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Toby S. Cubitt

Publications and source records attributed to Toby S. Cubitt.

At least 19 recordsLinked to original sources
arXiv2025

Robust Lindbladian Estimation for Quantum Dynamics

We revisit the problem of fitting Lindbladian models to the outputs of quantum process tomography. A sequence of prior theoretical works approached the problem by considering whether there exists a Lindbladian generator close to a matrix logarithm of the tomographically estimated transfer matrix. This technique must take into account the non-uniqueness of the matrix logarithm, so that in general multiple branches of the logarithm must be checked. In contrast, all practical demonstrations of Lindbladian fitting on real experimental data have to our knowledge eschewed logarithm search, instead adopting direct numerical optimisation or ad-hoc approaches tailored to a particular experimental realisation. In our work, we introduce algorithmic improvements to logarithm search, demonstrating that it can be applied in practice to settings relevant for current quantum computing hardware. We additionally augment the task of Lindbladian fitting with techniques from gate set tomography to improve robustness against state preparation and measurement (SPAM) errors, which can otherwise obfuscate estimates of the model underlying the process of interest. We benchmark our techniques extensively using simulated tomographic data employing a range of realistic error models, before demonstrating their application to tomographic data collected from real superconducting-qubit hardware.

quant-ph↗
arXiv2024

Quantum Phase Estimation without Controlled Unitaries

In this work we demonstrate the use of adapted classical phase retrieval algorithms to perform control-free quantum phase estimation. We eliminate the costly controlled time evolution and Hadamard test commonly required to access the complex time-series needed to reconstruct the spectrum. This significant reduction of the number of coherent controlled-operations lowers the circuit depth and considerably simplifies the implementation of statistical quantum phase estimation in near-term devices. This seemingly impossible task can be achieved by extending the problem that one wishes to solve to one with a larger set of input signals while exploiting natural constraints on the signal and/or the spectrum. We leverage well-established algorithms that are widely used in the context of classical signal processing, demonstrating two complementary methods to do this, vectorial phase retrieval and two-dimensional phase retrieval. We numerically investigate the feasibility of both approaches for estimating the spectrum of the Fermi-Hubbard model and discuss their resilience to inherent statistical noise.

quant-ph↗
arXiv2023

Fitting quantum noise models to tomography data

The presence of noise is currently one of the main obstacles to achieving large-scale quantum computation. Strategies to characterise and understand noise processes in quantum hardware are a critical part of mitigating it, especially as the overhead of full error correction and fault-tolerance is beyond the reach of current hardware. Non-Markovian effects are a particularly unfavourable type of noise, being both harder to analyse using standard techniques and more difficult to control using error correction. In this work we develop a set of efficient algorithms, based on the rigorous mathematical theory of Markovian master equations, to analyse and evaluate unknown noise processes. In the case of dynamics consistent with Markovian evolution, our algorithm outputs the best-fit Lindbladian, i.e., the generator of a memoryless quantum channel which best approximates the tomographic data to within the given precision. In the case of non-Markovian dynamics, our algorithm returns a quantitative and operationally meaningful measure of non-Markovianity in terms of isotropic noise addition. We provide a Python implementation of all our algorithms, and benchmark these on a range of 1- and 2-qubit examples of synthesised noisy tomography data, generated using the Cirq platform. The numerical results show that our algorithms succeed both in extracting a full description of the best-fit Lindbladian to the measured dynamics, and in computing accurate values of non-Markovianity that match analytical calculations.

quant-ph↗
arXiv2023

Dissipative ground state preparation and the Dissipative Quantum Eigensolver

For any local Hamiltonian H, I construct a local CPT map and stopping condition which converges to the ground state subspace of H. Like any ground state preparation algorithm, this algorithm necessarily has exponential run-time in general (otherwise BQP=QMA), even for gapped, frustration-free Hamiltonians (otherwise BQP is in NP). However, this dissipative quantum eigensolver has a number of interesting characteristics, which give advantages over previous ground state preparation algorithms. - The entire algorithm consists simply of iterating the same set of local measurements repeatedly. - The expected overlap with the ground state subspace increases monotonically with the length of time this process is allowed to run. - It converges to the ground state subspace unconditionally, without any assumptions on or prior information about the Hamiltonian. - The algorithm does not require any variational optimisation over parameters. - It is often able to find the ground state in low circuit depth in practice. - It has a simple implementation on certain types of quantum hardware, in particular photonic quantum computers. - The process is immune to errors in the initial state. - It is inherently error- and noise-resilient, i.e. to errors during execution of the algorithm and also to faulty implementation of the algorithm itself, without incurring any computational overhead: the overlap of the output with the ground state subspace degrades smoothly with the error rate, independent of the algorithm's run-time. I give rigorous proofs of the above claims, and benchmark the algorithm on some concrete examples numerically.

quant-ph↗
arXiv2023

Fitting time-dependent Markovian dynamics to noisy quantum channels

Understanding how to characterise and mitigate errors is a key challenge in developing reliable quantum architecture for near-term applications. Recent work (arXiv:2103.17243) provides an efficient set of algorithms for analysing unknown noise processes requiring only tomographic snapshots of the quantum operator under consideration, without the need of any a-priori information on the noise model, nor necessitating a particular experimental setup. The only assumption made is that the observed channel can be approximated by a time-independent Markovian map, which is a typically reasonable framework when considering short time scales. In this note we lift the time-independent assumption, presenting an extension of the scheme now able to analyse noisy dynamics with time-dependent generators from a sequence of snapshots. We hence provide a diagnostic tool for a wider spectrum of instances while inheriting all the favourable features from the previous protocol. On the theoretical side, the problem of characterising time-dependent Markovian channels has been an open problem for many decades. This work gives an approach to tackle this characterisation problem rigorously.

quant-ph↗
arXiv2021

Computational Complexity of the Ground State Energy Density Problem

We study the complexity of finding the ground state energy density of a local Hamiltonian on a lattice in the thermodynamic limit of infinite lattice size. We formulate this rigorously as a function problem, in which we request an estimate of the ground state energy density to some specified precision; and as an equivalent promise problem, $\mathsf{GSED}$, in which we ask whether the ground state energy density is above or below specified thresholds. The ground state energy density problem is unusual, in that it concerns a single, fixed Hamiltonian in the thermodynamic limit, whose ground state energy density is just some fixed, real number. The only input to the computational problem is the precision to which to estimate this fixed real number, corresponding to the ground state energy density. Hardness of this problem for a complexity class therefore implies that the solutions to all problems in the class are encoded in this single number (analogous to Chaitin's constant in computability theory). This captures computationally the type of question most commonly encountered in condensed matter physics, which is typically concerned with the physical properties of a single Hamiltonian in the thermodynamic limit. We show that for classical, translationally invariant, nearest neighbour Hamiltonians on a 2D square lattice, $\mathsf{P}^{\mathsf{NEEXP}}\subseteq\mathsf{EXP}^{\mathsf{GSED}}\subseteq \mathsf{EXP}^{\mathsf{NEXP}}$, and for quantum Hamiltonians $\mathsf{P}^{\mathsf{NEEXP}}\subseteq\mathsf{EXP}^{\mathsf{GSED}}\subseteq \mathsf{EXP}^{\mathsf{QMA}_{EXP}}$. With some technical caveats on the oracle definitions, the $\mathsf{EXP}$ in some of these results can be strengthened to $\mathsf{PSPACE}$. We also give analogous complexity bounds for the function version of $\mathsf{GSED}$.

quant-ph↗
arXiv2021

A Note on the Second Spectral Gap Incompleteness Theorem

Pick a formal system. Any formal system. Whatever your favourite formal system is, as long as it's capable of reasoning about elementary arithmetic. The First Spectral Gap Incompleteness Theorem of [CPGW15] proved that there exist Hamiltonians whose spectral gap is independent of that system; your formal system is incapable of proving that the Hamiltonian is gapped, and equally incapable of proving that it's gapless. In this note, I prove a Second Spectral Gap Incompleteness Theorem: I show how to explicitly construct, within the formal system, a concrete example of a Hamiltonian whose spectral gap is independent of that system. Just to be sure, I prove this result three times. Once with G\"odel's help. Once with Zermelo and Fraenkel's help. And finally, doing away with these high-powered friends, I give a simple, direct argument which reveals the inherent self-referential structure at the heart of these results, by asking the Hamiltonian about its own spectral gap.

quant-ph↗
arXiv2021

Uncomputably Complex Renormalisation Group Flows

Renormalisation group (RG) methods provide one of the most important techniques for analysing the physics of many-body systems, both analytically and numerically. By iterating an RG map, which "course-grains" the description of a many-body system and generates a flow in the parameter space, physical properties of interest can be extracted even for complex models. RG analysis also provides an explanation of physical phenomena such as universality. Many systems exhibit simple RG flows, but more complicated -- even chaotic -- behaviour is also known. Nonetheless, the structure of such RG flows can still be analysed, elucidating the physics of the system, even if specific trajectories may be highly sensitive to the initial point. In contrast, recent work has shown that important physical properties of quantum many-body systems, such as its spectral gap and phase diagram, can be uncomputable. In this work, we show that such undecidable systems exhibit a novel type of RG flow, revealing a qualitatively different and more extreme form of unpredictability than chaotic RG flows. In contrast to chaotic RG flows in which initially close points can diverge exponentially, trajectories under these novel uncomputable RG flows can remain arbitrarily close together for an uncomputable number of iterations, before abruptly diverging to different fixed points that are in separate phases. The structure of such uncomputable RG flows is so complex that it cannot be computed or approximated, even in principle. We give a mathematically rigorous construction of the block-renormalisation-group map for the original undecidable many-body system that appeared in the literature (Cubitt, Pérez-Garcia, Wolf, Nature 528, 207-211 (2015)). We prove that each step of this RG map is computable, and that it converges to the correct fixed points, yet the resulting RG flow is uncomputable.

quant-ph↗
arXiv2021

Uncomputability of Phase Diagrams

The phase diagram of a material is of central importance to describe the properties and behaviour of a condensed matter system. We prove that the general task of determining the quantum phase diagram of a many-body Hamiltonian is uncomputable, by explicitly constructing a one-parameter family of Hamiltonians for which this is the case. This work builds off recent results from Cubitt et al. and Bausch et al., proving undecidability of the spectral gap problem. However, in all previous constructions, the Hamiltonian was necessarily a discontinuous function of its parameters, making it difficult to derive rigorous implications for phase diagrams or related condensed matter questions. Our main technical contribution is to prove undecidability of the spectral gap for a continuous, single-parameter family of translationally invariant, nearest-neighbour spin-lattice Hamiltonians on a 2D square lattice: $H(φ)$ where $φ\in \mathbb R$. As well as implying uncomputablity of phase diagrams, our result also proves that undecidability can hold for a set of positive measure of a Hamiltonian's parameter space, whereas previous results only implied undecidability on a zero measure set.

quant-ph↗
arXiv2018

History-state Hamiltonians are critical

All Hamiltonian complexity results to date have been proven by constructing a local Hamiltonian whose ground state -- or at least some low-energy state -- is a "computational history state", encoding a quantum computation as a superposition over the history of the computation. We prove that all history-state Hamiltonians must be critical. More precisely, for any circuit-to-Hamiltonian mapping that maps quantum circuits to local Hamiltonians with low-energy history states, there is an increasing sequence of circuits that maps to a growing sequence of Hamiltonians with spectral gap closing at least as fast as O(1/n) with the number of qudits n in the circuit. This result holds for very general notions of history state, and also extends to quasi-local Hamiltonians with exponentially-decaying interactions. This suggests that QMA-hardness for gapped Hamiltonians (and also BQP-completeness of adiabatic quantum computation with constant gap) either require techniques beyond history state constructions. Or gapped Hamiltonians cannot be QMA-hard (respectively, BQP-complete).

quant-ph↗
arXiv2018

Size-Driven Quantum Phase Transitions

Can the properties of the thermodynamic limit of a many-body quantum system be extrapolated by analysing a sequence of finite-size cases? We present a model for which such an approach gives completely misleading results: a translationally invariant, local Hamiltonian on a square lattice with open boundary conditions and constant spectral gap, which has a classical product ground state for all system sizes smaller than a particular threshold size, but a ground state with topological degeneracy for all system sizes larger than this threshold. Starting from a minimal case with spins of dimension 6 and threshold lattice size 15 x 15, we show that the latter grows faster than any computable function with increasing local spin dimension. The resulting effect may be viewed as a new type of quantum phase transition that is driven by the size of the system rather than by an external field or coupling strength. We prove that the construction is thermally robust, opening the possibility that these effects are accessible to experimental observation.

quant-ph↗
arXiv2016

Simple universal models capture all classical spin physics

Spin models are used in many studies of complex systems---be it condensed matter physics, neural networks, or economics---as they exhibit rich macroscopic behaviour despite their microscopic simplicity. Here we prove that all the physics of every classical spin model is reproduced in the low-energy sector of certain `universal models'. This means that (i) the low energy spectrum of the universal model reproduces the entire spectrum of the original model to any desired precision, (ii) the corresponding spin configurations of the original model are also reproduced in the universal model, (iii) the partition function is approximated to any desired precision, and (iv) the overhead in terms of number of spins and interactions is at most polynomial. This holds for classical models with discrete or continuous degrees of freedom. We prove necessary and sufficient conditions for a spin model to be universal, and show that one of the simplest and most widely studied spin models, the 2D Ising model with fields, is universal.

cond-mat.stat-mech↗
arXiv2016

Comment on "On the uncomputability of the spectral gap"

The aim of this short note is to clarify some of the claims made in the comparison made in [S. Lloyd, On the uncomputability of the spectral gap, arXiv:1602.05924] between our recent result [T.S. Cubitt, D. Perez-Garcia, M.M. Wolf, Undecidability of the spectral gap, Nature 528, 207-211 (2015), arXiv:1502.04573] and his 1994 paper [S. Lloyd, Necessary and sufficient conditions for quantum computation, J. Mod. Opt. 41(12), 2503-2520 (1994)].

quant-ph↗
arXiv2016

Universal Refocusing of Systematic Quantum Noise

Refocusing of a quantum system in NMR and quantum information processing can be achieved by application of short pulses according to the methods of spin echo and dynamical decoupling. However, these methods are strongly limited by the requirement that the evolution of the system between pulses be suitably small. Here we show how refocusing may be achieved for arbitrary (but time-independent) evolution of the system between pulses. We first illustrate the procedure with one-qubit systems, and then generalize to $d$-dimensional quantum systems. We also give an application of this result to quantum computation, proving a new version of the Solovay-Kitaev theorem that does not require inverse gates.

quant-ph↗
arXiv2015

Stability of local quantum dissipative systems

Open quantum systems weakly coupled to the environment are modeled by completely positive, trace preserving semigroups of linear maps. The generators of such evolutions are called Lindbladians. In the setting of quantum many-body systems on a lattice it is natural to consider Lindbladians that decompose into a sum of local interactions with decreasing strength with respect to the size of their support. For both practical and theoretical reasons, it is crucial to estimate the impact that perturbations in the generating Lindbladian, arising as noise or errors, can have on the evolution. These local perturbations are potentially unbounded, but constrained to respect the underlying lattice structure. We show that even for polynomially decaying errors in the Lindbladian, local observables and correlation functions are stable if the unperturbed Lindbladian has a unique fixed point and a mixing time which scales logarithmically with the system size. The proof relies on Lieb-Robinson bounds, which describe a finite group velocity for propagation of information in local systems. As a main example, we prove that classical Glauber dynamics is stable under local perturbations, including perturbations in the transition rates which may not preserve detailed balance.

quant-ph↗
arXiv2015

Rapid mixing and stability of quantum dissipative systems

The physics of many materials is modeled by quantum many-body systems with local interactions. If the model of the system is sensitive to noise from the environment, or small perturbations to the original interactions, it will not properly model the robustness of the real physical system it aims to describe, or be useful when engineering novel systems for quantum information processing. We show that local observables and correlation functions of local Liouvillians are stable to local perturbations if the dynamics is rapidly mixing and has a unique fixed point. No other condition is required.

quant-ph↗
arXiv2013

A constructive commutative quantum Lovasz Local Lemma, and beyond

The recently proven Quantum Lovasz Local Lemma generalises the well-known Lovasz Local Lemma. It states that, if a collection of subspace constraints are "weakly dependent", there necessarily exists a state satisfying all constraints. It implies e.g. that certain instances of the kQSAT quantum satisfiability problem are necessarily satisfiable, or that many-body systems with "not too many" interactions are always frustration-free. However, the QLLL only asserts existence; it says nothing about how to find the state. Inspired by Moser's breakthrough classical results, we present a constructive version of the QLLL in the setting of commuting constraints, proving that a simple quantum algorithm converges efficiently to the required state. In fact, we provide two different proofs, one using a novel quantum coupling argument, the other a more explicit combinatorial analysis. Both proofs are independent of the QLLL. So these results also provide independent, constructive proofs of the commutative QLLL itself, but strengthen it significantly by giving an efficient algorithm for finding the state whose existence is asserted by the QLLL. We give an application of the constructive commutative QLLL to convergence of CP maps. We also extend these results to the non-commutative setting. However, our proof of the general constructive QLLL relies on a conjecture which we are only able to prove in special cases.

quant-ph↗