arXiv subjects

David Jennings

Publications and source records attributed to David Jennings.

At least 19 recordsLinked to original sources

arXiv2026

An end-to-end quantum algorithm for weakly nonlinear plasma physics with superquadratic speedup

Nonlinear kinetic plasma simulation is high-dimensional and classically demanding, while quantum algorithms face different bottlenecks: embedding nonlinear dynamics into a linear computation, loading dense field-interaction data, and efficiently extracting information. We present an end-to-end quantum algorithm, with rigorous convergence guarantees, for a weakly nonlinear kinetic plasma model. The system describes a 3D electron-ion plasma with adiabatic electrons, kinetic ions, Debye screening, and Krook relaxation. After Fourier-Hermite truncation, the dynamics reduces to a high-dimensional quadratic ordinary differential equation. To tackle quantum bottlenecks we combine three key ingredients. First, we use a plasma free energy to identify a Lyapunov transform under which a Carleman linear embedding converges exponentially in the truncation order within a certified weakly nonlinear regime. Second, we develop a hierarchical block-encoding protocol for dense matrices, exploiting the spatial decay of the field to avoid polynomial overhead from sparse access encodings. Third, we introduce a subroutine for information extraction that exploits nonlinear components encoded in the full Carleman history state to improve the estimation of linear observables. We construct a quantum algorithm to estimate the spacetime-averaged kinetic energy using $\widetilde{O}\!\left( N_F N_H^{1/2} \operatorname{polylog}\!\left(\frac{T}{\epsilon}\right)\frac{1}{\epsilon}\right)$ gates and $\widetilde{O}\!\left(\log\!\left(N_F N_H^{1/2}T\right)\log\!\left(\frac{1}{\epsilon}\right)\right)$ qubits, where $N_F$ and $N_H$ are the Fourier and Hermite cutoffs. Relative to a Fourier-Hermite spectral solver, this yields exponential memory savings and superquadratic improvements in time. Together, these results establish a controlled nonlinear plasma benchmark for quantum simulation.

quant-ph↗

arXiv2026

Quantum Koopman Algorithms

We define an observable-space framework of Quantum Koopman Algorithms (QKAs) for simulating the dynamics of both linear quantum and nonlinear classical systems, based on approximately closed sets of observables and efficient coherent encodings of their Koopman-driven evolution. QKAs have two strands: Dynamic-QKA for the initial-value problem of observables dynamics, and Spectral-QKA for the eigenvalue analysis of the Koopman operator. We demonstrate the scope of the framework through several applications. First, for classes of $N$ free fermions linearly coupled to a bath, we construct quantum algorithms with gate cost $O(\mathrm{polylog}(N))$, an exponential improvement over classical methods, and use them to reconstruct heat flows and decay rates. Second, for nonlinear classical dynamics, we introduce a novel nonlinear interaction-picture quantum algorithm that enables perturbative expansions around solvable nonlinear reference flows, going beyond existing approaches that only apply to weakly nonlinear systems. Third, we develop spectral methods for extracting eigen-frequencies of late-time nonlinear dynamics, introducing a windowed quantum ODE-solver. Our results identify the Koopman-quantum interface as a natural setting in which quantum algorithms can exploit observable-space structure to simulate both classical and quantum dynamics.

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arXiv2025

Simulating non-trivial incompressible flows with a quantum lattice Boltzmann algorithm

Quantum algorithms have been identified as a potential means to accelerate computational fluid dynamics (CFD) simulations, with the lattice Boltzmann method (LBM) being a promising candidate for realizing quantum speedups. Here, we extend the recent quantum algorithm for the incompressible LBM to account for realistic fluid dynamics setups by incorporating walls, inlets, outlets, and external forcing. We analyze the associated complexity cost and show that these modifications preserve the asymptotic scaling, and potential quantum advantage, of the original algorithm. Moreover, to support our theoretical analysis, we provide a classical numerical study illustrating the accuracy, complexity, and convergence of the algorithm for representative incompressible-flow cases, including the driven Taylor-Green vortex, the lid-driven cavity flow, and the flow past a cylinder. Our results provide a pathway to accurate quantum simulation of nonlinear fluid dynamics, and a framework for extending quantum LBM to more challenging flow configurations.

physics.flu-dyn↗

arXiv2025

An end-to-end quantum algorithm for nonlinear fluid dynamics with bounded quantum advantage

Computational fluid dynamics (CFD) is a cornerstone of classical scientific computing, and there is growing interest in whether quantum computers can accelerate such simulations. To date, the existing proposals for fault-tolerant quantum algorithms for CFD have almost exclusively been based on the Carleman embedding method, used to encode nonlinearities on a quantum computer. In this work, we begin by showing that these proposals suffer from a range of severe bottlenecks that negate conjectured quantum advantages: lack of convergence of the Carleman method, prohibitive time-stepping requirements, unfavorable condition number scaling, and inefficient data extraction. With these roadblocks clearly identified, we develop a novel algorithm for the incompressible lattice Boltzmann equation that circumvents these obstacles, and then provide a detailed analysis of our algorithm, including all potential sources of algorithmic complexity, as well as gate count estimates. We find that for an end-to-end problem, a modest quantum advantage may be preserved for selected observables in the high-error-tolerance regime. We lower bound the Reynolds number scaling of our quantum algorithm in dimension $D$ at Kolmogorov microscale resolution with $O(\mathrm{Re}^{\frac{3}{4}(1+\frac{D}{2})} \times q_M)$, where $q_M$ is a multiplicative overhead for data extraction with $q_M = O(\mathrm{Re}^{\frac{3}{8}})$ for the drag force. This upper bounds the scaling improvement over classical algorithms by $O(\mathrm{Re}^{\frac{3D}{8}})$. However, our numerical investigations suggest a lower speedup, with a scaling estimate of $O(\mathrm{Re}^{1.936} \times q_M)$ for $D=2$. Our results give robust evidence that small, but nontrivial, quantum advantages can be achieved in the context of CFD, and motivate the need for additional rigorous end-to-end quantum algorithm development.

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arXiv2025

Quantum algorithms for general nonlinear dynamics based on the Carleman embedding

Important nonlinear dynamics, such as those found in plasma and fluid systems, are typically hard to simulate on classical computers. Thus, if fault-tolerant quantum computers could efficiently solve such nonlinear problems, it would be a transformative change for many industries. In a recent breakthrough [Liu et al., PNAS 2021], the first efficient quantum algorithm for solving nonlinear differential equations was constructed, based on a single condition $R<1$, where $R$ characterizes the ratio of nonlinearity to dissipation. This result, however, is limited to the class of purely dissipative systems with negative log-norm, which excludes application to many important problems. In this work, we correct technical issues with this and other prior analysis, and substantially extend the scope of nonlinear dynamical systems that can be efficiently simulated on a quantum computer in a number of ways. Firstly, we extend the existing results from purely dissipative systems to a much broader class of stable systems, and show that every quadratic Lyapunov function for the linearized system corresponds to an independent $R$-number criterion for the convergence of the Carlemen scheme. Secondly, we extend our stable system results to physically relevant settings where conserved polynomial quantities exist. Finally, we provide extensive results for the class of non-resonant systems. With this, we are able to show that efficient quantum algorithms exist for a much wider class of nonlinear systems than previously known, and prove the BQP-completeness of nonlinear oscillator problems of exponential size. In our analysis, we also obtain several results related to the Poincar\'{e}-Dulac theorem and diagonalization of the Carleman matrix, which could be of independent interest.

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arXiv2023

The cost of solving linear differential equations on a quantum computer: fast-forwarding to explicit resource counts

How well can quantum computers simulate classical dynamical systems? There is increasing effort in developing quantum algorithms to efficiently simulate dynamics beyond Hamiltonian simulation, but so far exact resource estimates are not known. In this work, we provide two significant contributions. First, we give the first non-asymptotic computation of the cost of encoding the solution to general linear ordinary differential equations into quantum states -- either the solution at a final time, or an encoding of the whole history within a time interval. Second, we show that the stability properties of a large class of classical dynamics allow their fast-forwarding, making their quantum simulation much more time-efficient. From this point of view, quantum Hamiltonian dynamics is a boundary case that does not allow this form of stability-induced fast-forwarding. In particular, we find that the history state can always be output with complexity $O(T^{1/2})$ for any stable linear system. We present a range of asymptotic improvements over state-of-the-art in various regimes. We illustrate our results with a family of dynamics including linearized collisional plasma problems, coupled, damped, forced harmonic oscillators and dissipative nonlinear problems. In this case the scaling is quadratically improved, and leads to significant reductions in the query counts after inclusion of all relevant constant prefactors.

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arXiv2023

Randomized adiabatic quantum linear solver algorithm with optimal complexity scaling and detailed running costs

Solving linear systems of equations is a fundamental problem with a wide variety of applications across many fields of science, and there is increasing effort to develop quantum linear solver algorithms. [Suba\c{s}i et al., Phys. Rev. Lett. (2019)] proposed a randomized algorithm inspired by adiabatic quantum computing, based on a sequence of random Hamiltonian simulation steps, with suboptimal scaling in the condition number $\kappa$ of the linear system and the target error $\epsilon$. Here we go beyond these results in several ways. Firstly, using filtering~[Lin et al., Quantum (2019)] and Poissonization techniques [Cunningham et al., arXiv:2406.03972 (2024)], the algorithm complexity is improved to the optimal scaling $O(\kappa \log(1/\epsilon))$ -- an exponential improvement in $\epsilon$, and a shaving of a $\log \kappa$ scaling factor in $\kappa$. Secondly, the algorithm is further modified to achieve constant factor improvements, which are vital as we progress towards hardware implementations on fault-tolerant devices. We introduce a cheaper randomized walk operator method replacing Hamiltonian simulation -- which also removes the need for potentially challenging classical precomputations; randomized routines are sampled over optimized random variables; circuit constructions are improved. We obtain a closed formula rigorously upper bounding the expected number of times one needs to apply a block-encoding of the linear system matrix to output a quantum state encoding the solution to the linear system. The upper bound is $837 \kappa$ at $\epsilon=10^{-10}$ for Hermitian matrices.

quant-ph↗

arXiv2023

Unveiling the non-Abelian statistics of $D(S_3)$ anyons via photonic simulation

Simulators can realise novel phenomena by separating them from the complexities of a full physical implementation. Here we put forward a scheme that can simulate the exotic statistics of $D(S_3)$ non-Abelian anyons with minimal resources. The qudit lattice representation of this planar code supports local encoding of $D(S_3)$ anyons. As a proof-of-principle demonstration we employ a photonic simulator to encode a single qutrit and manipulate it to perform the fusion and braiding properties of non-Abelian $D(S_3)$ anyons. The photonic technology allows us to perform the required non-unitary operations with much higher fidelity than what can be achieved with current quantum computers. Our approach can be directly generalised to larger systems or to different anyonic models, thus enabling advances in the exploration of quantum error correction and fundamental physics alike.

quant-ph↗

arXiv2023

A simple formulation of no-cloning and no-hiding that admits efficient and robust verification

Incompatibility is a feature of quantum theory that sets it apart from classical theory, and the inability to clone an unknown quantum state is one of the most fundamental instances. The no-hiding theorem is another such instance that arises in the context of the black-hole information paradox, and can be viewed as being dual to no-cloning. Here, we formulate both of these fundamental features of quantum theory in a single form that is amenable to efficient verification, and that is robust to errors arising in state preparation and measurements. We extend the notion of unitarity - an average figure of merit that for quantum theory captures the coherence of a quantum channel - to general physical theories. Then, we introduce the notion of compatible unitarity pair (CUP) sets, that correspond to the allowed values of unitarities for compatible channels in the theory. We show that a CUP-set constitutes a simple 'fingerprint' of a physical theory, and that incompatibility can be studied through them. We derive information-disturbance constraints on quantum CUP-sets that encode both the no-cloning/broadcasting and no-hiding theorems of quantum theory. We then develop randomised benchmarking protocols that efficiently estimate quantum CUP-sets and provide simulations using IBMQ of the simplest instance. Finally, we discuss ways in which CUP-sets and quantum no-go theorems could provide additional information to benchmark quantum devices.

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arXiv2022

General entropic constraints on CSS codes within magic distillation protocols

Magic states are fundamental building blocks on the road to fault-tolerant quantum computing. CSS codes play a crucial role in the construction of magic distillation protocols. Previous work has cast quantum computing with magic states for odd dimension $d$ within a phase space setting in which universal quantum computing is described by the statistical mechanics of quasiprobability distributions. Here we extend this framework to the important $d=2$ qubit case and show that we can exploit common structures in CSS circuits to obtain distillation bounds capable of out-performing previous monotone bounds in regimes of practical interest. Moreover, in the case of CSS code projections, we arrive at a novel cut-off result on the code length $n$ of the CSS code in terms of parameters characterising a desired distillation, which implies that for fixed target error rate and acceptance probability, one needs only consider CSS codes below a threshold number of qubits. These entropic constraints are not due simply to the data-processing inequality but rely explicitly on the stochastic representation of such protocols.

quant-ph↗

arXiv2022

Faster Born probability estimation via gate merging and frame optimisation

Outcome probability estimation via classical methods is an important task for validating quantum computing devices. Outcome probabilities of any quantum circuit can be estimated using Monte Carlo sampling, where the amount of negativity present in the circuit frame representation quantifies the overhead on the number of samples required to achieve a certain precision. In this paper, we propose two classical sub-routines: circuit gate merging and frame optimisation, which optimise the circuit representation to reduce the sampling overhead. We show that the runtimes of both sub-routines scale polynomially in circuit size and gate depth. Our methods are applicable to general circuits, regardless of generating gate sets, qudit dimensions and the chosen frame representations for the circuit components. We numerically demonstrate that our methods provide improved scaling in the negativity overhead for all tested cases of random circuits with Clifford+$T$ and Haar-random gates, and that the performance of our methods compares favourably with prior quasi-probability simulators as the number of non-Clifford gates increases.

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arXiv2021

Towards quantum key distribution with noisy communication sources

A beam emitted by a displaced thermal source, incident on a beam splitter, is used as the source in a central broadcast quantum key distribution protocol. These displaced thermal states have parallels to signals produced through phase-shift keying, already commonly used in a variety of microwave-based communications equipment. Here, we analyse a thermal state quantum key distribution protocol for which there is no dependency on displacement. Our analysis suggests the ability to carry out QKD using widely available broadcasting equipment. We also consider the effects of the introduction of loss at several points in the protocol, and find that, while increasing loss in Alice's channel eventually results in advantage distillation being required, security is still maintained.

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arXiv2021

Infinitesimal reference frames suffice to determine the asymmetry properties of a quantum system

Symmetry principles are fundamental in physics, and while they are well understood within Lagrangian mechanics, their impact on quantum channels has a range of open questions. The theory of asymmetry grew out of information-theoretic work on entanglement and quantum reference frames, and allows us to quantify the degree to which a quantum system encodes coordinates of a symmetry group. Recently, a complete set of entropic conditions was found for asymmetry in terms of correlations relative to infinitely many quantum reference frames. However, these conditions are difficult to use in practice and their physical implications unclear. In the present theoretical work, we show that this set of conditions has extensive redundancy, and one can restrict to reference frames forming any closed surface in the state space that has the maximally mixed state in its interior. This in turn implies that asymmetry can be reduced to just a single entropic condition evaluated at the maximally mixed state. Contrary to intuition, this shows that we do not need macroscopic, classical reference frames to determine the asymmetry properties of a quantum system, but instead infinitesimally small frames suffice. Building on this analysis, we provide simple, closed conditions to estimate the minimal depolarization needed to make a given quantum state accessible under channels covariant with any given symmetry group.

quant-ph↗

arXiv2021

Constraints on magic state protocols from the statistical mechanics of Wigner negativity

Magic states are key ingredients in schemes to realize universal fault-tolerant quantum computation. Theories of magic states attempt to quantify this computational element via monotones and determine how these states may be efficiently transformed into useful forms. Here, we develop a statistical mechanical framework based on majorization to describe Wigner negative magic states for qudits of odd prime dimension processed under Clifford circuits. We show that majorization allows us to both quantify disorder in the Wigner representation and derive upper bounds for magic distillation. These bounds are shown to be tighter than other bounds, such as from mana and thauma, and can be used to incorporate hardware physics, such as temperature dependence and system Hamiltonians. We also show that a subset of single-shot R\'{e}nyi entropies remain well-defined on quasi-distributions, are fully meaningful in terms of data processing and can acquire negative values that signal magic. We find that the mana of a magic state is the measure of divergence of these R\'{e}nyi entropies as one approaches the Shannon entropy for Wigner distributions, and discuss how distillation lower bounds could be obtained in this setting. This use of majorization for quasi-distributions could find application in other studies of non-classicality, and raises novel questions in the context of classical statistical mechanics.

quant-ph↗

arXiv2021

Estimation of correlations and non-separability in quantum channels via unitarity benchmarking

The ability to transfer coherent quantum information between systems is a fundamental component of quantum technologies and leads to coherent correlations within the global quantum process. However correlation structures in quantum channels are less studied than those in quantum states. Motivated by recent techniques in randomized benchmarking, we develop a range of results for efficient estimation of correlations within a bipartite quantum channel. We introduce sub-unitarity measures that are invariant under local changes of basis, generalize the unitarity of a channel, and allow for the analysis of coherent information exchange within channels. Using these, we show that unitarity is monogamous, and provide a novel information-disturbance relation. We then define a notion of correlated unitarity that quantifies the correlations within a given channel. Crucially, we show that this measure is strictly bounded on the set of separable channels and therefore provides a witness of non-separability. Finally, we describe how such measures for effective noise channels can be efficiently estimated within different randomized benchmarking protocols. We find that the correlated unitarity can be estimated in a SPAM-robust manner for any separable quantum channel, and show that a benchmarking/tomography protocol with mid-circuit resets can reliably witness non-separability for sufficiently small reset errors. The tools we develop provide information beyond that obtained via simultaneous randomized benchmarking and so could find application in the analysis of cross-talk and coherent errors in quantum devices.

quant-ph↗

arXiv2021

Thermal state quantum key distribution

We analyse a central broadcast continuous variable quantum key distribution protocol in which a beam produced by a thermal source is used to create a secret key between two parties, Alice and Bob. A beam splitter divides the initial beam into a pair of output beams, which are sent to Alice and Bob, with Eve intercepting Bob's beam. We investigate the protocol in detail, calculating mutual informations through a pair of analytic methods and comparing the results to the outputs of a Monte Carlo simulation of the protocol. In a lossless system, we find that a lower bound on the key rate remains positive in the protocol under a beam splitter attack, provided Bob receives a nonzero proportion of the beam initially sent to him. This suggests that the thermal state protocol could be used experimentally to produce secure keys.

quant-ph↗

arXiv2019

The geometry of passivity for quantum systems and a novel elementary derivation of the Gibbs state

Passivity is a fundamental concept that constitutes a necessary condition for any quantum system to attain thermodynamic equilibrium, and for a notion of temperature to emerge. While extensive work has been done that exploits this, the transition from passivity at a single-shot level to the completely passive Gibbs state is technically clear but lacks a good over-arching intuition. Here, we re-formulate passivity for quantum systems in purely geometric terms. This description makes the emergence of the Gibbs state from passive states entirely transparent. Beyond clarifying existing results, it also provides novel analysis for non-equilibrium quantum systems. We show that, to every passive state, one can associate a simple convex shape in a 2-dimensional plane, and that the area of this shape measures the degree to which the system deviates from the manifold of equilibrium states. This provides a novel geometric measure of athermality with relations to both ergotropy and $\beta$-athermality.

quant-ph↗

arXiv2019

Decomposable coherence and quantum fluctuation relations

In Newtonian mechanics, any closed-system dynamics of a composite system in a microstate will leave all its individual subsystems in distinct microstates, however this fails dramatically in quantum mechanics due to the existence of quantum entanglement. Here we introduce the notion of a `coherent work process', and show that it is the direct extension of a work process in classical mechanics into quantum theory. This leads to the notion of `decomposable' and `non-decomposable' quantum coherence and gives a new perspective on recent results in the theory of asymmetry as well as early analysis in the theory of classical random variables. Within the context of recent fluctuation relations, originally framed in terms of quantum channels, we show that coherent work processes play the same role as their classical counterparts, and so provide a simple physical primitive for quantum coherence in such systems. We also introduce a pure state effective potential as a tool with which to analyze the coherent component of these fluctuation relations, and which leads to a notion of temperature-dependent mean coherence, provides connections with multi-partite entanglement, and gives a hierarchy of quantum corrections to the classical Crooks relation in powers of inverse temperature.

quant-ph↗