arXiv subjects

Toby Cubitt

Publications and source records attributed to Toby Cubitt.

At least 19 recordsLinked to original sources

arXiv2025

Fermionic dynamics on a trapped-ion quantum computer beyond exact classical simulation

Simulation of the time-dynamics of fermionic many-body systems has long been predicted to be one of the key applications of quantum computers. Such simulations -- for which classical methods are often inaccurate -- are critical to advancing our knowledge and understanding of quantum chemistry and materials, underpinning a wide range of fields, from biochemistry to clean-energy technologies and chemical synthesis. However, the performance of all previous digital quantum simulations of fermions has been matched by classical methods, and it has thus far remained unclear whether near-term, intermediate-scale quantum hardware could offer any computational advantage in this area. Here, we implement an efficient quantum simulation algorithm on Quantinuum's System Model H2 trapped-ion quantum computer for the time dynamics of a 56-qubit system that is too complex for exact classical simulation. We focus on the periodic spinful 2D Fermi-Hubbard model and present evidence of spin-charge separation, where the elementary electron's charge and spin decouple. In the limited cases where ground truth is available through exact classical simulation, we find that it agrees with the results we obtain from the quantum device. Employing long-range Wilson operators to study deconfinement of the effective gauge field between spinons and the effective potential between charge carriers, we find behaviour that differs from predictions made by classical tensor network methods. Our results herald the use of quantum computing for simulating strongly correlated electronic systems beyond the capacity of classical computing.

quant-ph↗

arXiv2025

Programmable digital quantum simulation of 2D Fermi-Hubbard dynamics using 72 superconducting qubits

Simulating the time-dynamics of quantum many-body systems was the original use of quantum computers proposed by Feynman, motivated by the critical role of quantum interactions between electrons in the properties of materials and molecules. Accurately simulating such systems remains one of the most promising applications of general-purpose digital quantum computers, in which all the parameters of the model can be programmed and any desired physical quantity output. However, performing such simulations on today's quantum computers at a scale beyond the reach of classical methods requires advances in the efficiency of simulation algorithms and error mitigation techniques. Here we demonstrate programmable digital quantum simulation of the dynamics of the 2D Fermi-Hubbard model -- one of the best-known simplified models of electrons in crystalline solids -- at a scale beyond exact classical state-vector simulation. We implement simulations of this model on lattice sizes up to ${6\times 6}$ using 72 qubits on Google's Willow quantum processor, across a range of physical parameters, including different on-site electron-electron interaction strengths and magnetic flux values, and study phenomena including formation of magnetic polarons (charge carriers surrounded by local magnetic polarisation), dynamical symmetry-breaking in stripe-ordered states, attraction of charge carriers on an entangled background state known as a valence bond solid, and the approach to equilibrium through thermalisation. We validate our results against exact calculations in parameter regimes where these are feasible, and compare them to approximate classical simulations performed using tensor network and operator propagation methods. Our results demonstrate that meaningful programmable digital quantum simulation of many-body interacting electron models is now feasible on state-of-the-art quantum hardware.

quant-ph↗

arXiv2025

A sharper Magnus expansion bound woven in binary branches

The Magnus expansion provides an exponential representation of one-parameter operator families, expressed as a series expansion in its generators. This is useful for example in quantum mechanics for expressing a unitary evolution determined by a time-dependent Hamiltonian generator of the dynamics. The solution is constructed as a series expansion in terms of increasingly complex nested commutators that rapidly become challenging to compute directly. This work establishes a universal upper bound, agnostic to the generator, on the error incurred when the Magnus expansion is truncated at an arbitrary given order. The main technical ingredient of the proof is the binary tree representation introduced by Iserles and Norsett from which we derive a recursion formula to delimit the magnitude of any term in the expansion. We complement our analytic results for the truncation error with explicit calculation of the first 24 terms in the Magnus series, illustrating that they follow the scaling behaviour we have derived. With these findings we aim to contribute to the understanding of the accuracy and limitations of the Magnus expansion technique, and to provide a sharper bound for approximating quantum dynamics without requiring assumptions on the structure of their generators.

quant-ph↗

arXiv2025

Fault-Resilience of Dissipative Processes for Quantum Computing

Dissipative processes have long been proposed as a means of performing computational tasks on quantum computers that may be intrinsically more robust to noise. In this work, we prove two main results concerning the error-resilience capabilities of two types of dissipative algorithms: dissipative ground state preparation in the form of the dissipative quantum eigensolver (DQE), and dissipative quantum computation (DQC). The first result is that under circuit-level depolarizing noise, a version of the DQE algorithm applied to the geometrically local, stabilizer-encoded Hamiltonians that arise naturally when fermionic Hamiltonians are represented in qubits, can suppress the additive error in the ground space overlap of the final output state exponentially in the code distance. This enables us to get closer to fault-tolerance for this task without the associated overhead. In contrast, for computation as opposed to ground state preparation, the second result proves that DQC is no more robust to noise than the standard quantum circuit model.

quant-ph↗

arXiv2025

Encodings of Observable Subalgebras

Simulating complex systems remains an ongoing challenge for classical computers, while being recognised as a task where a quantum computer has a natural advantage. In both digital and analogue quantum simulations the system description is first mapped onto qubits or the physical system of the analogue simulator by an encoding. Previously mathematical definitions and characterisations of encodings have focused on preserving the full physics of the system. In this work, we consider encodings that only preserve a subset of the observables of the system. We motivate that such encodings are best described as maps between formally real Jordan algebras describing the subset of observables. Our characterisation of encodings is general, but notably holds for maps between finite-dimensional and semisimple $C^{*}$-algebras. Fermionic encodings are a pertinent example where a mathematical characterisation was absent. Our work applies to encodings of the the full CAR algebra, but also to encodings of just the even parity sector, corresponding to the physically relevant fermionic operators.

quant-ph↗

arXiv2025

Myths around quantum computation before full fault tolerance: What no-go theorems rule out and what they don't

In this perspective article, we revisit and critically evaluate prevailing viewpoints on the capabilities and limitations of near-term quantum computing and its potential transition toward fully fault-tolerant quantum computing. We examine theoretical no-go results and their implications, addressing misconceptions about the practicality of quantum error mitigation techniques and variational quantum algorithms. By emphasizing the nuances of error scaling, circuit depth, and algorithmic feasibility, we highlight viable near-term applications and synergies between error mitigation and early fault-tolerant architectures. Our discussion explores strategies for addressing current challenges, such as barren plateaus in variational circuits and the integration of quantum error mitigation and quantum error correction techniques. We aim to underscore the importance of continued innovation in hardware and algorithmic design to bridge the gap between theoretical potential and practical utility, paving the way for meaningful quantum advantage in the era of late noisy intermediate scale and early fault-tolerant quantum devices.

quant-ph↗

arXiv2024

Undecidability in Physics: a Review

The study of undecidability in problems arising from physics has experienced a renewed interest, mainly in connection with quantum information problems. The goal of this review is to survey this recent development. After a historical introduction, we first explain the necessary results about undecidability in mathematics and computer science. Then we briefly review the first results about undecidability in physics which emerged mostly in the 80s and early 90s. Finally we focus on the most recent contributions, which we divide in two main categories: many body systems and quantum information problems.

math-ph↗

arXiv2024

Chaitin Phase Transition

We construct a family of Hamiltonians whose phase diagram is guaranteed to have a single phase transition, yet the location of this phase transition is uncomputable. The Hamiltonians $H(\phi)$ describe qudits on a two-dimensional square lattice with translationally invariant, nearest-neighbour interactions tuned by a continuous parameter $\phi\in(0,1]$. For all $\phi\in(0,1]$, $H(\phi)$ is in one of two phases, one a gapless phase, the other a gapped phase. The phase transition occurs when $\phi$ equals the Chaitin's constant $\Omega$, a well-defined real number that encodes the Halting problem, and hence is uncomputable for Turing machines and undecidable for any consistent recursive axiomatization of mathematics. Our result implies that no general algorithm exists to determine the phase diagrams even under the promise that the phase diagram is exceedingly simple, and illustrates how uncomputable numbers may manifest in physical systems.

quant-ph↗

arXiv2024

Enhancing density functional theory using the variational quantum eigensolver

Quantum computers open up new avenues for modelling the physical properties of materials and molecules. Density Functional Theory (DFT) is the gold standard classical algorithm for predicting these properties, but relies on approximations of the unknown universal functional, limiting its general applicability for many fundamental and technologically relevant systems. In this work we develop a hybrid quantum/classical algorithm called quantum enhanced DFT (QEDFT) that systematically constructs quantum approximations of the universal functional using data obtained from a quantum computer. We benchmark the QEDFT algorithm on the Fermi-Hubbard model, both numerically and on data from experiments on real quantum hardware. We find that QEDFT surpasses the quality of groundstate results obtained from Hartree-Fock DFT, as well as from direct application of conventional quantum algorithms such as VQE. Furthermore, we demonstrate that QEDFT works even when only noisy, low-depth quantum computation is available, by benchmarking the algorithm on data obtained from Google's quantum computer. We further show how QEDFT also captures quintessential properties of strongly correlated Mott physics for large Fermi-Hubbard systems using functionals generated on much smaller system sizes. Our results indicate that QEDFT can be applied to realistic materials and molecular systems, and has the potential to outperform the direct application of either DFT or VQE alone, without the requirement of large scale or fully fault-tolerant quantum computers.

quant-ph↗

arXiv2024

Security of quantum position-verification limits Hamiltonian simulation via holography

We investigate the link between quantum position-verification (QPV) and holography established in [MPS19] using holographic quantum error correcting codes as toy models. By inserting the "temporal" scaling of the AdS metric by hand via the bulk Hamiltonian interaction strength, we recover a toy model with consistent causality structure. This leads to an interesting implication between two topics in quantum information: if position-based verification is secure against attacks with small entanglement then there are new fundamental lower bounds for resources required for one Hamiltonian to simulate another.

quant-ph↗

arXiv2023

Quantum Error Transmutation

We introduce a generalisation of quantum error correction, relaxing the requirement that a code should identify and correct a set of physical errors on the Hilbert space of a quantum computer exactly, instead allowing recovery up to a pre-specified admissible set of errors on the code space. We call these quantum error transmuting codes. They are of particular interest for the simulation of noisy quantum systems, and for use in algorithms inherently robust to errors of a particular character. Necessary and sufficient algebraic conditions on the set of physical and admissible errors for error transmutation are derived, generalising the Knill-Laflamme quantum error correction conditions. We demonstrate how some existing codes, including fermionic encodings, have error transmuting properties to interesting classes of admissible errors. Additionally, we report on the existence of some new codes, including low-qubit and translation invariant examples.

quant-ph↗

arXiv2023

Dissipative Quantum Gibbs Sampling

Systems in thermal equilibrium at non-zero temperature are described by their Gibbs state. For classical many-body systems, the Metropolis-Hastings algorithm gives a Markov process with a local update rule that samples from the Gibbs distribution. For quantum systems, sampling from the Gibbs state is significantly more challenging. Many algorithms have been proposed, but these are more complex than the simple local update rule of classical Metropolis sampling, requiring non-trivial quantum algorithms such as phase estimation as a subroutine. Here, we show that a dissipative quantum algorithm with a simple, local update rule is able to sample from the quantum Gibbs state. In contrast to the classical case, the quantum Gibbs state is not generated by converging to the fixed point of a Markov process, but by the states generated at the stopping time of a conditionally stopped process. This gives a new answer to the long-sought-after quantum analogue of Metropolis sampling. Compared to previous quantum Gibbs sampling algorithms, the local update rule of the process has a simple implementation, which may make it more amenable to near-term implementation on suitable quantum hardware. This dissipative Gibbs sampler works for arbitrary quantum Hamiltonians, without any assumptions on or knowledge of its properties, and comes with certifiable precision and run-time bounds. We also show that the algorithm benefits from some measure of built-in resilience to faults and errors (``fault resilience''). Finally, we also demonstrate how the stopping statistics of an ensemble of runs of the dissipative Gibbs sampler can be used to estimate the partition function.

quant-ph↗

arXiv2022

A mathematical framework for quantum Hamiltonian simulation and duality

Analogue Hamiltonian simulation is a promising near-term application of quantum computing and has recently been put on a theoretical footing. In Hamiltonian simulation, a physical Hamiltonian is engineered to have identical physics to another - often very different - Hamiltonian. This is qualitatively similar to the notion of duality in physics, whereby two superficially different theories are mathematically equivalent in some precise sense. However, existing characterisations of Hamiltonian simulations are not sufficiently general to extend to all dualities in physics. In particular, they cannot encompass the important cases of strong/weak and high-temperature/low-temperature dualities. In this work, we give three physically motivated axiomatisations of duality, formulated respectively in terms of observables, partition functions and entropies. We prove that these axiomatisations are equivalent, and characterise the mathematical form that any duality satisfying these axioms must take. A building block in one of our results is a strengthening of earlier results on entropy-preserving maps to maps that are entropy-preserving up to an additive constant, which we prove decompose as a direct sum of unitary and anti-unitary components, which may be of independent mathematical interest.

quant-ph↗

arXiv2022

Towards near-term quantum simulation of materials

Simulation of materials is one of the most promising applications of quantum computers. On near-term hardware the crucial constraint on these simulations is circuit depth. Many quantum simulation algorithms rely on a layer of unitary evolutions generated by each term in a Hamiltonian. This appears in time-dynamics as a single Trotter step, and in variational quantum eigensolvers under the Hamiltonian variational ansatz as a single ansatz layer. We present a new quantum algorithm design for materials modelling where the depth of a layer is independent of the system size. This design takes advantage of the locality of materials in the Wannier basis and employs a tailored fermionic encoding that preserves locality. We analyse the circuit costs of this approach and present a compiler that transforms density functional theory data into quantum circuit instructions -- connecting the physics of the material to the simulation circuit. The compiler automatically optimises circuits at multiple levels, from the base gate level to optimisations derived from the physics of the specific target material. We present numerical results for materials spanning a wide structural and technological range. Our results demonstrate a reduction of many orders of magnitude in circuit depth over standard prior methods that do not consider the structure of the Hamiltonian. For example our results improve resource requirements for Strontium Vanadate (SrVO$_3$) from 864 to 180 qubits for a $3\times3\times3$ lattice, and the circuit depth of a single Trotter or variational layer from $7.5\times 10^8$ to depth $884$. Although this is still beyond current hardware, our results show that materials simulation may be feasible on quantum computers without necessarily requiring scalable, fault-tolerant quantum computers, provided quantum algorithm design incorporates understanding of the materials and applications.

quant-ph↗

arXiv2021

Phase Estimation of Local Hamiltonians on NISQ Hardware

In this work we investigate a binned version of Quantum Phase Estimation (QPE) set out by [Somma 2019] and known as the Quantum Eigenvalue Estimation Problem (QEEP). Specifically, we determine whether the circuit decomposition techniques we set out in previous work, [Clinton et al 2020], can improve the performance of QEEP in the NISQ regime. To this end we adopt a physically motivated abstraction of NISQ device capabilities as in [Clinton et al 2020]. Within this framework, we find that our techniques reduce the threshold at which it becomes possible to perform the minimum two-bin instance of this algorithm by an order of magnitude. This is for the specific example of a two dimensional spin Fermi-Hubbard model. For example, given a $10\%$ acceptable error on a $3\times 3$ spin Fermi-Hubbard model, with a depolarizing noise rate of $10^{-6}$, we find that the phase estimation protocol of [Somma 2019] could be performed with a bin width of approximately $1/9$ the total spectral range at the circuit depth where traditional gate synthesis methods would yield a bin width that covers the entire spectral range. We explore possible modifications to this protocol and propose an application, which we call Randomized Quantum Eigenvalue Estimation Problem (rQeep). rQeep outputs estimates on the fraction of eigenvalues which lie within randomly chosen bins and upper bounds the total deviation of these estimates from the true values. One use case we envision for this algorithm is resolving density of states features of local Hamiltonians, such as spectral gaps.

quant-ph↗

arXiv2021

Holographic duality between local Hamiltonians from random tensor networks

The AdS/CFT correspondence realises the holographic principle where information in the bulk of a space is encoded at its border. We are yet a long way from a full mathematical construction of AdS/CFT, but toy models in the form of holographic quantum error correcting codes (HQECC) have replicated some interesting features of the correspondence. In this work we construct new HQECCs built from random stabilizer tensors that describe a duality between models encompassing local Hamiltonians whilst exactly obeying the Ryu-Takayanagi entropy formula for all boundary regions. We also obtain complementary recovery of local bulk operators for any boundary bipartition. Existing HQECCs have been shown to exhibit these properties individually, whereas our mathematically rigorous toy models capture these features of AdS/CFT simultaneously, advancing further towards a complete construction of holographic duality.

hep-th↗

arXiv2021

General conditions for universality of Quantum Hamiltonians

Recent work has demonstrated the existence of universal Hamiltonians - simple spin lattice models that can simulate any other quantum many body system to any desired level of accuracy. Until now proofs of universality have relied on explicit constructions, tailored to each specific family of universal Hamiltonians. In this work we go beyond this approach, and completely classify the simulation ability of quantum Hamiltonians by their complexity classes. We do this by deriving necessary and sufficient complexity theoretic conditions characterising universal quantum Hamiltonians. Although the result concerns the theory of analogue Hamiltonian simulation - a promising application of near-term quantum technology - the proof relies on abstract complexity theoretic concepts and the theory of quantum computation. As well as providing simplified proofs of previous Hamiltonian universality results, and offering a route to new universal constructions, the results in this paper give insight into the origins of universality. For example, finally explaining the previously noted coincidences between families of universal Hamiltonian and classes of Hamiltonians appearing in complexity theory.

quant-ph↗

arXiv2020

Translationally-Invariant Universal Quantum Hamiltonians in 1D

Recent work has characterised rigorously what it means for one quantum system to simulate another, and demonstrated the existence of universal Hamiltonians -- simple spin lattice Hamiltonians that can replicate the entire physics of any other quantum many body system. Previous universality results have required proofs involving complicated `chains' of perturbative `gadgets'. In this paper, we derive a significantly simpler and more powerful method of proving universality of Hamiltonians, directly leveraging the ability to encode quantum computation into ground states. This provides new insight into the origins of universal models, and suggests a deep connection between universality and complexity. We apply this new approach to show that there are universal models even in translationally invariant spin chains in 1D. This gives as a corollary a new Hamiltonian complexity result, that the local Hamiltonian problem for translationally-invariant spin chains in one dimension with an exponentially-small promise gap is PSPACE-complete. Finally, we use these new universal models to construct the first known toy model of 2D--1D holographic duality between local Hamiltonians.

quant-ph↗