arXiv subjects

David Gross

Publications and source records attributed to David Gross.

At least 37 records · Page 2Linked to original sources

arXiv2022

A convergent inflation hierarchy for quantum causal structures

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 paper, we construct a first version of the quantum inflation hierarchy that is provably convergent. 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.

quant-ph↗

arXiv2022

The axiomatic and the operational approaches to resource theories of magic do not coincide

Stabiliser operations occupy a prominent role in fault-tolerant quantum computing. They are defined operationally: by the use of Clifford gates, Pauli measurements and classical control. These operations can be efficiently simulated on a classical computer, a result which is known as the Gottesman-Knill theorem. However, an additional supply of magic states is enough to promote them to a universal, fault-tolerant model for quantum computing. To quantify the needed resources in terms of magic states, a resource theory of magic has been developed. Stabiliser operations (SO) are considered free within this theory, however they are not the most general class of free operations. From an axiomatic point of view, these are the completely stabiliser-preserving (CSP) channels, defined as those that preserve the convex hull of stabiliser states. It has been an open problem to decide whether these two definitions lead to the same class of operations. In this work, we answer this question in the negative, by constructing an explicit counter-example. This indicates that recently proposed stabiliser-based simulation techniques of CSP maps are strictly more powerful than Gottesman-Knill-like methods. The result is analogous to a well-known fact in entanglement theory, namely that there is a gap between the operationally defined class of local operations and classical communication (LOCC) and the axiomatically defined class of separable channels.

quant-ph↗

arXiv2021

Certifying Numerical Decompositions of Compact Group Representations

We present a performant and rigorous algorithm for certifying that a matrix is close to being a projection onto an irreducible subspace of a given group representation. This addresses a problem arising when one seeks solutions to semi-definite programs (SDPs) with a group symmetry. Indeed, in this context, the dimension of the SDP can be significantly reduced if the irreducible representations of the group action are explicitly known. Rigorous numerical algorithms for decomposing a given group representation into irreps are known, but fairly expensive. To avoid this performance problem, existing software packages -- e.g. RepLAB, which motivated the present work -- use randomized heuristics. While these seem to work well in practice, the problem of to which extent the results can be trusted arises. Here, we provide rigorous guarantees applicable to finite and compact groups, as well as a software implementation that can interface with RepLAB. Under natural assumptions, a commonly used previous method due to Babai and Friedl runs in time O(n^5) for n-dimensional representations. In our approach, the complexity of running both the heuristic decomposition and the certification step is O(max{n^3 log n, D d^2 log d}), where d is the maximum dimension of an irreducible subrepresentation, and D is the time required to multiply elements of the group. A reference implementation interfacing with RepLAB is provided.

math.RT↗

arXiv2021

Proof methods for robust low-rank matrix recovery

Low-rank matrix recovery problems arise naturally as mathematical formulations of various inverse problems, such as matrix completion, blind deconvolution, and phase retrieval. Over the last two decades, a number of works have rigorously analyzed the reconstruction performance for such scenarios, giving rise to a rather general understanding of the potential and the limitations of low-rank matrix models in sensing problems. In this article, we compare the two main proof techniques that have been paving the way to a rigorous analysis, discuss their potential and limitations, and survey their successful applications. On the one hand, we review approaches based on descent cone analysis, showing that they often lead to strong guarantees even in the presence of adversarial noise, but face limitations when it comes to structured observations. On the other hand, we discuss techniques using approximate dual certificates and the golfing scheme, which are often better suited to deal with practical measurement structures, but sometimes lead to weaker guarantees. Lastly, we review recent progress towards analyzing descent cones also for structured scenarios -- exploiting the idea of splitting the cones into multiple parts that are analyzed via different techniques.

cs.IT↗

arXiv2021

Sketching with Kerdock's crayons: Fast sparsifying transforms for arbitrary linear maps

Given an arbitrary matrix $A\in\mathbb{R}^{n\times n}$, we consider the fundamental problem of computing $Ax$ for any $x\in\mathbb{R}^n$ such that $Ax$ is $s$-sparse. While fast algorithms exist for particular choices of $A$, such as the discrete Fourier transform, there is currently no $o(n^2)$ algorithm that treats the unstructured case. In this paper, we devise a randomized approach to tackle the unstructured case. Our method relies on a representation of $A$ in terms of certain real-valued mutually unbiased bases derived from Kerdock sets. In the preprocessing phase of our algorithm, we compute this representation of $A$ in $O(n^3\log n)$ operations. Next, given any unit vector $x\in\mathbb{R}^n$ such that $Ax$ is $s$-sparse, our randomized fast transform uses this representation of $A$ to compute the entrywise $ε$-hard threshold of $Ax$ with high probability in only $O(sn + ε^{-2}\|A\|_{2\to\infty}^2n\log n)$ operations. In addition to a performance guarantee, we provide numerical results that demonstrate the plausibility of real-world implementation of our algorithm.

cs.CC↗

arXiv2021

Stabilizer extent is not multiplicative

The Gottesman-Knill theorem states that a Clifford circuit acting on stabilizer states can be simulated efficiently on a classical computer. Recently, this result has been generalized to cover inputs that are close to a coherent superposition of logarithmically many stabilizer states. The runtime of the classical simulation is governed by the stabilizer extent, which roughly measures how many stabilizer states are needed to approximate the state. An important open problem is to decide whether the extent is multiplicative under tensor products. An affirmative answer would yield an efficient algorithm for computing the extent of product inputs, while a negative result implies the existence of more efficient classical algorithms for simulating largescale quantum circuits. Here, we answer this question in the negative. Our result follows from very general properties of the set of stabilizer states, such as having a size that scales subexponentially in the dimension, and can thus be readily adapted to similar constructions for other resource theories.

quant-ph↗

arXiv2021

Schur-Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations

Schur-Weyl duality is a ubiquitous tool in quantum information. At its heart is the statement that the space of operators that commute with the tensor powers of all unitaries is spanned by the permutations of the tensor factors. In this work, we describe a similar duality theory for tensor powers of Clifford unitaries. The Clifford group is a central object in many subfields of quantum information, most prominently in the theory of fault-tolerance. The duality theory has a simple and clean description in terms of finite geometries. We demonstrate its effectiveness in several applications: (1) We resolve an open problem in quantum property testing by showing that "stabilizerness" is efficiently testable: There is a protocol that, given access to six copies of an unknown state, can determine whether it is a stabilizer state, or whether it is far away from the set of stabilizer states. We give a related membership test for the Clifford group. (2) We find that tensor powers of stabilizer states have an increased symmetry group. We provide corresponding de Finetti theorems, showing that the reductions of arbitrary states with this symmetry are well-approximated by mixtures of stabilizer tensor powers (in some cases, exponentially well). (3) We show that the distance of a pure state to the set of stabilizers can be lower-bounded in terms of the sum-negativity of its Wigner function. This gives a new quantitative meaning to the sum-negativity (and the related mana) -- a measure relevant to fault-tolerant quantum computation. The result constitutes a robust generalization of the discrete Hudson theorem. (4) We show that complex projective designs of arbitrary order can be obtained from a finite number (independent of the number of qudits) of Clifford orbits. To prove this result, we give explicit formulas for arbitrary moments of random stabilizer states.

quant-ph↗

arXiv2020

Rapid characterisation of linear-optical networks via PhaseLift

Linear-optical circuits are elementary building blocks for classical and quantum information processing with light. In particular, due to its monolithic structure, integrated photonics offers great phase-stability and can rely on the large scale manufacturability provided by the semiconductor industry. New devices, based on such optical circuits, hold the promise of faster and energy-efficient computations in machine learning applications and even implementing quantum algorithms intractable for classical computers. However, this technological revolution requires accurate and scalable certification protocols for devices that can be comprised of thousands of optical modes. Here, we present a novel technique to reconstruct the transfer matrix of linear optical networks that is based on the recent advances in low-rank matrix recovery and convex optimisation problems known as PhaseLift algorithms. Conveniently, our characterisation protocol can be performed with a coherent classical light source and photodiodes. We prove that this method is robust to noise and scales efficiently with the number of modes. We experimentally tested the proposed characterisation protocol on a programmable integrated interferometer designed for quantum information processing. We compared the transfer matrix reconstruction obtained with our method against the one provided by a more demanding reconstruction scheme based on two-photon quantum interference. For 5-dimensional random unitaries, the average circuit fidelity between the matrices obtained from the two reconstructions is 0.993.

physics.optics↗

arXiv2020

Rank-deficient representations in the Theta correspondence over finite fields arise from quantum codes

Let V be a symplectic vector space and let $μ$ be the oscillator representation of Sp(V). It is natural to ask how the tensor power representation $μ^{\otimes t}$ decomposes. If V is a real vector space, then Howe-Kashiwara-Vergne (HKV) duality asserts that there is a one-one correspondence between the irreducible subrepresentations of Sp(V) and the irreps of an orthogonal group O(t). It is well-known that this duality fails over finite fields. Addressing this situation, Gurevich and Howe have recently assigned a notion of rank to each Sp(V) representation. They show that a variant of HKV duality continues to hold over finite fields, if one restricts attention to subrepresentations of maximal rank. The nature of the rank-deficient components was left open. Here, we show that all rank-deficient Sp(V)-subrepresentations arise from embeddings of lower-order tensor products of $μ$ and $\barμ$ into $μ^{\otimes t}$. The embeddings live on spaces that have been studied in quantum information theory as tensor powers of self-orthogonal Calderbank-Shor-Steane (CSS) quantum codes. We then find that the irreducible Sp(V) subrepresentations of $μ^{\otimes t}$ are labelled by the irreps of orthogonal groups O(r) acting on certain r-dimensional spaces for r <= t. The results hold in odd charachteristic and the "stable range" t <= 1/2 dim V. Our work has implications for the representation theory of the Clifford group. It can be thought of as a generalization of the known characterization of the invariants of the Clifford group in terms of self-dual codes.

math.RT↗

arXiv2020

From estimation of quantum probabilities to simulation of quantum circuits

Investigating the classical simulability of quantum circuits provides a promising avenue towards understanding the computational power of quantum systems. Whether a class of quantum circuits can be efficiently simulated with a probabilistic classical computer, or is provably hard to simulate, depends quite critically on the precise notion of "classical simulation" and in particular on the required accuracy. We argue that a notion of classical simulation, which we call epsilon-simulation, captures the essence of possessing "equivalent computational power" as the quantum system it simulates: It is statistically impossible to distinguish an agent with access to an epsilon-simulator from one possessing the simulated quantum system. We relate epsilon-simulation to various alternative notions of simulation predominantly focusing on a simulator we call a poly-box. A poly-box outputs 1/poly precision additive estimates of Born probabilities and marginals. This notion of simulation has gained prominence through a number of recent simulability results. Accepting some plausible computational theoretic assumptions, we show that epsilon-simulation is strictly stronger than a poly-box by showing that IQP circuits and unconditioned magic-state injected Clifford circuits are both hard to epsilon-simulate and yet admit a poly-box. In contrast, we also show that these two notions are equivalent under an additional assumption on the sparsity of the output distribution (poly-sparsity).

quant-ph↗

arXiv2019

Implications of pinned occupation numbers for natural orbital expansions. II: Rigorous derivation and extension to non-fermionic systems

We have explained and comprehensively illustrated in Part I that the generalized Pauli constraints suggest a natural extension of the concept of active spaces. In the present Part II, we provide rigorous derivations of the theorems involved therein. This will offer in particular deeper insights into the underlying mathematical structure and will explain why the saturation of generalized Pauli constraints implies a specific simplified structure of the corresponding many-fermion quantum state. Moreover, we extend the results of Part I to non-fermionic multipartite quantum systems, revealing that extremal single-body information has always strong implications for the multipartite quantum state. In that sense, our work also confirms that pinned quantum systems define new physical entities and the presence of pinnings reflect the existence of (possibly hidden) ground state symmetries.

quant-ph↗

arXiv2019

Guaranteed recovery of quantum processes from few measurements

Quantum process tomography is the task of reconstructing unknown quantum channels from measured data. In this work, we introduce compressed sensing-based methods that facilitate the reconstruction of quantum channels of low Kraus rank. Our main contribution is the analysis of a natural measurement model for this task: We assume that data is obtained by sending pure states into the channel and measuring expectation values on the output. Neither ancillary systems nor coherent operations across multiple channel uses are required. Most previous results on compressed process reconstruction reduce the problem to quantum state tomography on the channel's Choi matrix. While this ansatz yields recovery guarantees from an essentially minimal number of measurements, physical implementations of such schemes would typically involve ancillary systems. A priori, it is unclear whether a measurement model tailored directly to quantum process tomography might require more measurements. We establish that this is not the case. Technically, we prove recovery guarantees for three different reconstruction algorithms. The reconstructions are based on a trace, diamond, and $\ell_2$-norm minimization, respectively. Our recovery guarantees are uniform in the sense that with one random choice of measurement settings all quantum channels can be recovered equally well. Moreover, stability against arbitrary measurement noise and robustness against violations of the low-rank assumption is guaranteed. Numerical studies demonstrate the feasibility of the approach.

quant-ph↗

arXiv2019

Robustness of Magic and Symmetries of the Stabiliser Polytope

We give a new algorithm for computing the robustness of magic - a measure of the utility of quantum states as a computational resource. Our work is motivated by the magic state model of fault-tolerant quantum computation. In this model, all unitaries belong to the Clifford group. Non-Clifford operations are effected by injecting non-stabiliser states, which are referred to as magic states in this context. The robustness of magic measures the complexity of simulating such a circuit using a classical Monte Carlo algorithm. It is closely related to the degree negativity that slows down Monte Carlo simulations through the infamous sign problem. Surprisingly, the robustness of magic is submultiplicative. This implies that the classical simulation overhead scales subexponentially with the number of injected magic states - better than a naive analysis would suggest. However, determining the robustness of n copies of a magic state is difficult, as its definition involves a convex optimisation problem in a 4^n-dimensional space. In this paper, we make use of inherent symmetries to reduce the problem to n dimensions. The total run-time of our algorithm, while still exponential in n, is super-polynomially faster than previously published methods. We provide a computer implementation and give the robustness of up to 10 copies of the most commonly used magic states. Guided by the exact results, we find a finite hierarchy of approximate solutions where each level can be evaluated in polynomial time and yields rigorous upper bounds to the robustness. Technically, we use symmetries of the stabiliser polytope to connect the robustness of magic to the geometry of a low-dimensional convex polytope generated by certain signed quantum weight enumerators. As a by-product, we characterised the automorphism group of the stabiliser polytope, and, more generally, of projections onto complex projective 3-designs.

quant-ph↗

arXiv2018

Note on the saturation of the norm inequalities between diamond and nuclear norm

The diamond norm plays an important role in quantum information and operator theory. Recently, it has also been proposed as a regularizer for low-rank matrix recovery. The norm constants that bound the diamond norm in terms of the nuclear norm (also known as trace norm) are explicitly known. This note provides a simple characterization of all operators saturating the upper and the lower bound.

cs.IT↗

arXiv2018

Recovering quantum gates from few average gate fidelities

Characterising quantum processes is a key task in and constitutes a challenge for the development of quantum technologies, especially at the noisy intermediate scale of today's devices. One method for characterising processes is randomised benchmarking, which is robust against state preparation and measurement (SPAM) errors, and can be used to benchmark Clifford gates. A complementing approach asks for full tomographic knowledge. Compressed sensing techniques achieve full tomography of quantum channels essentially at optimal resource efficiency. So far, guarantees for compressed sensing protocols rely on unstructured random measurements and can not be applied to the data acquired from randomised benchmarking experiments. It has been an open question whether or not the favourable features of both worlds can be combined. In this work, we give a positive answer to this question. For the important case of characterising multi-qubit unitary gates, we provide a rigorously guaranteed and practical reconstruction method that works with an essentially optimal number of average gate fidelities measured respect to random Clifford unitaries. Moreover, for general unital quantum channels we provide an explicit expansion into a unitary 2-design, allowing for a practical and guaranteed reconstruction also in that case. As a side result, we obtain a new statistical interpretation of the unitarity -- a figure of merit that characterises the coherence of a process. In our proofs we exploit recent representation theoretic insights on the Clifford group, develop a version of Collins' calculus with Weingarten functions for integration over the Clifford group, and combine this with proof techniques from compressed sensing.

quant-ph↗

arXiv2018

Generalized Pauli constraints in small atoms

The natural occupation numbers of fermionic systems are subject to non-trivial constraints, which include and extend the original Pauli principle. A recent mathematical breakthrough has clarified their mathematical structure and has opened up the possibility of a systematic analysis. Early investigations have found evidence that these constraints are exactly saturated in several physically relevant systems, e.g., in a certain electronic state of the Beryllium atom. It has been suggested that in such cases, the constraints, rather than the details of the Hamiltonian, dictate the system's qualitative behaviour. Here, we revisit this question with state-of-the-art numerical methods for small atoms. We find that the constraints are, in fact, not exactly saturated, but that they lie much closer to the surface defined by the constraints than the geometry of the problem would suggest. While the results seem incompatible with the statement that the generalized Pauli constraints drive the behaviour of these systems, they suggest that the qualitatively correct wave-function expansions can in some systems already be obtained on the basis of a limited number of Slater determinants, which is in line with numerical evidence from quantum chemistry.

quant-ph↗

arXiv2018

Computational tools for solving a marginal problem with applications in Bell non-locality and causal modeling

Marginal problems naturally arise in a variety of different fields: basically, the question is whether some marginal/partial information is compatible with a joint probability distribution. To this aim, the characterization of marginal sets via quantifier elimination and polyhedral projection algorithms is of primal importance. In this work, before considering specific problems, we review polyhedral projection algorithms with focus on applications in information theory, and, alongside known algorithms, we also present a newly developed geometric algorithm which walks along the face lattice of the polyhedron in the projection space. One important application of this is in the field of quantum non-locality, where marginal problems arise in the computation of Bell inequalities. We apply the discussed algorithms to discover many tight entropic Bell inequalities of the tripartite Bell scenario as well as more complex networks arising in the field of causal inference. Finally, we analyze the usefulness of these inequalities as nonlocality witnesses by searching for violating quantum states.

quant-ph↗

arXiv2017

Error regions in quantum state tomography: computational complexity caused by geometry of quantum states

The outcomes of quantum mechanical experiments are inherently random. It is therefore necessary to develop stringent methods for quantifying the degree of statistical uncertainty about the results of quantum experiments. For the particularly relevant task of quantum state estimation, it has been shown that a significant reduction in uncertainty can be achieved by taking the positivity of quantum states into account. However -- the large number of partial results and heuristics notwithstanding -- no efficient general algorithm is known that produces an optimal uncertainty region from experimental data and the prior constraint of positivity. Here, we make this problem precise and show that the general case is NP-hard. Our result leaves room for the existence of efficient approximate solutions, and therefore does not yet imply that the practical task of quantum uncertainty quantification is intractable. However, it does show that there exists a non-trivial trade-off between optimality and computational efficiency for error regions. We prove two versions of the result: One for frequentist and one for Bayesian statistics.

quant-ph↗