Parallelization is a major challenge in quantum algorithms due to physical constraints like no-cloning. This is vividly illustrated by the conjecture of Moore and Nilsson from their seminal work on quantum circuit complexity [MN01, announced 1998]: unitaries of a deceptively simple form--controlled-unitary "staircases"--require circuits of minimum depth $Ω(n)$. If true, this lower bound would represent a major break from classical parallelism and prove a quantum-native analogue of the famous NC $\neq$ P conjecture. In this work we settle the Moore-Nilsson conjecture in the negative by compressing all circuits in the class to depth $O(\log n)$, which is the best possible. The parallelizations are exact, ancilla-free, and can be computed in poly($n$) time. We also consider circuits restricted to 2D connectivity, for which we derive compressions of optimal depth $O(\sqrt{n})$. More generally, we make progress on the project of quantum parallelization by introducing a quantum blockwise precomputation technique somewhat analogous to the method of Arlazarov, Dinič, Kronrod, and Faradžev [Arl+70] in classical dynamic programming, often called the "Four-Russians method." We apply this technique to more-general "cascade" circuits as well, obtaining for example polynomial depth reductions for staircases of controlled $\log(n)$-qubit unitaries.

A K-XORGAME system corresponds to a K-XORSAT system with the additional restriction that the variables divide uniformly into K blocks. This forms a system of m equations with K n unknowns over ℤ_2, and a perfect strategy corresponds to a solution to these equations. Equivalently, such equations correspond to colorings of a K-uniform K-partite hypergraph. This paper proves that the satisfiability threshold of m/n for K-XORGAME problems exists and equals the satisfiability threshold for K-XORSAT.

Free spectrahedra are dimension free solution sets to linear matrix inequalities of the form L-A(X) = I-d circle times I-n+A(1)circle times X-1+A(2)circle times X-2+center dot center dot center dot+A(g)circle times X-g >= 0, where the Ai and Xi are symmetric matrices and the Xi have any size n x n. Free spectrahedra are ubiquitous in systems engineering, operator algebras, and the theory of matrix convex sets. Matrix and free extreme points of free spectrahedra are particularly important. We present theoretical, algorithmic, and experimental results illuminating basic properties of extreme points. For example, though many authors have studied matrix and free extreme points, it has until now been unknown if these two types of extreme points are actually different. This paper settles that issue.We also present and analyze several algorithms. Namely, we perfect an algorithm for computing an expansion of an element of a free spectrahedron in terms of free extreme points. We also give algorithms for testing if a point is matrix extreme and for computing matrix extreme points that are not free extreme.

We study synchronous values of games, especially synchronous games. It is known that a synchronous game has a perfect strategy if and only if it has a perfect synchronous strategy. However, we give examples of synchronous games, in particular graph colouring games, with synchronous value that is strictly smaller than their ordinary value. Thus, the optimal strategy for a synchronous game need not be synchronous. We derive a formula for the synchronous value of an XOR game as an optimization problem over a spectrahedron involving a matrix related to the cost matrix. We give an example of a game such that the synchronous value of repeated products of the game is strictly increasing. We show that the synchronous quantum bias of the XOR of two XOR games is not multiplicative. Finally, we derive geometric and algebraic conditions that a set of projections that yields the synchronous value of a game must satisfy.

The Quantum Max Cut (QMC) problem has emerged as a test-problem for designing approximation algorithms for local Hamiltonian problems. In this paper we attack this problem using the algebraic structure of QMC, in particular the relationship between the quantum max cut Hamiltonian and the representation theory of the symmetric group. The first major contribution of this paper is an extension of non-commutative Sum of Squares (ncSoS) optimization techniques to give a new hierarchy of relaxations to Quantum Max Cut. The hierarchy we present is based on optimizations over polynomials in the qubit swap operators. This is in contrast to the "standard" quantum Lasserre Hierarchy, which is based on polynomials expressed in terms of the Pauli matrices. To prove correctness of this hierarchy, we exploit a finite presentation of the algebra generated by the qubit swap operators. This presentation allows for the use of computer algebraic techniques to manipulate and simplify polynomials written in terms of the swap operators, and may be of independent interest. Surprisingly, we find that level-2 of this new hierarchy is numerically exact (up to tolerance 10^(-7)) on all QMC instances with uniform edge weights on graphs with at most 8 vertices. The second major contribution of this paper is a polynomial-time algorithm that computes (in exact arithmetic) the maximum eigenvalue of the QMC Hamiltonian for certain graphs, including graphs that can be "decomposed" as a signed combination of cliques. A special case of the latter are complete bipartite graphs with uniform edge-weights, for which exact solutions are known from the work of Lieb and Mattis. Our methods, which use representation theory of the symmetric group, can be seen as a generalization of the Lieb-Mattis result.

Semidefinite programming is based on optimization of linear functionals over convex sets defined by linear matrix inequalities, namely, inequalities of the form L-A(X)=I-A(1)X(1)- horizontal ellipsis -A(g)X(g) >= 0. Here, the X-j are real numbers and the set of solutions is called a spectrahedron. These inequalities make sense when the X-i are symmetric matrices of any size, n x n, and enter the formula though tensor product Ai circle times Xi: The solution set of LA(X)>= 0 is called a free spectrahedron since it contains matrices of all sizes and the defining "linear pencil" is "free" of the sizes of the matrices. In this article, we report on empirically observed properties of optimizers obtained from optimizing linear functionals over free spectrahedra restricted to matrices X-i of fixed size n x n. The optimizers we find are always classical extreme points. Surprisingly, in many reasonable parameter ranges, over 99.9% are also free extreme points. Moreover, the dimension of the active constraint, ker(L-A(X-l)), is about twice what we expected. Another distinctive pattern regards reducibility of optimizing tuples (X-1(l), ... ,X-g(l)). We give an algorithm for representing elements of a free spectrahedron as matrix convex combinations of free extreme points; these representations satisfy a very low bound on the number of free extreme points needed.

We consider 3XOR games with perfect commuting operator strategies. Given any 3XOR game, we show existence of a perfect commuting operator strategy for the game can be decided in polynomial time. Previously this problem was not known to be decidable. Our proof leads to a construction, showing a 3XOR game has a perfect commuting operator strategy iff it has a perfect tensor product strategy using a 3 qubit (8 dimensional) GHZ state. This shows that for perfect 3XOR games the advantage of a quantum strategy over a classical strategy (defined by the quantum-classical bias ratio) is bounded. This is in contrast to the general 3XOR case where the optimal quantum strategies can require high dimensional states and there is no bound on the quantum advantage. To prove these results, we first show equivalence between deciding the value of an XOR game and solving an instance of the subgroup membership problem on a class of right angled Coxeter groups. We then show, in a proof that consumes most of this paper, that the instances of this problem corresponding to 3XOR games can be solved in polynomial time.

The foundations of classical Algebraic Geometry and Real Algebraic Geometry are the Nullstellensatz and Positivstellensatz. Over the last two decades the basic analogous theorems for matrix and operator theory (noncommutative variables) have emerged. This paper concerns commuting operator strategies for nonlocal games, recalls NC Nullstellensatz which are helpful, extends these, and applies them to a very broad collection of games. In the process it brings together results spread over different literatures, hence rather than being terse, our style is fairly expository. The main results of this paper are two characterizations, based on Nullstellensatz, which apply to games with perfect commuting operator strategies. The first applies to all games and reduces the question of whether or not a game has a perfect commuting operator strategy to a question involving left ideals and sums of squares. Previously, Paulsen and others translated the study of perfect synchronous games to problems entirely involving a $*$-algebra.The characterization we present is analogous, but works for all games. The second characterization is based on a new Nullstellensatz we derive in this paper. It applies to a class of games we call torically determined games, special cases of which are XOR and linear system games. For these games we show the question of whether or not a game has a perfect commuting operator strategy reduces to instances of the subgroup membership problem and, for linear systems games, we further show this subgroup membership characterization is equivalent to the standard characterization of perfect commuting operator strategies in terms of solution groups. Both the general and torically determined games characterizations are amenable to computer algebra techniques, which we also develop.

Recent results showed it was possible to determine if a modest size 3XOR game has a perfect quantum strategy. We build on these and give an explicit polynomial time algorithm which constructs such a perfect strategy or refutes its existence. This new tool lets us numerically study the behavior of randomly generated 3XOR games with large numbers of questions. A key issue is: how common are pseudotelephathy games (games with perfect quantum strategies but no perfect classical strategies)? Our experiments strongly indicate that the probability of a randomly generated game being pseudotelpathic stays far from 1, indeed it is bounded below 0.15. We also find strong evidence that randomly generated 3XOR games undergo both a quantum and classical "phase transition", transitioning from almost certainly perfect to almost certainly imperfect as the ratio of number of clauses ($m$) to number of questions ($n$) increases. The locations of these two phase transitions appear to coincide at $m/n \approx 2.74$.

Unlimited access to a motorway network can, in overloaded conditions, cause a loss of throughput. Ramp metering, by controlling access to the motorway at onramps, can help avoid this loss of throughput. The queues that form at onramps are dependent on the metering rates chosen at the onramps, and these choices affect how the capacities of different motorway sections are shared amongst competing flows. In this paper we perform an analytical study of a fluid, or differential equation, model of a linear network topology with onramp queues. The model allows for adaptive arrivals, in the sense that the rate at which external traffic enters the queue at an onramp can depend on the current perceived delay in that queue. The model also includes a ramp metering policy which uses global onramp queue length information to determine the rate at which traffic enters the motorway from each onramp. This ramp metering policy minimizes the maximum delay over all onramps and produces equal delay times over many onramps. The paper characterizes both the dynamics and the equilibrium behavior of the system under this policy. While we consider an idealized model that leaves out many practical details, an aim of the paper is to develop analytical methods that yield interesting qualitative insights and might be adapted to more general contexts. The paper can be considered as a step in developing an analytical approach towards studying more complex network topologies and incorporating other model features.

This paper analyses a Waring type decomposition of a noncommuting (NC) polynomial p with respect to the goal of evaluating p efficiently on tuples of matrices. Such a decomposition can reduce the number of matrix multiplications needed to evaluate a noncommutative polynomial and is valuable when a single polynomial must be evaluated on many matrix tuples. In pursuit of this goal we examine a noncommutative analog of the classical Waring problem and various related decompositions. For example, we consider a "Waring decomposition" in which each product of linear terms is actually a power of a single linear NC polynomial or more generally a power of a homogeneous NC polynomial. We describe how NC polynomials compare to commutative ones with regard to these decompositions, describe a method for computing the NC decompositions and compare the effect of various decompositions on the speed of evaluation of generic NC polynomials.

The free closed semialgebraic set $${\mathcal {D}}_f$$ determined by a hermitian noncommutative polynomial $$f\in {\text {M}}_{{\delta }}({\mathbb {C}}\mathop {<}x,x^*\mathop {>})$$ is the closure of the connected component of $$\{(X,X^*)\mid f(X,X^*)\succ 0\}$$ containing the origin. When L is a hermitian monic linear pencil, the free closed semialgebraic set $${\mathcal {D}}_L$$ is the feasible set of the linear matrix inequality $$L(X,X^*)\succeq 0$$ and is known as a free spectrahedron. Evidently these are convex and it is well known that a free closed semialgebraic set is convex if and only it is a free spectrahedron. The main result of this paper solves the basic problem of determining those f for which $${\mathcal {D}}_f$$ is convex. The solution leads to an efficient algorithm that not only determines if $${\mathcal {D}}_f$$ is convex, but if so, produces a minimal hermitian monic pencil L such that $${\mathcal {D}}_f={\mathcal {D}}_L$$ . Of independent interest is a subalgorithm based on a Nichtsingulärstellensatz presented here: given a linear pencil $${\widetilde{L}}$$ and a hermitian monic pencil L, it determines if $${\widetilde{L}}$$ takes invertible values on the interior of $${\mathcal {D}}_L$$ . Finally, it is shown that if $${\mathcal {D}}_f$$ is convex for an irreducible hermitian $$f\in {\mathbb {C}}\mathop {<}x,x^*\mathop {>}$$ , then f has degree at most two, and arises as the Schur complement of an L such that $${\mathcal {D}}_f={\mathcal {D}}_L$$ .

Given a tuple $E=(E_1,dots,E_g)$ of $dtimes d$ matrices, the collection of those tuples of matrices $X=(X_1,dots,X_g)$ (of the same size) such that $| sum E_jotimes X_j|le 1$ is called a spectraball $mathcal B_E$. Likewise, given a tuple $B=(B_1,dots,B_g)$ of $etimes e$ matrices the collection of tuples of matrices $X=(X_1,dots,X_g)$ (of the same size) such that $I + sum B_jotimes X_j +sum B_j^* otimes X_j^*succeq 0$ is a free spectrahedron $mathcal D_B$. Assuming $E$ and $B$ are irreducible, plus an additional mild hypothesis, there is a free bianalytic map $p:mathcal B_Eto mathcal D_B$ normalized by $p(0)=0$ and $pu0027(0)=I$ if and only if $mathcal B_E=mathcal B_B$ and $B$ spans an algebra. Moreover $p$ is unique, rational and has an elegant algebraic representation.

A noncommutative (nc) function in x1,…,xg,x1⁎,…,xg⁎ is called plurisubharmonic (plush) if its nc complex Hessian takes only positive semidefinite values on an nc neighborhood of 0. The main result of this paper shows that an nc rational function is plush if and only if it is a composite of a convex rational function with an analytic (no xj⁎) rational function. The proof is entirely constructive. Further, a simple computable necessary and sufficient condition for an nc rational function to be plush is given in terms of its minimal realization.

An operator C on a Hilbert space H dilates to an operator T on a Hilbert space K if there is an isometry V from H to K such that C=V^*TV. A main result of this paper is, for a positive integer d, the simultaneous dilation, up to a sharp factor $\vartheta(d)$, of all d-by-d symmetric matrices of operator norm at most one to a collection of commuting self-adjoint contraction operators on a Hilbert space. An analytic formula for $\vartheta(d)$ is derived, which as a by-product gives new probabilistic results for the binomial and beta distributions. Dilating to commuting operators has consequences for the theory of linear matrix inequalities (LMIs). Given a tuple A=(A_1,...,A_g) of symmetric matrices of the same size, L(x):=I-\sum A_j x_j is a monic linear pencil. The solution set S_L of the corresponding linear matrix inequality, consisting of those x in R^g for which L(x) is positive semidefinite (PsD), is a spectrahedron. The set D_L of tuples X=(X_1,...,X_g) of symmetric matrices (of the same size) for which L(X):=I-\sum A_j \otimes X_j is PsD, is a free spectrahedron. A result here is: any tuple X of d-by-d symmetric matrices in a bounded free spectrahedron D_L dilates, up to a scale factor, to a tuple T of commuting self-adjoint operators with joint spectrum in the corresponding spectrahedron S_L. From another viewpoint, the scale factor measures the extent that a positive map can fail to be completely positive. Given another monic linear pencil M, the inclusion D_L \subset D_M obviously implies the inclusion S_L \subset S_M and thus can be thought of as its free relaxation. Determining if one free spectrahedron contains another can be done by solving an explicit LMI and is thus computationally tractable. The scale factor for commutative dilation of D_L gives a precise measure of the worst case error inherent in the free relaxation, over all monic linear pencils M of size d.

Let SM_n( ℝ )^g denote g -tuples of n × n real symmetric matrices. Given tuples X=(X_1, … , X_g) ∈SM_n_1( ℝ )^g and Y=(Y_1, … , Y_g) ∈SM_n_2( ℝ )^g , a matrix convex combination of X and Y is a sum of the form V_1^* XV_1+V_2^* Y V_2 V_1^* V_1+V_2^* V_2=I_n where V_1: ℝ ^n →ℝ ^n_1 and V_2: ℝ ^n →ℝ ^n_2 are contractions. Matrix convex sets are sets which are closed under matrix convex combinations. A key feature of matrix convex combinations is that the g -tuples X , Y , and V_1^* XV_1+V_2^* Y V_2 do not need to have the same size. As a result, matrix convex sets are a dimension free analog of convex sets. While in the classical setting there is only one notion of an extreme point, there are three main notions of extreme points for matrix convex sets: ordinary, matrix, and absolute extreme points. Absolute extreme points are closely related to the classical Arveson boundary. A central goal in the theory of matrix convex sets is to determine if one of these types of extreme points for a matrix convex set minimally recovers the set through matrix convex combinations. This article shows that every real compact matrix convex set which is defined by a linear matrix inequality is the matrix convex hull of its absolute extreme points, and that the absolute extreme points are the minimal set with this property. Furthermore, we give an algorithm which expresses a tuple as a matrix convex combination of absolute extreme points with optimal bounds. Similar results hold when working over the field of complex numbers rather than the reals.

We associate to each synchronous game an algebra whose representations determine if the game has a perfect deterministic strategy, perfect quantum strategy or one of several other perfect strategies. when applied to the graph coloring game, this leads to characterizations in terms of properties of an algebra of various quantum chromatic numbers that have been studied in the literature. This allows us to develop a correspondence between various chromatic numbers of a graph and ideals in this algebra which can then be approached via various Grobner basis methods.

Linear matrix inequalities (LMIs) \(I_d + \sum _{j=1}^g A_jx_j + \sum _{j=1}^g A_j^*x_j^* \succeq 0\) play a role in many areas of applications. The set of solutions of an LMI is a spectrahedron. LMIs in (dimension-free) matrix variables model most problems in linear systems engineering, and their solution sets are called free spectrahedra. Free spectrahedra are exactly the free semialgebraic convex sets. This paper studies free analytic maps between free spectrahedra and, under certain (generically valid) irreducibility assumptions, classifies all those that are bianalytic. The foundation of such maps turns out to be a very small class of birational maps we call convexotonic. The convexotonic maps in g variables sit in correspondence with g-dimensional algebras. If two bounded free spectrahedra \({\mathcal {D}}_A\) and \({\mathcal {D}}_B\) meeting our irreducibility assumptions are free bianalytic with map denoted p, then p must (after possibly an affine linear transform) extend to a convexotonic map corresponding to a g-dimensional algebra spanned by \((U-I)A_1,\ldots ,(U-I)A_g\) for some unitary U. Furthermore, B and UA are unitarily equivalent. The article also establishes a Positivstellensatz for free analytic functions whose real part is positive semidefinite on a free spectrahedron and proves a representation for a free analytic map from \({\mathcal {D}}_A\) to \({\mathcal {D}}_B\) (not necessarily bianalytic). Another result shows that a function analytic on any radial expansion of a free spectrahedron is approximable by polynomials uniformly on the spectrahedron. These theorems are needed for classifying free bianalytic maps.

The free singularity locus of a noncommutative polynomial f is defined to be the sequence of hypersurfaces Zn(f)={X∈Mn(k)g:det⁡f(X)=0}. The main theorem of this article shows that f is irreducible if and only if Zn(f) is eventually irreducible. A key step in the proof is an irreducibility result for linear pencils. Arising from this is a free singularity locus Nullstellensatz for noncommutative polynomials. Apart from consequences to factorization in a free algebra, the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry.

For matrix convex sets, a unified geometric interpretation of notions of extreme points and of Arveson boundary points is given. These notions include, in increasing order of strength, the core notions of “Euclidean” extreme points, “matrix” extreme points, and “absolute” extreme points. A seemingly different notion, the “Arveson boundary”, has by contrast a dilation-theoretic flavor. An Arveson boundary point is an analog of a (not necessarily irreducible) boundary representation for an operator system. This article provides and explores dilation-theoretic formulations for the above notions of extreme points. The scalar solution set of a linear matrix inequality (LMI) is known as a spectrahedron. The matricial solution set of an LMI is a free spectrahedron. Spectrahedra (resp. free spectrahedra) lie between general convex sets (resp. matrix convex sets) and convex polyhedra (resp. free polyhedra). As applications of our theorems on extreme points, it is shown that the polar dual of a matrix convex set K is generated, as a matrix convex set, by finitely many Arveson boundary points if and only if K is a free spectrahedron, and if the polar dual of a free spectrahedron K is again a free spectrahedron, then at the scalar level K is a polyhedron.