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Annual Symposium on Foundations of Computer Science

Efficient Unitary Designs from Random Sums and Permutations

Chi-Fang Chen, Jordan Docter, Michelle Xu, Adam Bouland, Fernando G. S. L. Brandão, Patrick M. Hayden

Published Oct 27, 2024• 13 citations
Publisher

Abstract

A unitary k-design is an ensemble of unitaries that matches the first$k$moments of the Haar measure. In this work, we provide two efficient constructions of k-designs on n-qubits using new random matrix theory techniques. Our first construction is based on exponentiating sums of random i.i.d. Hermitian matrices and uses O(k2n2)-many gates. In the spirit of central limit theorems, we show that this random sum approximates the Gaussian Unitary Ensemble (GUE). We then show that the product of just two exponentiated GUE matrices is already approximately Haar random. Our second construction is based on products of exponentiated sums of random permutations and uses Õ($k$poly ($n$)) many gates. The$k$dependence is optimal (up to polylogarithmic factors) and is inherited from the efficiency of existing k-wise independent permutations. Furthermore, replacing random permutations with quantum-secure pseudorandom permutations (PRPs), we also obtain a pseudorandom unitary (PRU) ensemble that is secure under nonadaptive queries. A central feature of both proofs is a new connection between the polynomial method in quantum query complexity and the large-dimension ($N$) expansion in random matrix theory. In particular, the first construction uses the polynomial method to control high moments of certain random matrix ensembles without requiring delicate Weingarten calculations. In doing so, we define and solve a moment problem on the unit circle, asking whether a finite number of equally weighted points can reproduce a given set of moments. In our second construction, the key step is to exhibit an orthonormal basis for irreducible representations of the partition algebra that has a low-degree large-$N$expansion. This allows us to show that the distinguishing probability is a low-degree rational polynomial of the dimension$N$.

DOI
10.1109/focs61266.2024.00037

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A unitary k-design is an ensemble of unitaries that matches the firstkkkmoments of the Haar measure. In this work, we provide two efficient constructions of k-designs on n-qubits using new random matrix theory techniques. Our first construction is based on exponentiating sums of random i.i.d. Hermitian matrices and uses O(k2n2)-many gates. In the spirit of central limit theorems, we show that this random sum approximates the Gaussian Unitary Ensemble (GUE). We then show that the product of just two exponentiated GUE matrices is already approximately Haar random. Our second construction is based on products of exponentiated sums of random permutations and uses Õ(kkkpoly (nnn)) many gates. Thekkkdependence is optimal (up to polylogarithmic factors) and is inherited from the efficiency of existing k-wise independent permutations. Furthermore, replacing random permutations with quantum-secure pseudorandom permutations (PRPs), we also obtain a pseudorandom unitary (PRU) ensemble that is secure under nonadaptive queries. A central feature of both proofs is a new connection between the polynomial method in quantum query complexity and the large-dimension (NNN) expansion in random matrix theory. In particular, the first construction uses the polynomial method to control high moments of certain random matrix ensembles without requiring delicate Weingarten calculations. In doing so, we define and solve a moment problem on the unit circle, asking whether a finite number of equally weighted points can reproduce a given set of moments. In our second construction, the key step is to exhibit an orthonormal basis for irreducible representations of the partition algebra that has a low-degree large-NNNexpansion. This allows us to show that the distinguishing probability is a low-degree rational polynomial of the dimensionNNN.
In this work we give an efficient construction of unitary kkk-designs using O~(k⋅poly(n))\tilde{O}(k\cdot poly(n))O~(k⋅poly(n)) quantum gates, as well as an efficient construction of a parallel-secure pseudorandom unitary (PRU). Both results are obtained by giving an efficient quantum algorithm that lifts random permutations over S(N)S(N)S(N) to random unitaries over U(N)U(N)U(N) for N=2nN=2^nN=2n. In particular, we show that products of exponentiated sums of S(N)S(N)S(N) permutations with random phases approximately match the first 2Ω(n)2^{Ω(n)}2Ω(n) moments of the Haar measure. By substituting either O~(k)\tilde{O}(k)O~(k)-wise independent permutations, or quantum-secure pseudorandom permutations (PRPs) in place of the random permutations, we obtain the above results. The heart of our proof is a conceptual connection between the large dimension (large-NNN) expansion in random matrix theory and the polynomial method, which allows us to prove query lower bounds at finite-NNN by interpolating from the much simpler large-NNN limit. The key technical step is to exhibit an orthonormal basis for irreducible representations of the partition algebra that has a low-degree large-NNN expansion. This allows us to show that the distinguishing probability is a low-degree rational polynomial of the dimension NNN.
Unitary TTT-designs play an important role in quantum information, with diverse applications in quantum algorithms, benchmarking, tomography, and communication. Until now, the most efficient construction of unitary TTT-designs for nnn-qudit systems has been via random local quantum circuits, which have been shown to converge to approximate TTT-designs in the diamond norm using O(T5+o(1)n2)O(T^{5+o(1)} n^2)O(T5+o(1)n2) quantum gates. In this work, we provide a new construction of TTT-designs via random matrix theory using O~(T2n2)\tilde{O}(T^2 n^2)O~(T2n2) quantum gates. Our construction leverages two key ideas. First, in the spirit of central limit theorems, we approximate the Gaussian Unitary Ensemble (GUE) by an i.i.d. sum of random Hermitian matrices. Second, we show that the product of just two exponentiated GUE matrices is already approximately Haar random. Thus, multiplying two exponentiated sums over rather simple random matrices yields a unitary TTT-design, via Hamiltonian simulation. A central feature of our proof is a new connection between the polynomial method in quantum query complexity and the large-dimension (NNN) expansion in random matrix theory. In particular, we show that the polynomial method provides exponentially improved bounds on the high moments of certain random matrix ensembles, without requiring intricate Weingarten calculations. In doing so, we define and solve a new type of moment problem on the unit circle, asking whether a finite number of equally weighted points, corresponding to eigenvalues of unitary matrices, can reproduce a given set of moments.
Can simple processes appear highly complex? Gowers (Comb. Prob. Comp. '96) conjectured that repeatedly composing local random reversible operations can yield global permutations that are indistinguishable from random. In this work, we study the unitary quantum analog of this question, in an attempt to make new progress on this longstanding conjecture. Our first result shows that statistical moment matching in the form of unitary designs does not generically lead to pseudorandomness---even for the simplest quantum processes: for every fixed ttt, we give an efficiently samplable family {νn}n\{ν_n\}_n{νn​}n​ of distributions on one- and two-qubit gates such that, after T=Ot(n2log⁡2n)T=O_t(n^2\log^2 n)T=Ot​(n2log2n) independent steps, the resulting nnn-qubit ensemble is an approximate unitary ttt-design with negligible error exp⁡(−Ω(log⁡2n))\exp(-Ω(\log^2 n))exp(−Ω(log2n)), yet an efficient quantum algorithm distinguishes it from random using only Ot(log⁡2n)O_t(\log^2 n)Ot​(log2n) queries. This refutes the unitary analog of the Hoory--Magen--Myers--Rackoff conjecture (ICALP '04) for permutations. Our second result is a stronger separation between unitary designs and pseudorandom unitaries at polynomially bounded moments; our counterexample, however, requires highly structured ensembles, in contrast with the simple local walks from before. This suggests caution when using unitary designs to model information scrambling in black-hole physics, as even maximally scrambled systems can exhibit structure which is accessible to efficient experiments. Motivated by these findings, we then propose new conjectures for how pseudorandomness can plausibly emerge within simple quantum processes, such as random quantum circuits.
Unitary designs provide finite-moment approximations to Haar-random unitaries, with wide-ranging applications across physics and quantum information, from scrambling and black-hole dynamics to foundational primitives in quantum algorithms. Strong unitary designs capture a more demanding operational notion of approximation, requiring indistinguishability from Haar randomness even for quantum algorithms that may access a unitary not only in the forward direction, but also through its inverse, transpose, and complex conjugate. Motivated by the physical requirement that scrambling arise within the system itself, Schuster, Ma, Lombardi, Brandão, and Huang (arXiv:2509.26310) left open whether strong unitary designs can be generated in logarithmic depth using only the system qubits. For every fixed design order kkk and measurable-error tolerance, we construct strong approximate unitary kkk-designs in optimal Θ(log⁡n)Θ(\log n)Θ(logn) all-to-all circuit depth using only the nnn original system qubits. Our new ingredient is a logarithmic-depth Pauli-mixing bound for the perfect-matching ensemble, whose layers pair the qubits uniformly at random and apply independent random two-qubit gates. This bound controls the mixed forward-reverse two-query case, which we combine with existing design and gluing results to obtain strong unitary designs of arbitrary fixed order.
Pseudorandom unitaries (PRUs) are ensembles of efficiently implementable unitary operators that cannot be distinguished from Haar random unitaries by any quantum polynomial-time algorithm with query access to the unitary. We present a simple PRU construction that is a concatenation of a random Clifford unitary, a pseudorandom binary phase operator, and a pseudorandom permutation operator. We prove that this PRU construction is secure against non-adaptive distinguishers assuming the existence of quantum-secure one-way functions. This means that no efficient quantum query algorithm that is allowed a single application of U⊗poly(n)U^{\otimes \mathrm{poly}(n)}U⊗poly(n) can distinguish whether an nnn-qubit unitary UUU was drawn from the Haar measure or our PRU ensemble. We conjecture that our PRU construction remains secure against adaptive distinguishers, i.e. secure against distinguishers that can query the unitary polynomially many times in sequence, not just in parallel.
We construct ε\varepsilonε-approximate unitary kkk-designs on nnn qubits in circuit depth O(log⁡klog⁡log⁡nk/ε)O(\log k \log \log n k / \varepsilon)O(logkloglognk/ε). The depth is exponentially improved over all known results in all three parameters nnn, kkk, ε\varepsilonε. We further show that each dependence is optimal up to exponentially smaller factors. Our construction uses O~(nk)\tilde{O}(nk)O~(nk) ancilla qubits and O(nk){O}(nk)O(nk) bits of randomness, which are also optimal up to log⁡(nk)\log(n k)log(nk) factors. An alternative construction achieves a smaller ancilla count O~(n)\tilde{O}(n)O~(n) with circuit depth O(klog⁡log⁡nk/ε){O}(k \log \log nk/\varepsilon)O(kloglognk/ε). To achieve these efficient unitary designs, we introduce a highly-structured random unitary ensemble that leverages long-range two-qubit gates and low-depth implementations of random classical hash functions. We also develop a new analytical framework for bounding errors in quantum experiments involving many queries to random unitaries. As an illustration of this framework's versatility, we provide a succinct alternative proof of the existence of pseudorandom unitaries.