Some representation-theoretic multiplicities, such as the Kostka and the Littlewood-Richardson coefficients, admit a combinatorial interpretation that places their computation in the complexity class #𝖯. Whether this holds more generally is considered an important open problem in mathematics and computer science, with relevance for geometric complexity theory and quantum information. Recent work has investigated the quantum complexity of particular multiplicities, such as the Kronecker coefficients and certain special cases of the plethysm coefficients. Here, we show that a broad class of representation-theoretic multiplicities is in #BQP. This includes the result that plethysm coefficients are in #BQP, which was only known in certain cases. It also implies all known results on the quantum complexity of previously studied coefficients as special cases, thus unifying, simplifying, and extending prior work. We obtain our result by multiple applications of the Schur transform; recent work has improved its dependence on the local dimension, which is crucial for our work. We further describe a general approach for showing that representation-theoretic multiplicities are in #BQP that captures the approaches of our and previous work. We complement the above by showing that the same multiplicities are also naturally in GapP and obtain polynomial-time classical algorithms when certain parameters are fixed.

A wide range of natural computational problems in computer science, mathematics, physics, and other sciences amounts to deciding if two objects are equivalent. Very often this equivalence is defined in terms of group actions. A natural question is to ask when two objects can be distinguished by polynomial functions that are invariant under the group action. For finite groups, this is just the usual notion of equivalence, but for continuous groups such as the general linear groups it gives rise to a new notion, called orbit closure intersection. This new notion has recently seen substantial interest in the community, as it captures, among others, the graph isomorphism problem, noncommutative polynomial identity testing, null cone problems in invariant theory, equivalence problems for tensor networks, and the classification of multiparty quantum states. Despite remarkable recent algorithmic progress in celebrated special cases, the computational complexity of general orbit closure intersection problems is currently quite unclear. In particular, tensors seem to give rise to the most difficult problems.In this work we start a systematic study of orbit closure intersection problems from the complexity-theoretic viewpoint. Our key contributions include:•We define a complexity class TOCI that captures the power of orbit closure intersection problems for general tensor actions.•We give an appropriate notion of algebraic reductions that imply polynomial-time reductions in the usual sense, but are amenable to invariant-theoretic techniques.•We identify several natural tensor problems that are complete for TOCI, one of which is the equivalence of PEPS tensor networks considered by Acuaviva et al (FOCS’23).•We show that the graph isomorphism problem can be reduced to these complete problems and hence $\mathbf{G I} \subseteq$ TOCI.As such, our work establishes the first lower bound on the computational complexity of orbit closure intersection problems, and it explains the difficulty of finding unconditional polynomialtime algorithms beyond special cases, as has been observed in the recent literature.

Tensors play a central role in various areas of computer science and mathematics, such as algebraic complexity theory (matrix multiplication), quantum information theory (entanglement), and additive combinatorics (slice rank). Fundamental problems about tensors are strongly tied to well-known questions in computational complexity - such as the problem of determining the matrix multiplication exponent via asymptotic rank, and the stronger Strassen asymptotic rank conjecture, which has recently been intimately linked to a whole range of computational problems. Unlike matrices, which are often well understood through their rank, tensors have such intricate structure that understanding them (and aforementioned problems) requires information of a more subtle nature. The moment polytope, going back decades to work in symplectic geometry, invariant theory, and representation theory, is a mathematical object associated to any tensor that collects such "rank-like"information. Their relevance has become apparent in several areas: (1) through applications in geometric complexity theory (GCT), (2) in the construction of functions in Strassen's asymptotic spectrum of tensors, (3) as entanglement polytopes in quantum information theory, and (4) in optimization via scaling algorithms. Despite their fundamental role and interest from many angles, little is known about these polytopes, and in particular for tensors beyondC2λ-λ.,⊗2λ-λ.,⊗2 andC2λ-λ.,⊗2λ-λ.,⊗2λ-λ.,⊗2 only sporadically have they been computed. Even less is known about the polytopes' inclusions and separations (which are particularly relevant for applications). We give a new algorithm for computing moment polytopes of tensors (and in fact moment polytopes for a natural general class of reductive algebraic groups) based on a mathematical characterization of moment polytopes by Franz. This algorithm enables us to compute moment polytopes of tensors of dimension an order of magnitude larger than previous methods, allowing us to compute with certainty, for the first time, all moment polytopes of tensors inC3λ-λ.,⊗3λ-λ.,⊗3, and with high probability those inC4λ-λ.,⊗4λ-λ.,⊗4. Towards an open problem in geometric complexity theory, we prove (guided by moment polytopes computed with our algorithm) separations between the moment polytopes of matrix multiplication tensors and unit tensors, showing in particular that the matrix multiplication moment polytopes are not maximal (i.e., not equal to the corresponding Kronecker polytopes). As a consequence of the above, we obtain a no-go result for a certain operational characterization of moment polytope inclusion, by proving that Strassen's asymptotic restriction on tensors does not imply moment polytope inclusion. Finally, based on our algorithmic observations, we construct explicit (concise) non-free tensors in every formatCn λ-Cn λ-Cn, thus solving a "hay in a haystack"problem for this generic property that plays an important role in Strassen's theory of asymptotic spectra.

Thanks to the rapid progress and growing complexity of quantum algorithms, correctness of quantum programs has become a major concern. Pioneering research over the past years has proposed various approaches to formally verify quantum programs using proof systems such as quantum Hoare logic. All these prior approaches are post-hoc: one first implements a program and only then verifies its correctness. Here we propose Quantum Correctness by Construction (QbC) : an approach to constructing quantum programs from their specification in a way that ensures correctness. We use pre- and postconditions to specify program properties, and propose sound and complete refinement rules for constructing programs in a quantum while language from their specification. We validate QbC by constructing quantum programs for idiomatic problems and patterns. We find that the approach naturally suggests how to derive program details, highlighting key design choices along the way. As such, we believe that QbC can play a role in supporting the design and taxonomization of quantum algorithms and software.

When a group acts on a set, it naturally partitions it into orbits, giving rise to orbit problems. These are natural algorithmic problems, as symmetries are central in numerous questions and structures in physics, mathematics, computer science, optimization, and more. Accordingly, it is of high interest to understand their computational complexity. Recently, Bürgisser et al. gave the first polynomial-time algorithms for orbit problems of torus actions, that is, actions of commutative continuous groups on Euclidean space. In this work, motivated by theoretical and practical applications, we study the computational complexity of robust generalizations of these orbit problems, which amount to approximating the distance of orbits in $\mathbb{C}^n$ up to a factor $γ>1$. In particular, this allows deciding whether two inputs are approximately in the same orbit or far from being so. On the one hand, we prove the NP-hardness of this problem for $γ= n^{Ω(1/\log\log n)}$ by reducing the closest vector problem for lattices to it. On the other hand, we describe algorithms for solving this problem for an approximation factor $γ= \exp(\mathrm{poly}(n))$. Our algorithms combine tools from invariant theory and algorithmic lattice theory, and they also provide group elements witnessing the proximity of the given orbits (in contrast to the algebraic algorithms of prior work). We prove that they run in polynomial time if and only if a version of the famous number-theoretic $abc$-conjecture holds -- establishing a new and surprising connection between computational complexity and number theory.

Undirected st-connectivity is important both for its applications in network problems, and for its theoretical connections with logspace complexity. Classically, a long line of work led to a time-space tradeoff of T = Oe(n2/S) for any S such that S = Ω(log(n)) and S = O(n2/m). Surprisingly, we show that quantumly there is no nontrivial time-space tradeoff: there is a quantum algorithm that achieves both optimal time Oe(n) and space O(log(n)) simultaneously. This improves on previous results, which required either O(log(n)) space and Oe(n1.5) time, or Oe(n) space and time. To complement this, we show that there is a nontrivial time-space tradeoff when given a lower bound on the spectral gap of a corresponding random walk.

Interior-point methods offer a highly versatile framework for convex optimization that is effective in theory and practice. A key notion in their theory is that of a self-concordant barrier. We give a suitable generalization of self-concordance to Riemannian manifolds and show that it gives the same structural results and guarantees as in the Euclidean setting, in particular local quadratic convergence of Newton’s method. We analyze a path-following method for optimizing compatible objectives over a convex domain for which one has a self-concordant barrier, and obtain the standard complexity guarantees as in the Euclidean setting. We provide general constructions of barriers, and show that on the space of positive-definite matrices and other symmetric spaces, the squared distance to a point is self-concordant. To demonstrate the versatility of our framework, we give algorithms with state-of-the-art complexity guarantees for the general class of scaling and non-commutative optimization problems, which have been of much recent interest, and we provide the first algorithms for efficiently finding high-precision solutions for computing minimal enclosing balls and geometric medians in non-positive curvature.

Tensor networks have a gauge degree of freedom on the virtual degrees of freedom that are contracted. A canonical form is a choice of fixing this degree of freedom. For matrix product states, choosing a canonical form is a powerful tool, both for theoretical and numerical purposes. On the other hand, for tensor networks in dimension two or greater there is only limited understanding of the gauge symmetry. Here we introduce a new canonical form, the minimal canonical form, which applies to projected entangled pair states (PEPS) in any dimension, and prove a corresponding fundamental theorem. Already for matrix product states this gives a new canonical form, while in higher dimensions it is the first rigorous definition of a canonical form valid for any choice of tensor. We show that two tensors have the same minimal canonical forms if and only if they are gauge equivalent up to taking limits; moreover, this is the case if and only if they give the same quantum state for any geometry. In particular, this implies that the latter problem is decidable – in contrast to the well-known undecidability for equality of PEPS on grids. We also provide rigorous algorithms for computing minimal canonical forms. To achieve this we draw on geometric invariant theory and recent progress in theoretical computer science in non-commutative group optimization.

An action of a group on a vector space partitions the latter into a set of orbits. We consider three natural and useful algorithmic "isomorphism" or "classification" problems, namely, orbit equality, orbit closure intersection, and orbit closure containment. These capture and relate to a variety of problems within mathematics, physics and computer science, optimization and statistics. These orbit problems extend the more basic null cone problem, whose algorithmic complexity has seen significant progress in recent years. In this paper, we initiate a study of these problems by focusing on the actions of commutative groups (namely, tori). We explain how this setting is motivated from questions in algebraic complexity, and is still rich enough to capture interesting combinatorial algorithmic problems. While the structural theory of commutative actions is well understood, no general efficient algorithms were known for the aforementioned problems. Our main results are polynomial time algorithms for all three problems. We also show how to efficiently find separating invariants for orbits, and how to compute systems of generating rational invariants for these actions (in contrast, for polynomial invariants the latter is known to be hard). Our techniques are based on a combination of fundamental results in invariant theory, linear programming, and algorithmic lattice theory.

Matrix scaling and matrix balancing are two basic linear-algebraic problems with a wide variety of applications, such as approximating the permanent, and pre-conditioning linear systems to make them more numerically stable. We study the power and limitations of quantum algorithms for these problems. We provide quantum implementations of two classical (in both senses of the word) methods: Sinkhorn’s algorithm for matrix scaling and Osborne’s algorithm for matrix balancing. Using amplitude estimation as our main tool, our quantum implementations both run in time Õ(√{mn}/ε⁴) for scaling or balancing an n × n matrix (given by an oracle) with m non-zero entries to within 𝓁₁-error ε. Their classical analogs use time Õ(m/ε²), and every classical algorithm for scaling or balancing with small constant ε requires Ω(m) queries to the entries of the input matrix. We thus achieve a polynomial speed-up in terms of n, at the expense of a worse polynomial dependence on the obtained 𝓁₁-error ε. Even for constant ε these problems are already non-trivial (and relevant in applications). Along the way, we extend the classical analysis of Sinkhorn’s and Osborne’s algorithm to allow for errors in the computation of marginals. We also adapt an improved analysis of Sinkhorn’s algorithm for entrywise-positive matrices to the 𝓁₁-setting, obtaining an Õ(n^{1.5}/ε³)-time quantum algorithm for ε-𝓁₁-scaling. We also prove a lower bound, showing our quantum algorithm for matrix scaling is essentially optimal for constant ε: every quantum algorithm for matrix scaling that achieves a constant 𝓁₁-error w.r.t. uniform marginals needs Ω(√{mn}) queries.

We consider the problem of computing succinct encodings of lists of generators for invariant rings for group actions. Mulmuley conjectured that there are always polynomial sized such encodings for invariant rings of SL_n(ℂ)-representations. We provide simple examples that disprove this conjecture (under standard complexity assumptions). We develop a general framework, denoted algebraic circuit search problems, that captures many important problems in algebraic complexity and computational invariant theory. This framework encompasses various proof systems in proof complexity and some of the central problems in invariant theory as exposed by the Geometric Complexity Theory (GCT) program, including the aforementioned problem of computing succinct encodings for generators for invariant rings.

Optimally encoding classical information in a quantum system is one of the oldest and most fundamental challenges of quantum information theory. Holevo's bound places a hard upper limit on such encodings, while the Holevo-Schumacher-Westmoreland (HSW) theorem addresses the question of how many classical messages can be “packed” into a given quantum system. In this article, we use Sen's recent quantum joint typicality results to prove a one-shot multiparty quantum packing lemma generalizing the HSW theorem. The lemma is designed to be easily applicable in many network communication scenarios. As an illustration, we use it to straightforwardly obtain quantum generalizations of well-known classical coding schemes for the relay channel: multihop, coherent multihop, decode-forward, and partial decode-forward. We provide both finite blocklength and asymptotic results, the latter matching existing classical formulas. Given the key role of the classical packing lemma in network information theory, our packing lemma should help open the field to direct quantum generalization.

This paper initiates a systematic development of a theory of non-commutative optimization, a setting which greatly extends ordinary (Euclidean) convex optimization. It aims to unify and generalize a growing body of work from the past few years which developed and analyzed algorithms for natural geodesically convex optimization problems on Riemannian manifolds that arise from the symmetries of non-commutative groups. More specifically, these are algorithms to minimize the moment map (a noncommutative notion of the usual gradient), and to test membership in moment polytopes (a vast class of polytopes, typically of exponential vertex and facet complexity, which quite magically arise from this apriori non-convex, non-linear setting). The importance of understanding this very general setting of geodesic optimization, as these works unveiled and powerfully demonstrate, is that it captures a diverse set of problems, many non-convex, in different areas of CS, math, and physics. Several of them were solved efficiently for the first time using noncommutative methods; the corresponding algorithms also lead to solutions of purely structural problems and to many new connections between disparate fields. In the spirit of standard convex optimization, we develop two general methods in the geodesic setting, a first order and a second order method, which respectively receive first and second order information on the “derivatives” of the function to be optimized. These in particular subsume all past results. The main technical work, again unifying and extending much of the previous work, goes into identifying the key parameters of the underlying group actions which control convergence to the optimum in each of these methods. These non-commutative analogues of “smoothness” in the commutative case are far more complex, and require significant algebraic and analytic machinery (much existing and some newly developed here). Despite this complexity, the way in which these parameters control convergence in both methods is quite simple and elegant. We also bound these parameters in several general cases. Our work points to intriguing open problems and suggests further research directions. We believe that extending this theory, namely understanding geodesic optimization better, is both mathematically and computationally fascinating; it provides a great meeting place for ideas and techniques from several very different research areas, and promises better algorithms for existing and yet unforeseen applications.

We present a polynomial time algorithm to approximately scale tensors of any format to arbitrary prescribed marginals (whenever possible). This unifies and generalizes a sequence of past works on matrix, operator and tensor scaling. Our algorithm provides an efficient weak membership oracle for the associated moment polytopes, an important family of implicitly-defined convex polytopes with exponentially many facets and a wide range of applications. These include the entanglement polytopes from quantum information theory (in particular, we obtain an efficient solution to the notorious one-body quantum marginal problem) and the Kronecker polytopes from representation theory (which capture the asymptotic support of Kronecker coefficients). Our algorithm can be applied to succinct descriptions of the input tensor whenever the marginals can be efficiently computed, as in the important case of matrix product states or tensor-train decompositions, widely used in computational physics and numerical mathematics. Beyond these applications, the algorithm enriches the arsenal of "numerical" methods for classical problems in invariant theory that are significantly faster than "symbolic" methods which explicitly compute invariants or covariants of the relevant action. We stress that (like almost all past algorithms) our convergence rate is polynomial in the approximation parameter; it is an intriguing question to achieve exponential convergence rate, beating symbolic algorithms exponentially, and providing strong membership and separation oracles for the problems above. We strengthen and generalize the alternating minimization approach of previous papers by introducing the theory of highest weight vectors from representation theory into the numerical optimization framework. We show that highest weight vectors are natural potential functions for scaling algorithms and prove new bounds on their evaluations to obtain polynomial-time convergence. Our techniques are general and we believe that they will be instrumental to obtain efficient algorithms for moment polytopes beyond the ones consider here, and more broadly, for other optimization problems possessing natural symmetries.

Alternating minimization heuristics seek to solve a (difficult) global optimization task through iteratively solving a sequence of (much easier) local optimization tasks on different parts (or blocks) of the input parameters. While popular and widely applicable, very few examples of this heuristic are rigorously shown to converge to optimality, and even fewer to do so efficiently. In this paper we present a general framework which is amenable to rigorous analysis, and expose its applicability. Its main feature is that the local optimization domains are each a group of invertible matrices, together naturally acting on tensors, and the optimization problem is minimizing the norm of an input tensor under this joint action. The solution of this optimization problem captures a basic problem in Invariant Theory, called the null-cone problem. This algebraic framework turns out to encompass natural computational problems in combinatorial optimization, algebra, analysis, quantum information theory, and geometric complexity theory. It includes and extends to high dimensions the recent advances on (2-dimensional) operator scaling. Our main result is a fully polynomial time approximation scheme for this general problem, which may be viewed as a multi-dimensional scaling algorithm. This directly leads to progress on some of the problems in the areas above, and a unified view of others. We explain how faster convergence of an algorithm for the same problem will allow resolving central open problems. Our main techniques come from Invariant Theory, and include its rich non-commutative duality theory, and new bounds on the bitsizes of coefficients of invariant polynomials. They enrich the algorithmic toolbox of this very computational field of mathematics, and are directly related to some challenges in geometric complexity theory (GCT).

We show that the problem of deciding membership in the moment polytope associated with a finite-dimensional unitary representation of a compact, connected Lie group is in NP and coNP. This is the first nontrivial result on the computational complexity of this problem, which naively amounts to a quadratically constrained program. Our result applies in particular to the Kronecker polytopes, and therefore to the problem of deciding positivity of the stretched Kronecker coefficients. In contrast, it has recently been shown that deciding positivity of a single Kronecker coefficient is NP-hard, in general [C. Ikenmeyer, K. D. Mulmuley, and M. Walter, preprint, arXiv:1507.02955, 2015]. We discuss the consequences of our work in the context of complexity theory and the quantum marginal problem.

We study universal quantum codes for entanglement-assisted quantum communication over compound quantum channels. In this setting, sender and receiver do not know the specific channel that will be used for communication, but only know the set that the channel is selected from. We investigate different variations of the problem: uninformed users, informed receiver, informed sender, and feedback assistance. We derive single-letter formulas for all corresponding channel capacities. Our proofs are based on one-shot decoupling bounds and properties of smooth entropies.

The laws of quantum mechanics place fundamental limits on the accuracy of measurements and therefore on the estimation of physical parameters by a quantum system. In this work, we prove lower bounds on the size of confidence regions reported by any region estimator for a given ensemble of probe states and probability of success. Our bounds are derived from a previously unnoticed connection between the size of confidence regions and the error probabilities of a corresponding binary hypothesis test. In group-covariant scenarios, we find that there is an ultimate bound for any estimation scheme which depends only on the representation-theoretic data of the probe system, and we evaluate its asymptotics in the limit of many systems, establishing a general “Heisenberg limit” for region estimation. We apply our results to several scenarios, in particular to phase estimation, where our bounds strengthen the well-known Heisenberg and shot-noise scaling.

The laws of quantum mechanics place fundamental limits on the accuracy of measurements and, therefore, on the estimation of unknown parameters of a quantum system. In this paper, we prove lower bounds on the size of confidence regions reported by any region estimator for a given ensemble of probe states and probability of success. Our bounds are derived from a previously unnoticed connection between the size of confidence regions and the error probabilities of a corresponding binary hypothesis test. In group-covariant scenarios, we find that there is an ultimate bound for any estimation scheme, which depends only on the representation-theoretic data of the probe system, and we evaluate its asymptotics in the limit of many systems, establishing a general Heisenberg limit for region estimation. We apply our results to several examples, in particular, to phase estimation, where our bounds allow us to recover the well-known Heisenberg and shot-noise scaling.

For fixed compact connected Lie groups H ⊆ G, we provide a polynomial time algorithm to compute the multiplicity of a given irreducible representation of H in the restriction of an irreducible representation of G. Our algorithm is based on a finite difference formula which makes the multiplicities amenable to Barvinok's algorithm for counting integral points in polytopes. The Kronecker coefficients of the symmetric group, which can be seen to be a special case of such multiplicities, play an important role in the geometric complexity theory approach to the P vs. NP problem. Whereas their computation is known to be #P-hard for Young diagrams with an arbitrary number of rows, our algorithm computes them in polynomial time if the number of rows is bounded. We complement our work by showing that information on the asymptotic growth rates of multiplicities in the coordinate rings of orbit closures does not directly lead to new complexity-theoretic obstructions beyond what can be obtained from the moment polytopes of the orbit closures. Nonasymptotic information on the multiplicities, such as provided by our algorithm, may therefore be essential in order to find obstructions in geometric complexity theory.