed states. A brief overview of the theory of GBS, recent experiments and other types of methods are also presented. Although this is not an exhaustive review, we aim to provide a brief introduction to phase-space methods applied to linear photonic networks to encourage further theoretical investigations.

quant-ph↗

arXiv2023

Validation tests of GBS quantum computers give evidence for quantum advantage with a decoherent target

Computational validation is vital for all large-scale quantum computers. One needs computers that are both fast and accurate. Here we apply precise, scalable, high order statistical tests to data from large Gaussian boson sampling (GBS) quantum computers that claim quantum computational advantage. These tests can be used to validate the output results for such technologies. Our method allows investigation of accuracy as well as quantum advantage. Such issues have not been investigated in detail before. Our highly scalable technique is also applicable to other applications of linear bosonic networks. We utilize positive-P phase-space simulations of grouped count probabilities (GCP) as a fingerprint for verifying multi-mode data. This is exponentially more efficient than other phase-space methods, due to much lower sampling errors. We randomly generate tests from exponentially many high-order, grouped count tests. Each of these can be efficiently measured and simulated, providing a quantum verification method that is hard to replicate classically. We give a detailed comparison of theory with a 144-channel GBS experiment, including grouped correlations up to the largest order measured. We show how one can disprove faked data, and apply this to a classical count algorithm. There are multiple distance measures for evaluating the fidelity and computational complexity of a distribution. We compute these and explain them. The best fit to the data is a partly thermalized Gaussian model, which is neither the ideal case, nor the model that gives classically computable counts. Even with this model, discrepancies of $Z>100$ were observed from some $χ^2$ tests, indicating likely parameter estimation errors. Total count distributions were much closer to a thermalized quantum model than the classical model, giving evidence consistent with quantum computational advantage for a modified target problem.

quant-ph↗

arXiv2022

Massive particle interferometry with lattice solitons

We discuss an interferometric scheme employing interference of bright solitons formed as specific bound states of attracting bosons on a lattice. We revisit the proposal of Castin and Weiss [Phys. Rev. Lett. vol. 102, 010403 (2009)] for using the scattering of a quantum matter-wave soliton on a barrier in order to create a coherent superposition state of the soliton being entirely to the left of the barrier and being entirely to the right of the barrier. In that proposal, it was assumed that the scattering is perfectly elastic, i.e.\ that the center-of-mass kinetic energy of the soliton is lower than the chemical potential of the soliton. Here we relax this assumption: By employing a combination of Bethe ansatz and DMRG based analysis of the dynamics of the appropriate many-body system, we find that the interferometric fringes persist even when the center-of-mass kinetic energy of the soliton is above the energy needed for its complete dissociation into constituent atoms.

cond-mat.quant-gas↗

arXiv2022

A midpoint projection algorithm for stochastic differential equations on manifolds

Stochastic differential equations projected onto manifolds occur in physics, chemistry, biology, engineering, nanotechnology and optimization, with interdisciplinary applications. Intrinsic coordinate stochastic equations on the manifold are often computationally impractical, and numerical projections are useful in many cases. We show that the Stratonovich interpretation of the stochastic calculus is obtained using adiabatic elimination with a constraint potential. We derive intrinsic stochastic equations for spheroidal and hyperboloidal surfaces for comparison purposes, and review some earlier projection algorithms. In this paper, a combined midpoint projection algorithm is proposed that uses a midpoint projection onto a tangent space, combined with a subsequent normal projection to satisfy the constraints. Numerical examples are given for a range of manifolds, including circular, spheroidal, hyperboloidal, and catenoidal cases, as well as higher-order polynomial constraints and a ten-dimensional hypersphere. We show that in all cases the combined midpoint method has greatly reduced errors compared to methods using a combined Euler projection approach or purely tangential projection. Our technique can handle multiple constraints. This allows manifolds that embody several conserved quantities. The algorithm is accurate, simple and efficient. An order of magnitude error reduction in diffusion distance is typically found compared to the other methods, with reductions of several orders of magnitude in constraint errors.

math.NA↗

arXiv2022

Massive particle interferometry with lattice solitons: robustness against ionization

We revisit the proposal of Castin and Weiss [Phys. Rev. Lett. vol. 102, 010403 (2009)] for using the scattering of a quantum matter-wave soliton on a barrier in order to create a coherent superposition state of the soliton being entirely to the left of the barrier and being entirely to the right of the barrier. In that proposal, is was assumed that the scattering is perfectly elastic, i.e. that the center-of-mass kinetic energy of the soliton is lower than the chemical potential of the soliton. Here we relax this assumption. Also, we introduce an interferometric scheme, which uses interference of soltions, that can be used to detect the degree of coherence between the reflected and transmitted part of the soliton. Using exact diagonalization, we numerically simulate a complete interferometric cycle for a soliton consisting of six atoms. We find that the interferometric fringes persist even when the center-of-mass kinetic energy of the soliton is above the energy needed for complete dissociation of the soliton into constituent atoms.

cond-mat.quant-gas↗

arXiv2022

Simulating complex networks in phase space: Gaussian boson sampling

We show how phase-space simulations of Gaussian quantum states in a photonic network permit verification of measurable correlations of Gaussian boson sampling (GBS) quantum computers. Our results agree with experiments for up to 100-th order correlations, provided decoherence is included. We extend this to more than 16,000 modes, and describe how to simulate genuine multipartite entanglement.

quant-ph↗

arXiv2022

Mesoscopic and macroscopic quantum correlations in photonic, atomic and optomechanical systems

This paper reviews the progress that has been made in our knowledge of quantum correlations at the mesoscopic and macroscopic level. We begin by summarizing the Einstein-Podolsky-Rosen (EPR) argument and the Bell correlations that cannot be explained by local hidden variable theories. It was originally an open question as to whether (and how) such quantum correlations could occur on a macroscopic scale, since this would seem to counter the correspondence principle. The purpose of this review is to examine how this question has been answered over the decades since the original papers of EPR and Bell. We first review work relating to higher spin measurements which revealed that macroscopic quantum states could exhibit Bell correlations. This covers higher dimensional, multi-particle and continuous-variable EPR and Bell states where measurements on a single system give a spectrum of outcomes, and also multipartite states where measurements are made at multiple separated sites. It appeared that the macroscopic quantum observations were for an increasingly limited span of measurement settings and required a fine resolution of outcomes. Motivated by this, we next review correlations for macroscopic superposition states, and examine predictions for the violation of Leggett-Garg inequalities for dynamical quantum systems. These results reveal Bell correlations for coarse-grained measurements which need only distinguish between macroscopically distinct states, thus bringing into question the validity of certain forms of macroscopic realism. Finally, we review progress for massive systems, including Bose-Einstein condensates and optomechanical oscillators, where EPR-type correlations have been observed between massive systems. Experiments are summarized, which support the predictions of quantum mechanics in mesoscopic regimes.

quant-ph↗

arXiv2021

Simulating macroscopic quantum correlations in linear networks

Many developing quantum technologies make use of quantum networks of different types. Even linear quantum networks are nontrivial, as the output photon distributions can be exponentially complex. Despite this, they can still be computationally simulated. The methods used are transformations into equivalent phase-space representations, which can then be treated probabilistically. This provides an exceptionally useful tool for the prediction and validation of experimental results, including decoherence. As well as experiments in Gaussian boson sampling, which are intended to demonstrate quantum computational advantage, these methods are applicable to other types of entangled linear quantum networks as well. This paper provides a tutorial and review of work in this area, to explain quantum phase-space techniques using the positive-P and Wigner distributions.

quant-ph↗

arXiv2020

Time evolution with symmetric stochastic action

Quantum dynamical time-evolution of bosonic fields is shown to be equivalent to a stochastic trajectory in space-time, corresponding to samples of a statistical mechanical steady-state in a higher dimensional quasi-time. This is proved using the Q-function of quantum theory with time-symmetric diffusion, that is equivalent to a forward-backward stochastic process in both the directions of time. The resulting probability distribution has a positive, time-symmetric action principle and path integral, whose solution corresponds to a classical field equilibrating in an additional dimension. Comparisons are made to stochastic quantization and other higher dimensional physics proposals. Five-dimensional space-time was originally introduced by Kaluza and Klein, and is now widely proposed in cosmology and particle physics. Time-symmetric action principles for quantum fields are also related to electrodynamical absorber theory, which is known to be capable of violating a Bell inequality. We give numerical methods and examples of solutions to the resulting stochastic partial differential equations in a higher time-dimension, giving agreement with exact solutions for soluble boson field quantum dynamics. This approach may lead to useful computational techniques for quantum field theory, as the action principle is real, as well as to ontological models of physical reality.

quant-ph↗

arXiv2020

Q-functions as models of physical reality

We show that one may interpret physical reality as random fields in space-time. These have a probability given by the expectation of a coherent state projection operator, called the Q-function. The resulting dynamical evolution includes retrocausal effects. This suggests that a physical universe exists without requiring observers, but with a well-defined probability for its field configuration. By including the meter dynamics, we show that field trajectories have quantum measurement properties without wave-function collapse, including sharp measured eigenvalues. We treat continuous and discrete measurements, and show that this model predicts Bell inequality violations for measurements on correlated spins. A discussion is give of a number of well-known quantum paradoxes, showing how these can be treated in a realistic model of measurement. Our theory resolves a number of practical and philosophical issues in quantum measurement, and we compare it with earlier theories.

quant-ph↗

arXiv2020

The fate of the false vacuum: Finite temperature, entropy and topological phase in quantum simulations of the early universe

Despite being at the heart of the theory of the "Big Bang" and cosmic inflation, the quantum field theory prediction of false vacuum tunneling has not been tested. To address the exponential complexity of the problem, a table-top quantum simulator in the form of an engineered Bose-Einstein condensate (BEC) has been proposed to give dynamical solutions of the quantum field equations. In this paper, we give a numerical feasibility study of the BEC quantum simulator under realistic conditions and temperatures, with an approximate truncated Wigner (tW) phase-space method. We report the observation of false vacuum tunneling in these simulations, and the formation of multiple bubble 'universes' with distinct topological properties. The tunneling gives a transition of the relative phase of coupled Bose fields from a metastable to a stable 'vacuum'. We include finite temperature effects that would be found in a laboratory experiment and also analyze the cut-off dependence of modulational instabilities in Floquet space. Our numerical phase-space model does not use thin-wall approximations, which are inapplicable to cosmologically interesting models. It is expected to give the correct quantum treatment including superpositions and entanglement during dynamics. By analyzing a nonlocal observable called the topological phase entropy (TPE), our simulations provide information about phase structure in the true vacuum. We observe a cooperative effect in which the true vacua bubbles representing distinct universes each have one or the other of two distinct topologies. The TPE initially increases with time, reaching a peak as the multiple universes are formed, and then decreases with time to the phase-ordered vacuum state. This gives a model for the formation of universes with one of two distinct phases, which is a possible solution to the problem of particle-antiparticle asymmetry.

quant-ph↗

arXiv2019

Initial states and apodisation for quantum field simulations in phase-space

Bosonic quantum fields can be simulated with `quantum software' in phase-space. The positive-P, Wigner and Q-function phase-space methods are reviewed. Initial quantum states and boundaries for infinite domains are considered in detail. The quantum initial conditions treated include both multi-mode Gaussian states and number states, which are sampled using phase-shifted weighting techniques. This is more efficient than direct probabilistic sampling. Algorithms for treating periodic boundaries within infinite domains are developed via apodisation, or absorbers near the boundary. Similar truncation or anti-aliasing issues arise in the frequency domain with Fourier transform methods. Complex apodisation, in which there are phase-shifts as well as absorption, improves accuracy by reducing unwanted errors, and this is illustrated numerically. The complex power law apodisation techniques described here can be applied to numerical calculations or even directly in optics. Phase-space simulations, used for both quantum optical and Bose-Einstein condensate quantum dynamical calculations, require quantum noise terms to be included. A method is described to analyze number conservation in apodised calculations. Some techniques developed here are potentially applicable to all digital simulations, whether using classical or quantum logic.

quant-ph↗