Entanglement-assisted perfect discrimination of quantum measurements

Connections between measurement incompatibility and quantum coherence

A no go theorem for local Gaussian work extraction for multimode bosonic systems

Avoiding local minima in variational quantum eigensolvers with the natural gradient optimizer

Upper bounds on the rate in device-independent quantum key distribution

Universal protocols for transforming unitary quantum operations:
Exponential advantage with adaptive protocols and the power of indefinite causality

Morphophoric quantum measurements, generalised qplexes, and 2-designs

On the universal constraints for relaxation rates for quantum dynamical semigroup

Measuring the projective-unitary invariant properties of a set of states, and applications

Quantifying causal influences in the presence of a quantum common cause

General Measurements with Limited Resources and Their Application to Quantum Unambiguous State Discrimination

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

In the era of Noisy Intermediate-Scale Quantum (NISQ) computing, the design of space-efficient and fault-tolerant quantum algorithms is inevitable. Considering the problems defined over permutations, it becomes harder to reach the optimal solutions as for some problems the feasible solutions constitute only a small fraction of the whole space. Addressing these issues, we propose Encoding-Changing Quantum Approximate Optimization Algorithm (EC-QAOA), which bases on a different ansatz compared to the original QAOA. We demonstrate the effectiveness of the proposed method through the Travelling Salesman Problem. Furthermore, we show that the proposed approach enables quantum error mitigation using mid-circuit measurements. We compare the performance of the proposed method with the existing approaches by numerical studies.

Bistochastic and unitary matrices are similar concepts in respectively classical and quantum domains. We investigate the connection between them, in particular the problem of determining which bistochastic matrix has its unitary counterpart, i.e. the unistochasticity problem, with applications ranging from quantum walks to particle physics. The talk shall be composed of an exposition of the results concerning new algebraical and geometrical structures inside bistochastic matrices.

Once a quantum computation or simulation setup gains in size and coherence new research challenges appear when trying to get a handle on what the experimental device is doing. One of the most immediate questions that emerges is simply: How can I know that the system performs correctly if quantum state tomography is out of reach? Broadly speaking, we would like to devise ways for certifying successful implementations of quantum protocols despite large numbers of degrees of freedom involved in the process. In the talk I will describe the notion of fidelity witnesses which offer an experimentally-friendly way for circumventing the difficulty of measuring the fidelity between a known target quantum state and an unknown experimental preparation. Part of the talk will be devoted to specific fidelity witnesses useful for certifying high-fidelity state preparations of fermionic Gaussian states which recently found an application on the Sycamore quantum processor. Besides giving an overview of what is the current understanding of the range of applicability of the fidelity witness approach to verifying quantum simulators, I will also hint at comparisons to other post-tomographic certification methods.

The optimal transport problem, established by Monge and refined by Kantorovich and Wasserstein, has ubiquitous applications in statistics, machine learning, computer vision and early Universe reconstruction. Recently, several approaches towards its quantum version have been developed. In the talk I will provide an overview of the quantum optimal transport problem. The general results will be illustrated with a more detailed study of the single-qubit transport problem. In particular, I will show that the quantum optimal transport induces a new metric on the Bloch ball with intriguing properties. The talk is based on a joint work with S. Cole, S. Friedland and K. Życzkowski - arXiv:2102.07787, 2105.06922.

My presentation will be focused on various approaches towards the task of discrimination of quantum measurements and channels. I will elaborate on minimum error discrimination, unambiguous discrimination, and asymmetric discrimination, which is also known as certification. All of these approaches will be considered in the single-shot scheme as well as multiple-shot scenarios. I will also discuss when parallel strategies are optimal for discrimination and when the use of additional processing in adaptive scheme gives an advantage over the parallel one.

Designing locally maximally entangled quantum states with arbitrary local symmetries

One of the key ingredients of many LOCC protocols in quantum information is a multiparticle (locally) maximally entangled quantum state, aka a critical state, that possesses local symmetries. It is natural to ask if such states exist in a given quantum system and how do they look like. During my talk, I will briefly explain why such states are important. Then, I will introduce some relevant notions, such as diagonal H-symmetries, and move to the main technical result which states that the Nth tensor power of any irreducible representation of SU(N) contains a copy of the trivial representation. The rest of the talk will be devoted to the applications of the main result with its corollaries and examples of designing critical states with large local unitary symmetry. In particular, I will explain that critical states with large local symmetries can be realised in a quantum system of distinguishable traps with bosons or fermions occupying a finite number of modes. I will also link our results with the existence of so-called strictly semistable states with particular asymptotic diagonal symmetries.

The impact of measurement imperfections on quantum metrology protocols has been largely ignored, even though these are inherent to any sensing platform in which the detection process exhibits noise that neither can be eradicated, nor translated onto the sensing stage and interpreted as decoherence. In this seminar, we report our recent work in addressing the issue of imperfect measurement in quantum metrology, in a systematic manner (arXiv:2109.01160). Specifically, we demonstrate how the quantum Fisher information must be modified to account for noisy detection, and we propose tractable methods allowing for its approximate evaluation. Using this general expression, we then prove a go-theorem, which states that in canonical scenarios involving N probes with local measurements undergoing readout noise, the effect of noisy detection can be counterbalanced, by implementing a suitable global control unitary operation before the readout stage. This shows that the ideal, quantum-enhanced sensitivity (e.g. Heisenberg scaling) can always be recovered given a large enough number of probes. On the contrary, we also prove a no-go theorem, which states that such a feat cannot however be achieved with just local control operations, where the optimal sensitivity will then be limited to just a constant factor improvement over the classical results, which scales linearly with the probe size. We illustrate our results with a relevant example of an NV-centre used to sense a magnetic field, as well as schemes involving spin-1/2 probes (qubits) with bit-flip errors affecting their two-outcome measurements. We also provide the input states and control unitary operations sufficient to attain the ultimate asymptotic precision.

I will discuss the following task: given an unknown unitary gate U as a black box, implement the controlled-unitary* gate. Araújo et al. showed that a quantum circuit that makes one call to U cannot implement controlled-U. I will show that the task remains impossible even if the quantum circuit is allowed any number of calls to U. Our result also excludes circuits that use postselection and only approximate the task. Handling approximation and postselection simultaneously requires a new notion: diamond distance for the postselected setting. * Up to a certain relative phase. The papers: arXiv:2011.10031 arXiv:2011.08487

Unlike classical states, quantum states cannot necessarily be extended in such a way that the two-particle reduced states are all identical. More precisely, only the separable states are those that can be extended in such a way. The so-called shareability or extendibility number describes to how many parties a given state can be extended to. This is a good entanglement measure (i.e., a LOCC-monotone function), however, it has been calculated only for a few types of states. The talk presents the (k,l)-shareable states for a set Werner-like states, and the set of (1,2)-shareable OO-states.

Device-independent certification schemes have gained a lot of interest lately. In this regard, we explore quantum steering for certifying higher-dimensional quantum systems in a one-sided device-independent way. In the first part of the talk, I would discuss our proposal of a one-sided device-independent protocol that could certify any set of d-outcome projective measurements which do not share any common invariant subspace which we termed as “genuinely incompatible measurements” which includes mutually unbiased bases which are an important resource for quantum cryptography. We also find the robustness of our protocol for a class of mutually unbiased bases towards experimental imperfections. In the second part of the talk, I would discuss our proposal of a one-sided device-independent protocol that could certify any bipartite entangled state using a minimal number of measurements possible, that is, two per subsystem. Using the certified state, we were able to certify every extremal POVM which in turn can be used to certify randomness of amount 2logd bits, which is the maximum amount that can be achieved using quantum systems of dimension d.

The disturbance caused by measurements in quantum mechanics depends on the interaction between system and apparatus. If this interaction obeys a conservation law, the observables that may be non-disturbed will be restricted. We obtain general bounds that indicate the necessary conditions for non-disturbance in the presence of a conservation law and show that an observable not commuting with the conserved quantity admits a repeatable measurement – a special instance of a non-disturbing measurement – only if it is unsharp, and the apparatus is prepared in a state with a large uncertainty in the conserved quantity. This generalises the well-known Wigner-Araki-Yanase theorem.

We show that it is impossible to perform ideal projective measurements on quantum systems using finite resources. We identify three fundamental features of ideal projective measurements and show that when limited by finite resources only one of these features can be salvaged. Our framework is general enough to accommodate any system and measuring device (pointer) models, but for illustration we use an explicit model of an N-particle pointer. For a pointer that perfectly reproduces the statistics of the system, we provide tight analytic expressions for the energy cost of performing the measurement. This cost may be broken down into two parts: first, the cost of preparing the pointer in a suitable state, and second, the cost of a global interaction between the system and pointer in order to correlate them. It turns out that even under the assumption that the interaction can be controlled perfectly, achieving perfect correlation is infinitely expensive. We provide protocols for achieving optimal correlation given finite resources for the most general system and pointer Hamiltonians, phrasing our results as fundamental bounds in terms of the dimensions of these systems. Finally, we show on how our results affect Jarzynski and Crook’s relations in the context of the two point measurement scheme.

In this talk I will discuss the concept of self-testing which aims to answer the fundamental question of how do we certify proper functioning of black-box quantum devices. We will see that there is a close link between self-testing and representations of algebraic relations. We will leverage this link to propose a family of protocols capable of certifying quantum states and measurements of arbitrarily large dimension with just four binary-outcome measurements. This is a joint work with Chris Schafhauser and Jitendra Prakash.

A causal structure is a description of the functional dependencies between random variables. A distribution is compatible with a given causal structure if it can be realized by a process respecting these dependencies. Deciding whether a distribution is compatible with a structure is a practically and fundamentally relevant, yet very difficult problem. Only recently has a general class of algorithms been proposed: These so-called inflation techniques associate to any causal structure a hierarchy of increasingly strict compatibility tests, where each test can be formulated as a computationally efficient convex optimization problem. Remarkably, it has been shown that in the classical case, this hierarchy is complete in the sense that each non-compatible distribution will be detected at some level of the hierarchy. An inflation hierarchy has also been formulated for causal structures that allow for the observed classical random variables to arise from measurements on quantum states - however, no proof of completeness of this quantum inflation hierarchy has been supplied. In this presentation, I will talk about causal structures and the inflation technique and our recent paper (https://arxiv.org/abs/2110.14659) in which we construct a first version of the quantum inflation hierarchy that is provably convergent. It takes an additional parameter, r, which can be interpreted as an upper bound on the Schmidt rank of the observables involved. For each r, it provides a family of increasingly strict and ultimately complete compatibility tests for correlations that are compatible with a given causal structure under this Schmidt rank constraint. From a technical point of view, convergence proofs are built on de Finetti Theorems, which show that certain symmetries (which can be imposed in convex optimization problems) imply independence of random variables (which is not directly a convex constraint). A main technical ingredient to our proof is a Quantum de Finetti Theorem that holds for general tensor products of C*-algebras, generalizing previous work that was restricted to minimal tensor products.

Device-independent lower bounds on the conditional von Neumann entropy

The rates of several device-independent (DI) protocols, including quantum key-distribution (QKD) and randomness expansion (RE), can be computed via an optimization of the conditional von Neumann entropy over a particular class of quantum states. In this work we introduce a numerical method to compute lower bounds on such rates. We derive a sequence of optimization problems that converge to the conditional von Neumann entropy of systems defined on general separable Hilbert spaces. Using the Navascués-Pironio-Acín hierarchy we can then relax these problems to semidefinite programs, giving a computationally tractable method to compute lower bounds on the rates of DI protocols. Applying our method to compute the rates of DI-RE and DI-QKD protocols we find substantial improvements over all previous numerical techniques, demonstrating significantly higher rates for both DI-RE and DI-QKD. In particular, for DI-QKD we show a new minimal detection efficiency threshold which is within the realm of current capabilities. Moreover, we demonstrate that our method is capable of converging rapidly by recovering instances of known tight analytical bounds. Finally, we note that our method is compatible with the entropy accumulation theorem and can thus be used to compute rates of finite round protocols and subsequently prove their security. Based on joint work with Peter Brown and Hamza Fawzi available https://arxiv.org/abs/2106.13692

The Universality Problem asks whether a finite set of one-qudit gates is universal, i.e. we can approximate arbitrarily well any other one-qudit gate by composing such gates. We describe two solutions to this problem: the first uses the adjoint representation and some additional tool, whereas the second exploits only a large enough representation (depending on the value of d).

I will discuss several protocols designed for the verification and the characterization of quantum devices. I will focus one a recently developed protocol to verify the output of a quantum computer using only classical means. I will present a minimal example for realizing such a verification protocol and discuss its experimental implementation.

Hessian-based toolbox for interpretable and reliable machines learning physics

machine learning; phase transition; quantum many-body physics; Fermi-Hubbard model

Identifying phase transitions is one of the key problems in quantum many-body physics. The challenge is the exponential growth of the complexity of quantum systems’ description with the number of studied particles, which quickly renders exact numerical analysis impossible. A promising alternative is to harness the power of machine learning (ML) methods designed to deal with large datasets [1]. However, ML models, and especially neural networks (NNs), are known for their black-box construction, i.e., they usually hinder any insight into the reasoning behind their predictions. As a result, if we apply ML to novel problems, neither we can fully trust their predictions (lack of reliability) nor learn what the ML model learned (lack of interpretability). I will present a set of Hessian-based methods opening the black box of ML models, increasing their interpretability and reliability. We demonstrate how these methods can guide physicists in understanding patterns responsible for the phase transition. We also show that influence functions allow checking that the NN, trained to recognize known quantum phases, can predict new unknown ones. We present this power both for the numerically simulated data from the one-dimensional extended spinless Fermi-Hubbard model [2] and experimental topological data [3]. We also show how we can generate error bars for the NN’s predictions and check whether the NN predicts using extrapolation instead of extracting information from the training data [4]. The presented toolbox is entirely independent of the ML model’s architecture and is thus applicable to various physical problems. [1] J. Carrasquilla. (2020). Machine learning for quantum matter, Advances in Physics: X, 5:1. [2] A. Dawid et al. (2020). Phase detection with neural networks: interpreting the black box. New J. Phys. 22, 115001. [3] N. Käming, A. Dawid, K. Kottmann et al. (2021). Unsupervised machine learning of topological phase transitions from experimental data. Mach. Learn.: Sci. Technol. 2, 035037. [4] A. Dawid et al. (2022). Hessian-based toolbox for interpretable and reliable machine learning in physics. Mach. Learn.: Sci. Technol. 3, 015002.

Boson Sampling; Gaussian; Quantum Computing; computational advantage

Gaussian boson sampling is a model of photonic quantum computing that has attracted attention as a platform for quantum devices capable of performing tasks that are out of reach of their classical counterparts. Most recent photonic quantum computational advantage experiments were performed within this Gaussian variant of bosonsampling, having observed events with over 100 photons and seriously challenged the capabilities of competing classical algorithms. Thus, there is significant interest in solidifying the mathematical and complexity-theoretic foundations for the hardness of simulating these devices. We show that there is no efficient classical algorithm to approximately sample from the output of an ideal Gaussian boson sampling device unless the polynomial hierarchy collapses, under the same two conjectures as the original bosonsampling proposal by Aaronson and Arkhipov. Crucial to the proof is a new method for programming a Gaussian boson sampling device such that the output probabilities are proportional to permanents of (submatrices of) an arbitrary matrix. This provides considerable flexibility in programming, and likely has applications much beyond those discussed here. We leverage this to make progress towards the goal of proving hardness in the regime where there are fewer than quadratically more modes than photons (i.e., in the high-collision regime). Our reduction suffices to prove that GBS is hard in the constant-collision regime, though we believe some ingredients of it can be used to push this direction further.

Causal model; Quantum foundations; directed acyclic graphs; hidden variables; inequality constraints

Directed acyclic graphs (DAGs) with hidden variables are often used to characterize causal relations between variables in a system. When some variables are unobserved, DAGs imply a notoriously complicated set of constraints on the distribution of observed variables. I will discuss how to construct inequality constraints implied by graphical criterions that can be applied to hidden variable DAGs and learn from them about the true causal model. The focus of my presentation will be on causal models that exhibit d-separation relations and with a promise that unobserved variables have known cardinalities (arXiv:2109.05656), and on causal models that exhibit e-separation relations (arXiv:2107.07087). In addition, I will discuss the relationship between causal inference and quantum foundations, and the possibility of leveraging our results to study causal influence in models that involve quantum systems.

Hybrid quantum-classical computation; shuffling oracles; O2H lemma; quantum advice

An important theoretical problem in the study of quantum computation, that is also practically relevant in the context of near-term quantum devices, is to understand the computational power of hybrid models that combine polynomial-time classical computation with short-depth quantum computation. Here, we consider two such models: CQ_d which captures the scenario of a polynomial-time classical algorithm that queries a 𝑑-depth quantum computer many times; and QC_d which is more analogous to measurement-based quantum computation and captures the scenario of a 𝑑-depth quantum computer with the ability to change the sequence of gates being applied depending on measurement outcomes processed by a classical computation. Chia, Chung and Lai (STOC 2020) and Coudron and Menda (STOC 2020) showed that these models (with 𝑑 = log^O(1) (𝑛)) are strictly weaker than BQP (the class of problems solvable by polynomial-time quantum computation), relative to an oracle, disproving a conjecture of Jozsa in the relativised world. In this talk, we will show that, despite the similarities between CQ_d and QC_d, the two models are incomparable, i.e. CQ_d \nsubseteq QC_d and QC_d \nsubseteq CQ_d relative to an oracle. In other words, we show that there exist problems that one model can solve but not the other and vice versa.

variational quantum linear solver; variational quantum algorithms; quantum singular value decomposition

Current quantum computers typically have a few tens of qubits and are prone to errors due to imperfect gate implementations or undesired coupling with the environment. Among many of the proposed near-term applications to overcome these inconveniences, the field of Variational Quantum Algorithms (VQAs) is considered one of the most promising approaches. Thus, it seems natural to explore the use of VQAs for different applications, and more specifically, for linear algebra. In this talk, we focus on a few applications that make use of variational approaches: (1) the Quantum Singular Value Decomposer, which produces the singular value decomposition of pure bipartite states, (2) the Variational Quantum Linear Solver, for solving linear systems of equations, and (3) Quantum generative models via adversarial learning, to learn underlying distribution functions.

Institute for Theoretical Physics, University of Innsbruck

Quantum theory: From the whole to the parts, magic squares and shadows of infinity

non-commutative spaces; quantum magic squares; shadows of infinity; hyperreal numbers

The foundations of quantum theory rest on non-commutative spaces, where composition is given by the tensor product, positive elements play a distinguished role, and so do complex numbers. I will present our recent investigations on the interplay of these characters from three perspectives. First, I will present a framework to go from the whole to the parts, that is, to decompose elements of tensor product spaces while transferring invariance and positivity to the parts, including approximations. Second, I will talk about quantum magic squares and the fact that they cannot be “purified”, that is, dilated to quantum permutation matrices. Finally, I will talk about shadows of infinity: one is the shadow of the hyperreal numbers to the reals, where I will show that the existence of bound entangled states with a negative partial transpose can be solved in the hyperreals, and the other is the undecidability of a related problem, which can be seen as an infinite set not admitting a finite description.

Multiple-phase quantum interferometry: real and apparent gains of measuring all the phases simultaneously

We characterize operationally meaningful quantum gains in a paradigmatic model of lossless multiple-phase interferometry and stress insufficiency of the analysis based solely on the concept of quantum Fisher information (QFI). We show that the advantage of the optimal simultaneous estimation scheme amounts to a constant factor improvement when compared with schemes where each phase is estimated separately -- contrary to the results obtained by a naive application of QFI, which leads to a better precision scaling in terms of the number of phases involved.

Towards reconciliation of completely positive open system dynamics with equilibration postulate

master equation, thermalization, mean-force Hamiltonian, cumulant equation, steady-state coherences

Almost every quantum system interacts with a large environment, so the exact quantum mechanical description of its evolution is impossible. One has to resort to approximate description, usually in the form of a master equation. There are at least two basic requirements for such description: first of all, it should preserve positivity of probabilities; second, it should reproduce the wisdom coming from thermodynamics - systems coupled to a single thermal bath tend to the equilibrium. Existing two widespread descriptions of evolution fail to satisfy at least one of those conditions. The so-called Davies master equation, while preserving positivity of probabilities (due to Gorini-Kossakowski-Sudarshan-Lindblad form), fails to describe thermalization properly. On the other hand, the Bloch-Redfield master equation violates the positivity of probabilities, but it correctly describes equilibration at least for off-diagonal elements for several important scenarios. However, is it possible to have a description of open system dynamics that would share both features? In this talk, I will show our recent research that partially resolves this problem. (i) We provide a general form, up to second order, of the proper thermal equilibrium state (which is nontrivial even in the weak coupling limit). (ii) Next, we derive the steady-state coherences for a whole class of master equations, and in particular, we show that the solution for the Bloch-Redfield equation coincides with the equilibrium state. (iii) We consider the so-called cumulant equation, which is explicitly completely positive, and we show that up to second order, its steady-state coherences are the same as one of the Bloch-Redfield dynamics. (iv) We solve the second-order correction to the diagonal part of the stationary-state for a two-level system for both the Bloch-Redfield and cumulant master equation, showing that the solution of the cumulant is very close to the equilibrium state, whereas the Bloch-Redfield differs significantly.

Genuine multipartite entanglement and nonlocality in pair-entangled network states

The study of entanglement and nonlocality in multipartite quantum states plays a major role in quantum information theory and genuine multipartite entanglement (GME) and nonlocality (GMNL) signal some of its strongest forms for applications. However, their characterization for general (mixed) states is a highly nontrivial problem and its experimental preparation faces the formidable challenge of controlling quantum states with many constituents. In this talk I introduce a subclass of multipartite states, which I term pair-entangled network (PEN) states, as those that can be created by distributing exclusively bipartite entanglement in a connected network, and I study how their entanglement and nonlocality properties are affected by noise and the geometry of the graph that provides the connection pattern. The motivation is twofold. First, this class represents arguably the most feasible way to prepare GME and GMNL states in practice. Second, the class of PEN states provides an operationally motivated subset of multipartite states in which the well-developed theory of bipartite entanglement can be exploited to analyze entanglement in the multipartite scenario. I will show that all pure PEN states are GME and GMNL independently of the amount of entanglement shared and the network (as long as it is connected). In contrast, in the case of mixed PEN states these properties depend both on the level of noise and the network topology and they are not guaranteed by the mere distribution of mixed bipartite entangled states. In particular, the amount of connectivity in the network determines whether GME is robust to noise for any system size or whether it is completely washed out under the slightest form of noise for a sufficiently large number of parties. This latter case implies fundamental limitations for the application of certain networks in realistic scenarios, where the presence of some form of noise is unavoidable. In addition to this, if time allows, to illustrate the applicability of PEN states to study the complex phenomenology behind multipartite entanglement I will present three more results which use them as a proof ingredient: (i) all pure GME states are GMNL in the multiple-copy scenario, (ii) GMNL can be superactivated for any number of parties, and (iii) the set of GME-activatable states can be characterized as those states that are not partially separable.

Certifying quantum properties with minimal assumptions is a fundamental problem in quantum information science. Self-testing is a method to infer the underlying physics of a quantum experiment only from the measured statistics. While all bipartite pure entangled states can be self-tested, little is known about how to self-test quantum states of an arbitrary number of systems. Here, we introduce a framework for network-assisted self-testing and use it to self-test any pure entangled quantum state of an arbitrary number of systems. The scheme requires the preparation of a number of singlets that scales linearly with the number of systems, and the implementation of standard projective and Bell measurements, all feasible with current technology. When all the network constraints are exploited, the obtained self-testing certification is stronger than what is achievable in any Bell-type scenario. Our work does not only solve an open question in the field, but also shows how properly designed networks offer new opportunities for the certification of quantum phenomena.

Learning theory, quantum circuits, distribution learning

The output distributions of quantum circuits are an interesting class of probability distributions. They feature very prominently in the quest for quantum advantage via sampling experiments. Recently, they have also been of great interest in Quantum Machine Learning: they constitute a very versatile class of models for unsupervised machine learning that can be trained by updating the parameters of a parametrized quantum circuit. But how hard is it to learn these distributions from data? Can the training be done efficiently? In this talk, I will introduce you to a learning theory based perspective on these questions. This perspective allows us to deduce rigorous and quite strong answers while also bringing up many further questions.

Quantum cellular automata (QCAs) are systems with causal dynamics in discrete spacetimes; one example being time-periodic quantum circuits. On one hand, we show that all quasi-free fermionic QCAs in one dimension can be generated by quasi-local Hamiltonians, that is, Hamiltonians with interactions which decay with the distance. On the other hand, we prove that some QCAs cannot be generated with quasi-local Hamiltonians. This also implies the existence of integrable system with no local, quasi-local nor low-weight constants of motion; a result that challenges the standard definition of integrability. Finally, we see that QCAs with disorder display a new form of many-body localisation.

Training variational quantum circuits with CoVaR: covariance root finding with classical shadows

Exploiting near-term quantum computers and achieving practical value is a considerable and exciting challenge. Most prominent candidates as variational algorithms typically aim to find the ground state of a Hamiltonian by minimising a single classical (energy) surface which is sampled from by a quantum computer. Here we introduce a method we call CoVaR , an alternative means to exploit the power of variational circuits: We find eigenstates by finding joint roots of a polynomially growing number of properties of the quantum state as covariance functions between the Hamiltonian and an operator pool of our choice. The most remarkable feature of our CoVaR approach is that it allows us to fully exploit the extremely powerful classical shadow techniques, i.e., we simultaneously estimate a very large number > 10^4 − 10^7 of covariances. We randomly select covariances and estimate analytical derivatives at each iteration applying a stochastic Levenberg-Marquardt step via a large but tractable linear system of equations that we solve with a classical computer. We prove that the cost in quantum resources per iteration is comparable to a standard gradient estimation, however, we observe in numerical simulations a very significant improvement by many orders of magnitude in convergence speed. CoVaR is directly analogous to stochastic gradient-based optimisations of paramount importance to classical machine learning while we also offload significant but tractable work onto the classical processor.

The Algebraic Bethe Ansatz (ABA) is a highly successful analytical method used to exactly solve several physical models in both statistical mechanics and condensed-matter physics. Here we bring the ABA to unitary form, for its direct implementation on a quantum computer. This is achieved by distilling the non-unitary R matrices that make up the ABA into unitaries using the QR decomposition. Our algorithm is deterministic and works for both real and complex roots of the Bethe equations. We illustrate our method in the spin- 1 2 XX and XXZ models. We show that using this approach one can efficiently prepare eigenstates of the XX model on a quantum computer with quantum resources that match previous state-of-the-art approaches. We run numerical simulations, preparing eigenstates of the XXZ model for systems of up to 24 qubits and 12 magnons. Furthermore, we run small-scale error-mitigated implementations on the IBM quantum computers, including the preparation of the ground state for the XX and XXZ models in 4 sites. Finally, we derive a new form of the Yang-Baxter equation using unitary matrices, and also verify it on a quantum computer.