arXiv subjects

Dominic W. Berry

Publications and source records attributed to Dominic W. Berry.

At least 19 recordsLinked to original sources

arXiv2026

Cascading amplifiers can create exponentially large coherence

A standard laser beam has photon degeneracy, or coherence, $\mathfrak{C}$, of at most $8\mu^2$, where $\mu$ is the number of photons in the laser itself. Even quantum-engineered lasers, if required to produce a beam with the standard statistical properties, have limited coherence, scaling as $\mu^4$. Moreover, such lasers (still unrealised) require very unconventional gain and output-coupling mechanisms. Here, we propose a different path to increasing $\mathfrak{C}$: cascaded linear amplifiers, with conventional couplings. By dropping the requirement on the beam's properties, this approach can, in theory, achieve $\mathfrak{C}$ scaling exponentially in $\mu$. Here $\mu$ is the total source excitation number, across all the amplifiers. Two amplifiers suffice to surpass the standard $\mu^2$ scaling.

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arXiv2026

Spectral amplification for ground-state energy estimation of electronic structure in first quantization

We demonstrate an asymptotic gate complexity improvement in first-quantized ground-state energy estimation of electronic structure Hamiltonians in a plane wave basis by employing the sum-of-squares spectral gap amplification protocol. The improvement relies on identifying a sum-of-squares representation of the Hamiltonian which provides a lower bound certificate and low cost block encoding that leads to a provably lower quantum phase estimation gate cost. This is achieved by using a sum-of-squares operator generated by the total charge density operator resulting in a block encoding normalization improvement of $\lambda = \mathcal{O}\left(\eta\Delta^{-1.5}+\eta^{1.5}\Delta^{-1} \right)$ compared to prior work $\lambda = \mathcal{O}(\eta\Delta^{-2}+\eta^2\Delta^{-1})$ where $\eta$ is the number of electrons and $\Delta$ is the simulation grid spacing. The asymptotic reduction in block encoding normalization and similar block encoding costs to prior work is demonstrated to reduce resource estimates for materials and chemical systems by a factor of $2 - 44\times$ corresponding to the lowest cost estimates for ab initio materials simulation.

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arXiv2026

A Denser Planar Surface Code

We present a quantum code implementable on a regular $2$D hex grid with an estimated encoding rate up to $4.5\times$ of that of a rotated surface code patch using circuit-level noise in a one- and two-qubit $10^{-3}$ error uniform depolarizing model. Our approach is based on yoking a dense packing of surface code twist defects, enabled by new stabilizer measurement cycles with an optimal four layers of nearest-neighbor two-qubit gates, almost no distance-reducing hook errors, and efficient decoding. We demonstrate a space-efficient architecture for computing on densely packed logical qubits, including new padding-free lattice surgery protocols in an optimal bounding box of $2d^2$ data and measurement qubits per patch. Assuming a $1\mu$s surface code cycle time and a $10\mu$s reaction time, these developments enable chemically accurate ground state phase estimation of a broad class of `utility-scale' electronic structure simulation problems such as the $108$ spin-orbital FeMoco-based nitrogen fixation catalyst in under a month with $89$k noisy superconducting qubits. We elucidate a Pareto frontier of space-time trade-offs and find a minimum physical quantum volume of $1.3$ mega-qubit-hours. These correspond to a $36\times$ space and $6.6\times$ spacetime improvement, respectively, over our previous state-of-the-art minimum-Toffoli resource estimates (Phys. Rev. X 15, 041016).

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arXiv2026

Efficient quantum algorithm for linear matrix differential equations and applications to open quantum systems

We present an efficient, nearly optimal quantum algorithm for solving linear matrix differential equations, with applications to the simulation of open quantum systems and beyond. For unitary or dissipative dynamics, the algorithm computes an entry of the solution matrix with query complexity $\widetilde{\mathcal{O}}(\nu \mathcal{L} t/\epsilon)$, where the constant $\nu$ depends on the problem parameters, $\mathcal{L}$ involves a time integral of upper bounds on the norms of evolution operators, and $\epsilon$ is the error. In particular, $\nu \mathcal{L}$ is linear in $t$ for unitary dynamics and can be a constant for dissipative dynamics. Our result contrasts prior quantum approaches for differential equations that typically require exponential time for this problem due to the encoding in a quantum state, which can lead to exponentially small amplitudes. We demonstrate the utility of the algorithm through an end-to-end application, namely the simulation of dissipative dynamics for non-interacting fermions, which can be extended to other quantum and classical systems. We compare with classical algorithms and give evidence of polynomial quantum speedups for systems in a lattice, which become more pronounced for systems with long-range interactions and can be shown to be exponential in general. We also provide a lower bound of $\Omega(\nu \mathcal{L} t/\epsilon)$ for unitary or dissipative dynamics that proves our algorithm is optimal up to logarithmic factors.

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arXiv2026

Quantum algorithm for solving differential equations using SLAC derivatives

In numerical approaches to solving differential equations on a lattice, a representation of the derivative operator that correctly matches the continuum behaviour of functions of momentum up to the band limit must be non-local. We present the construction of efficient linear-combination-of-unitaries ($\mathrm{LCU}$)-based block-encodings for the first-order derivative and Laplacian operators in the non-local $N=2^n$-dimensional SLAC representation. We use state-preparation techniques designed for smoothly decaying functions to prepare the dense $\mathrm{LCU}$ amplitudes with high success probability and low gate cost. Furthermore, we demonstrate how Shannon wavelet transforms can be applied to these block-encodings to obtain multiscale representations of the SLAC derivative operators. We then show how to apply a diagonal preconditioner that reduces the condition number of these matrices in the multiscale wavelet basis to a small constant. This enables the solution of partial differential equations (PDEs) with SLAC-discretised derivative operators on a finite lattice using the quantum linear solving algorithm ($\mathrm{QLSA}$). For a $d$-dimensional PDE, after projection away from the nullspace, the resulting quantum linear-system algorithm has overall gate complexity ${O}(dn^3\alpha^{(k)}\log(1/\varepsilon))$, where $\alpha^{(k)}$ is the subnormalisation factor of the order-$k$ SLAC block-encoding and $\varepsilon$ denotes the algorithmic approximation error.

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arXiv2026

Constant Factor Analysis of Optimal Quantum Linear Solvers in Practice

Optimal quantum linear equation solvers provide complexity $O(\kappa\log(1/\epsilon))$, where $\kappa$ is the condition number and $\epsilon$ is the allowable error. The optimal solver using a discrete adiabatic approach [PRX Quantum 3, 040303 (2022)] has large analytically proven constant factors for the upper bound on the complexity. The constant factors were later found to be about 1,200 times smaller in numerical testing [Quantum 9, 1887 (2025)]. This meant it is about an order of magnitude more efficient than using a randomised approach from [PRX Quantum 6, 040373 (2025)], which has far smaller analytically proven constant factors. Recently, a ``Shortcut'' method has been found to provide an optimal solver which also has small proven constant factors. In the present work, we conduct a comprehensive numerical analysis comparing this method with the adiabatic solver for two families of random linear systems. We find that, in the case where the solution norm is unknown, the adiabatic solver provides slightly better performance. If the solution norm is known, then the shortcut method provides significantly better performance for non-Hermitian matrices.

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arXiv2026

End-to-End Simulation of Chemical Dynamics on a Quantum Computer

Simulations of chemical dynamics are a powerful means for understanding chemistry. However, classical computers struggle to simulate many chemical processes, especially non-adiabatic ones, where the Born-Oppenheimer approximation breaks down. Quantum computers could simulate quantum-chemical dynamics more efficiently than classical computers, but there is currently no complete quantum algorithm for calculating dynamical observables to within a known error. Here, we develop an efficient, end-to-end quantum algorithm for simulating chemical dynamics that avoids all uncontrolled approximations (including the Born-Oppenheimer approximation) and whose error is bounded subject to mild assumptions. To do so, we treat the nuclei and the electrons on an equal footing and simulate the full molecular wavefunction on a momentum-space grid in first quantization, including all algorithmic steps: initial-state preparation, time evolution using qubitization, and measurement of chemical observables such as reaction yields and rates. Our work gives the first algorithm for quantum simulation of chemistry whose end-to-end complexity achieves sublinear scaling in the size of the grid. We achieve this by developing an exponentially faster method for initial-state-preparation. Photochemistry is a likely early application of our algorithm and we estimate resources required for end-to-end simulations of non-adiabatic dynamics of atmospherically important molecules. Classically intractable photochemical computations could be performed using resources comparable to those required for other chemical applications of quantum computing.

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arXiv2026

Quantum phase estimation with optimal confidence interval using three control qubits

Quantum phase estimation is an important routine in many quantum algorithms, particularly for estimating the ground state energy in quantum chemistry simulations. This estimation involves applying powers of a unitary to the ground state, controlled by an auxiliary state prepared on a control register. In many applications the goal is to provide a confidence interval for the phase estimate, and optimal performance is provided by a discrete prolate spheroidal sequence. We show how to prepare the corresponding state in a far more efficient way than prior work. We find that a matrix product state representation with a bond dimension of 4 is sufficient to give a highly accurate approximation for all dimensions tested, up to $2^{24}$. This matrix product state can be efficiently prepared using a sequence of simple three-qubit operations. When the dimension is a power of 2, the phase estimation can be performed with only three qubits for the control register, making it suitable for early-generation fault-tolerant quantum computers with a limited number of logical qubits.

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arXiv2025

Quantum simulation of electronic structure via quantum fast multipole method

Here we describe an approach for simulating electronic structure on quantum computers with significantly lower asymptotic complexity than prior work. The approach uses a real-space first-quantised representation of the molecular Hamiltonian which we propagate using high-order product formulae. Essential for this low complexity is the use of a technique similar to the fast multipole method for computing the Coulomb operator with $\widetilde{\cal O}(\eta)$ complexity for a simulation with $\eta$ particles. We show how to modify this algorithm so that it can be implemented on a quantum computer. We ultimately demonstrate an approach with $t(\eta^{4/3}N^{1/3} + \eta^{1/3} N^{2/3} ) (\eta Nt/\epsilon)^{o(1)}$ gate complexity, where $N$ is the number of grid points, $\epsilon$ is target precision, and $t$ is the duration of time evolution. This is roughly a speedup by ${\cal O}(\eta)$ over most prior algorithms. We provide lower complexity than all prior work for $N<\eta^7$ (the regime of practical interest), with only first-quantised interaction-picture simulations providing better performance for $N>\eta^7$. As with the classical fast multipole method, large particle numbers $\eta\gtrsim 10^3$ would be needed to realise this advantage.

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arXiv2025

Large time-step discretisation of adiabatic quantum dynamics

Adiabatic quantum computing is a general framework for preparing eigenstates of Hamiltonians on quantum devices. However, its digital implementation requires an efficient Hamiltonian simulation subroutine, which may introduce extra computational overhead or complicated quantum control logic. In this work, we show that the time step sizes in time discretization can be much larger than expected, and the overall complexity is greatly reduced. Remarkably, regardless of the general convergence order of the numerical method, we can choose a uniform time step size independent of tolerated error and evolution time for sufficiently accurate simulation. Furthermore, with the boundary cancellation condition where the continuous diabatic errors are exponentially suppressed, we provide strong evidence on an exponential convergence of even first-order Trotter with uniform time step size. We apply our analysis to the example of adiabatic unstructured search and show several preferable features of the Trotterized adiabatic approach: it can match the Grover lower bound, it does not require a priori knowledge on the number of marked states, and its performance can be asymptotically comparable with that of the quantum approximate optimization algorithm.

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arXiv2025

Quantum algorithm for linear matrix equations

We describe an efficient quantum algorithm for solving the linear matrix equation AX+XB=C, where A, B, and C are given complex matrices and X is unknown. This is known as the Sylvester equation, a fundamental equation with applications in control theory and physics. Our approach constructs the solution matrix X/x in a block-encoding, where x is a rescaling factor needed for normalization. This allows us to obtain certain properties of the entries of X exponentially faster than would be possible from preparing X as a quantum state. The query and gate complexities of the quantum circuit that implements this block-encoding are almost linear in a condition number that depends on A and B, and depend logarithmically in the dimension and inverse error. We show how our quantum circuits can solve BQP-complete problems efficiently, discuss potential applications and extensions of our approach, its connection to Riccati equation, and comment on open problems.

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arXiv2025

Analytical results for laser models producing a beam with sub-Poissonian photon statistics and coherence scaling as the Heisenberg limit

Recent advances in laser theory have demonstrated that a quantum enhancement is possible for the production of coherence $\mathfrak{C}$ by a continuous-wave laser device. Curiously, natural families of laser models that achieve Heisenberg-limited scaling for coherence produce the most coherence when the beam exhibits sub-Poissonian photon statistics. In this work, we provide an analytical treatment of those novel families of laser models by specializing to a parameter regime that permits a linearization. We characterize the dynamics of each laser system, and find that some of the intuitions from standard laser theory may be applied here. Specifically, the intracavity number dynamics are well-described as an Ornstein-Uhlenbeck process, while the intracavity phase dynamics are well-described in terms of a physically realizable ensemble of pure states, which evolve according to pure phase diffusion. Unlike a standard laser, however, we find that the pure states comprising the ensemble in the Heisenberg-limited lasers are substantially phase squeezed. From our dynamical analysis, we deduce various quantities of the beam for each laser family, including the first- and second-order Glauber coherence functions, intensity noise spectrum, Mandel-Q parameter and coherence $\mathfrak{C}$. In addition, inspired from these phase diffusion dynamics, we derive an upper bound on laser coherence $\mathfrak{C} \lesssim 1.1156 \mu^4$ -- which is tighter by a factor of $3/8$ when compared to that derived in [Baker et al., Nat. Phys. 17 179 (2021)] -- by making one of the assumptions of that paper slightly stronger.

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arXiv2025

Fast quantum simulation of electronic structure by spectrum amplification

The most advanced techniques using fault-tolerant quantum computers to estimate the ground-state energy of a chemical Hamiltonian involve compression of the Coulomb operator through tensor factorizations, enabling efficient block-encodings of the Hamiltonian. A natural challenge of these methods is the degree to which block-encoding costs can be reduced. We address this challenge through the technique of spectrum amplification, which magnifies the spectrum of the low-energy states of Hamiltonians that can be expressed as sums of squares. Spectrum amplification enables estimating ground-state energies with significantly improved cost scaling in the block encoding normalization factor $Λ$ to just $\sqrt{2ΛE_{\text{gap}}}$, where $E_{\text{gap}} \ll Λ$ is the lowest energy of the sum-of-squares Hamiltonian. To achieve this, we show that sum-of-squares representations of the electronic structure Hamiltonian are efficiently computable by a family of classical simulation techniques that approximate the ground-state energy from below. In order to further optimize, we also develop a novel factorization that provides a trade-off between the two leading Coulomb integral factorization schemes -- namely, double factorization and tensor hypercontraction -- that when combined with spectrum amplification yields a factor of 4 to 195 speedup over the state of the art in ground-state energy estimation for models of Iron-Sulfur complexes and a CO$_{2}$-fixation catalyst.

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arXiv2025

Selection and improvement of product formulae for best performance of quantum simulation

Quantum algorithms for simulation of Hamiltonian evolution are often based on product formulae. The fractal methods give a systematic way to find arbitrarily high-order product formulae, but result in a large number of exponentials. On the other hand, product formulae with fewer exponentials can be found by numerical solution of simultaneous nonlinear equations. It is also possible to reduce the cost of long-time simulations by processing, where a kernel is repeated and a processor need only be applied at the beginning and end of the simulation. In this work, we found thousands of new product formulae, and numerically tested these formulae, together with many formulae from prior literature. We provide methods to fairly compare product formulae of different lengths and different orders. For the case of 8th order, we have found new product formulae with exceptional performance, about two orders of magnitude better accuracy than prior work, both in the processed and non-processed cases. The processed product formula provides the best performance due to being shorter than the non-processed product formula. It outperforms all other tested product formulae over a range of many orders of magnitude in system parameters $T$ (time) and $ε$ (allowable error). That includes reasonable combinations of parameters to be used in quantum algorithms, where the size of the simulation is large enough to be classically intractable, but not so large it takes an impractically long time on a quantum computer.

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arXiv2025

Quantum Linear System Solvers: A Survey of Algorithms and Applications

Solving linear systems of equations plays a fundamental role in numerous computational problems from different fields of science. The widespread use of numerical methods to solve these systems motivates investigating the feasibility of solving linear systems problems using quantum computers. In this work, we provide a survey of the main advances in quantum linear systems algorithms, together with some applications. We summarize and analyze the main ideas behind some of the algorithms for the quantum linear systems problem in the literature. The analysis begins by examining the Harrow-Hassidim-Lloyd (HHL) solver. We note its limitations and reliance on computationally expensive quantum methods, then highlight subsequent research efforts which aimed to address these limitations and optimize runtime efficiency and precision via various paradigms. We focus in particular on the post-HHL enhancements which have paved the way towards optimal lower bounds with respect to error tolerance and condition number. By doing so, we propose a taxonomy that categorizes these studies. Furthermore, by contextualizing these developments within the broader landscape of quantum computing, we explore the foundational work that have inspired and informed their development, as well as subsequent refinements. Finally, we discuss the potential applications of these algorithms in differential equations, quantum machine learning, and many-body physics.

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arXiv2024

Rapid initial state preparation for the quantum simulation of strongly correlated molecules

Studies on quantum algorithms for ground state energy estimation often assume perfect ground state preparation; however, in reality the initial state will have imperfect overlap with the true ground state. Here we address that problem in two ways: by faster preparation of matrix product state (MPS) approximations, and more efficient filtering of the prepared state to find the ground state energy. We show how to achieve unitary synthesis with a Toffoli complexity about $7 \times$ lower than that in prior work, and use that to derive a more efficient MPS preparation method. For filtering we present two different approaches: sampling and binary search. For both we use the theory of window functions to avoid large phase errors and minimise the complexity. We find that the binary search approach provides better scaling with the overlap at the cost of a larger constant factor, such that it will be preferred for overlaps less than about $0.003$. Finally, we estimate the total resources to perform ground state energy estimation of Fe-S cluster systems, including the FeMo cofactor by estimating the overlap of different MPS initial states with potential ground-states of the FeMo cofactor using an extrapolation procedure. {With a modest MPS bond dimension of 4000, our procedure produces an estimate of $\sim 0.9$ overlap squared with a candidate ground-state of the FeMo cofactor, producing a total resource estimate of $7.3 \times 10^{10}$ Toffoli gates; neglecting the search over candidates and assuming the accuracy of the extrapolation, this validates prior estimates that used perfect ground state overlap. This presents an example of a practical path to prepare states of high overlap in a challenging-to-compute chemical system.

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arXiv2024

Faster Algorithmic Quantum and Classical Simulations by Corrected Product Formulas

Hamiltonian simulation using product formulas is arguably the most straightforward and practical approach for algorithmic simulation of a quantum system's dynamics on a quantum computer. Here we present corrected product formulas (CPFs), a variation of product formulas achieved by injecting auxiliary terms called correctors into standard product formulas. We establish several correctors that improve the accuracy of standard product formulas by orders of magnitude when simulating Hamiltonians comprised of two exactly simulatable partitions, a common structure of lattice Hamiltonians. Importantly, injecting these correctors increases the overall simulation cost by only a small additive or multiplicative factor. We show that correctors are particularly advantageous for perturbed systems, where one partition has a relatively small norm compared to the other, as they allow the small norm to be utilized as an additional parameter for controlling the simulation error. We demonstrate the performance of CPFs by numerical simulations for several lattice Hamiltonians. Numerical results show that our theoretical error bounds for CPFs match or outperform the empirical error scaling of standard product formulas for these systems. We also demonstrate improvements offered by CPFs by implementing small-size systems on actual quantum hardware, as well as on noisy and noiseless quantum simulators. CPFs could be a valuable algorithmic tool for early fault-tolerant quantum computers with limited computing resources. As for standard product formulas, CPFs could also be used for simulations on a classical computer.

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arXiv2024

Quantum Simulation of Realistic Materials in First Quantization Using Non-local Pseudopotentials

This paper improves and demonstrates the usefulness of the first quantized plane-wave algorithms for the quantum simulation of electronic structure, developed by Babbush et al. and Su et al. We describe the first quantum algorithm for first quantized simulation that accurately includes pseudopotentials. We focus on the Goedecker-Tetter-Hutter (GTH) pseudopotential, which is among the most accurate and widely used norm-conserving pseudopotentials enabling the removal of core electrons from the simulation. The resultant screened nuclear potential regularizes cusps in the electronic wavefunction so that orders of magnitude fewer plane waves are required for a chemically accurate basis. Despite the complicated form of the GTH pseudopotential, we are able to block encode the associated operator without significantly increasing the overall cost of quantum simulation. This is surprising since simulating the nuclear potential is much simpler without pseudopotentials, yet is still the bottleneck. We also generalize prior methods to enable the simulation of materials with non-cubic unit cells, which requires nontrivial modifications. Finally, we combine these techniques to estimate the block-encoding costs for commercially relevant instances of heterogeneous catalysis (e.g. carbon monoxide adsorption on transition metals) and compare to the quantum resources needed to simulate materials in second quantization. We conclude that for computational cells with many particles, first quantization often requires meaningfully less spacetime volume.

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