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arXiv:2302.08083

The Computational Complexity of Quantum Determinants

Hung, Shih-Han, En-Jui Kuo

Published Feb 16, 2023• 0 citations
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Abstract

In this work, we study the computational complexity of quantum determinants, a $q$-deformation of matrix permanents: Given a complex number $q$ on the unit circle in the complex plane and an $n\times n$ matrix $X$, the $q$-permanent of $X$ is defined as $$\mathrm{Per}_q(X) = \sum_{σ\in S_n} q^{\ell(σ)}X_{1,σ(1)}\ldots X_{n,σ(n)},$$ where $\ell(σ)$ is the inversion number of permutation $σ$ in the symmetric group $S_n$ on $n$ elements. The function family generalizes determinant and permanent, which correspond to the cases $q=-1$ and $q=1$ respectively. For worst-case hardness, by Liouville's approximation theorem and facts from algebraic number theory, we show that for primitive $m$-th root of unity $q$ for odd prime power $m=p^k$, exactly computing $q$-permanent is $\mathsf{Mod}_p\mathsf{P}$-hard. This implies that an efficient algorithm for computing $q$-permanent results in a collapse of the polynomial hierarchy. Next, we show that computing $q$-permanent can be achieved using an oracle that approximates to within a polynomial multiplicative error and a membership oracle for a finite set of algebraic integers. From this, an efficient approximation algorithm would also imply a collapse of the polynomial hierarchy. By random self-reducibility, computing $q$-permanent remains to be hard for a wide range of distributions satisfying a property called the strong autocorrelation property. Specifically, this is proved via a reduction from $1$-permanent to $q$-permanent for $O(1/n^2)$ points $z$ on the unit circle. Since the family of permanent functions shares common algebraic structure, various techniques developed for the hardness of permanent can be generalized to $q$-permanents.

DOI
10.48550/arxiv.2302.08083

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In this work, we study the computational complexity of quantum determinants, a qqq-deformation of matrix permanents: Given a complex number qqq on the unit circle in the complex plane and an n×nn\times nn×n matrix XXX, the qqq-permanent of XXX is defined as Perq(X)=∑σ∈Snqℓ(σ)X1,σ(1)…Xn,σ(n),\mathrm{Per}_q(X) = \sum_{σ\in S_n} q^{\ell(σ)}X_{1,σ(1)}\ldots X_{n,σ(n)},Perq​(X)=∑σ∈Sn​​qℓ(σ)X1,σ(1)​…Xn,σ(n)​, where ℓ(σ)\ell(σ)ℓ(σ) is the inversion number of permutation σσσ in the symmetric group SnS_nSn​ on nnn elements. The function family generalizes determinant and permanent, which correspond to the cases q=−1q=-1q=−1 and q=1q=1q=1 respectively. For worst-case hardness, by Liouville's approximation theorem and facts from algebraic number theory, we show that for primitive mmm-th root of unity qqq for odd prime power m=pkm=p^km=pk, exactly computing qqq-permanent is ModpP\mathsf{Mod}_p\mathsf{P}Modp​P-hard. This implies that an efficient algorithm for computing qqq-permanent results in a collapse of the polynomial hierarchy. Next, we show that computing qqq-permanent can be achieved using an oracle that approximates to within a polynomial multiplicative error and a membership oracle for a finite set of algebraic integers. From this, an efficient approximation algorithm would also imply a collapse of the polynomial hierarchy. By random self-reducibility, computing qqq-permanent remains to be hard for a wide range of distributions satisfying a property called the strong autocorrelation property. Specifically, this is proved via a reduction from 111-permanent to qqq-permanent for O(1/n2)O(1/n^2)O(1/n2) points zzz on the unit circle. Since the family of permanent functions shares common algebraic structure, various techniques developed for the hardness of permanent can be generalized to qqq-permanents.
We investigate the dividing line between classical and quantum computational power in estimating properties of matrix functions. More precisely, we study the computational complexity of two primitive problems: given a function f and a Hermitian matrix A , we compute a matrix element of f ( A ) or a local measurement on f ( A ) | 0 ⟩ ⊗ n , with | 0 ⟩ ⊗ n being an n -qubit reference state vector, in both cases up to additive approximation error. We consider four functions—monomials, Chebyshev polynomials, the time evolution function, and the inverse function—and probe the complexity across a broad landscape covering different problem input regimes. Namely, we consider two types of matrix inputs (sparse and Pauli access), matrix properties (norm and sparsity), the approximation error, and function-specific parameters. We identify BQP- forms of both problems for each function and then toggle the problem parameters to easier regimes to see where hardness remains, or where the problem becomes classically easy. As part of our results, we make concrete a hierarchy of hardness across the functions; in parameter regimes where we have classically efficient algorithms for monomials, all three other functions remain robustly BQP-, or hard under usual computational complexity assumptions. In identifying classically easy regimes, among others, we show that for any polynomial of degree poly ( n ) , both problems can be efficiently classically simulated when A has O ( log  n ) nonzero coefficients in the Pauli basis. This contrasts with the fact that the problems are BQP- in the sparse access model even for constant row sparsity, whereas the stated Pauli access efficiently constructs sparse access with row sparsity O ( log  n ) . Our work provides a catalog of efficient quantum and classical algorithms for fundamental linear-algebra tasks.
We report on a computational and experimental study of permanents. On the computational side, we use the GPU to greaatly accelerate the computation of permanents over C,\mathbb{C},C, R,\mathbb{R},R, Fp\mathbb{F}_pFp​ and Q.\mathbb{Q}.Q. First, for Haar-distributed unitary matrices~UUU, the permanent $\perm(U)$ follows a circularly-symmetric complex Gaussian distribution CN(0,σ2)\mathcal{CN}(0,σ^2)CN(0,σ2) -- we confirm this via a number of tests for nnn up to~23 with 50,00050{,}00050,000 samples. The DFT matrix permanent is an extreme outlier for every prime n≥7n\ge 7n≥7. In contrast, for Haar-random \emph{orthogonal} matrices~OOO, the permanent $\perm(O)$ is approximately real Gaussian but with positive excess kurtosis that decays as~O(1/n)O(1/n)O(1/n), indicating slower convergence. For matrices with Gaussian entries (GUE, GOE, Ginibre), the permanent follows an ααα-stable distribution with stability index α≈1.0α\approx 1.0α≈1.0--1.41.41.4, well below the Gaussian value α=2α=2α=2. We test Aaronson's conjecture that $|\perm(X)|^2$ is asymptotically lognormal for Gaussian~XXX: it is plausible for the complex Ginibre and GOE ensembles, but appears to fail for GUE and real Ginibre, where the ααα-stable tails prevent convergence. Anti-concentration, however, holds for all Gaussian ensembles and is more robust than for Haar unitaries. Secondly, we study the permanent along geodesics on the unitary group. For the geodesic from the identity to the nnn-cycle permutation matrix, we find a universal scaling function $f(t)=\frac{1}{n}\ln|\perm(γ(t))|$ that is independent of~nnn in the large-nnn limit, with a midpoint value \[ \perm(γ({\textstyle\frac12})) = (-1)^{(n-1)/2}\cdot 2e^{-n}\bigl(1+\tfrac{1}{3n}+O(n^{-2})\bigr) \] for odd~nnn and zero for even~nnn. We also study the geodesic forom the identity to the DFT matrix.
We study the approximation of the number of solutions of Laurent polynomials over finite fields. For a Laurent polynomial f(x)=∑j=1sajxuj∈Fq[x1±1,…,xn±1],f(x)=\sum_{j=1}^{s}a_jx^{u_j}\in \mathbb{F}_q[x_1^{\pm1},\ldots,x_n^{\pm1}], f(x)=∑j=1s​aj​xuj​∈Fq​[x1±1​,…,xn±1​], let UUU be its augmented support matrix whose columns are (1,uj)(1,u_j)(1,uj​) with rank ρρρ and N(f):=#{x∈(Fq×)n∣f(x)=0}N(f) := \# \{x\in (\mathbb{F}_q^\times)^n \mid f(x)=0\}N(f):=#{x∈(Fq×​)n∣f(x)=0} be its torus point count. Our first main result is a quantum algorithm that outputs N^(f)\widehat{N}(f)N(f) satisfying ∣N^(f)−N(f)∣≤εqn+s/2−ρ |\widehat{N}(f) - N(f)| \le \varepsilon q^{n+s/2-ρ} ∣N(f)−N(f)∣≤εqn+s/2−ρ with success probability 1−δ1-δ1−δ. Provided that ρρρ and ∥U∥∞\|U\|_\infty∥U∥∞​ are bounded, the algorithm runs in both classical bit and quantum gate complexity poly(n,s,log⁡q,1/ε,log⁡(1/δ))\mathrm{poly}(n, s, \log q, 1/\varepsilon, \log(1/δ))poly(n,s,logq,1/ε,log(1/δ)). It provides finer resolution than relative-error approximations in general settings. To the best of our knowledge, in the explicit finite-field input model considered here, no previous algorithm achieves this additive accuracy with running time polynomial in log⁡q\log qlogq. Van Dam (arXiv:quant-ph/0405081) conjectured the existence of such an algorithm under the assumption of an oracle reflecting the algebraic properties of the polynomial. In contrast, by exploiting a point-counting formula derived from character sums over finite fields, we develop an alternative approach that efficiently approximates the number of points without assuming the existence of such an oracle. As a second main result, we prove that the same approximation problem becomes #\##P-hard under randomized polynomial-time Turing reductions when the support matrix UUU varies freely as part of the input. Thus, taken together, our results clarify how the effectiveness of the quantum approach depends on the tradeoff between the accuracy scale and the support parameters of the input polynomial.
Computing the distribution of permanents of random matrices has been an outstanding open problem for several decades. In quantum computing, "anti-concentration" of this distribution is an unproven input for the proof of hardness of the task of boson-sampling. We study permanents of random i.i.d. complex Gaussian matrices, and more broadly, submatrices of random unitary matrices. Using a hybrid representation-theoretic and combinatorial approach, we prove strong lower bounds for all moments of the permanent distribution. We provide substantial evidence that our bounds are close to being tight and constitute accurate estimates for the moments. Let U(d)k×kU(d)^{k\times k}U(d)k×k be the distribution of k×kk\times kk×k submatrices of d×dd\times dd×d random unitary matrices, and Gk×kG^{k\times k}Gk×k be the distribution of k×kk\times kk×k complex Gaussian matrices. (1) Using the Schur-Weyl duality (or the Howe duality), we prove an expansion formula for the 2t2t2t-th moment of ∣Perm(M)∣|Perm(M)|∣Perm(M)∣ when MMM is drawn from U(d)k×kU(d)^{k\times k}U(d)k×k or Gk×kG^{k\times k}Gk×k. (2) We prove a surprising size-moment duality: the 2t2t2t-th moment of the permanent of random k×kk\times kk×k matrices is equal to the 2k2k2k-th moment of the permanent of t×tt\times tt×t matrices. (3) We design an algorithm to exactly compute high moments of the permanent of small matrices. (4) We prove lower bounds for arbitrary moments of permanents of matrices drawn from Gk×kG^{ k\times k}Gk×k or U(k)U(k)U(k), and conjecture that our lower bounds are close to saturation up to a small multiplicative error. (5) Assuming our conjectures, we use the large deviation theory to compute the tail of the distribution of log-permanent of Gaussian matrices for the first time. (6) We argue that it is unlikely that the permanent distribution can be uniquely determined from the integer moments and one may need to supplement the moment calculations with extra assumptions to prove the anti-concentration conjecture.
A fundamental pursuit in complexity theory concerns reducing worst-case problems to average-case problems. There exist complexity classes such as PSPACE that admit worst-case to average-case reductions. However, for many other classes such as NP, the evidence so far is typically negative, in the sense that the existence of such reductions would cause collapses of the polynomial hierarchy(PH). Basing cryptographic primitives, e.g., the average-case hardness of inverting one-way permutations, on NP-completeness is a particularly intriguing instance. As there is evidence showing that classical reductions from NP-hard problems to breaking these primitives result in PH collapses, it seems unlikely to base cryptographic primitives on NP-hard problems. Nevertheless, these results do not rule out the possibilities of the existence of quantum reductions. In this work, we initiate a study of the quantum analogues of these questions. Aside from formalizing basic notions of quantum reductions and demonstrating powers of quantum reductions by examples of separations, our main result shows that if NP-complete problems reduce to inverting one-way permutations using certain types of quantum reductions, then coNP ⊆ QIP( 2 ) .
In contrast to the determinant, no algorithm is known for the exact determination of the permanent of a square matrix that runs in time polynomial in its dimension. Consequently, noninteracting fermions are classically efficiently simulatable while noninteracting bosons are not, underpinning quantum supremacy arguments for sampling the output distribution of photon interferometer arrays. This work introduces a graph-theoretic framework that bridges both the determinant and permanent. The only nonzero eigenvalues of a sparse non-Hermitian operator $\stackrel{\ifmmode \u\else \textasciibreve \fi{}}{M}$ for nnn spin-1/21/21/2 particles are the nthn\mathrm{th}nth roots of the permanent or determinant of an $n\ifmmode\times\else\texttimes\fi{}n$ matrix MMM, interpreting basis states as bosonic or fermionic occupation states, respectively. This operator can be used to design a simple and straightforward method for the classical determination of the permanent that matches the efficiency of the best-known algorithm. Gauss-Jordan elimination for the determinant of MMM is then equivalent to the successive removal of the generalized zero eigenspace of the fermionic $\stackrel{\ifmmode \u\else \textasciibreve \fi{}}{M}$, equivalent to the deletion of some nodes and reweighting of the remaining edges in the graph such that only nnn nodes survive after the last step. In the bosonic case, the successive removal of generalized zero eigenspaces for $\stackrel{\ifmmode \u\else \textasciibreve \fi{}}{M}$ is also equivalent to node deletion, but new edges are added during this process, which gives rise to the higher complexity of computing the permanent. Our analysis may point the way to new strategies for classical and quantum evaluation of the permanent.