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arXiv:2007.10885 · IEEE Transactions on Information Theory

Epsilon-Nets, Unitary Designs, and Random Quantum Circuits

Michał Oszmaniec, Adam Sawicki, Michał Horodecki

Published Nov 13, 2021• 27 citations
arXivHTMLPDF

Abstract

Epsilon-nets and approximate unitary$t$-designs are natural notions that capture properties of unitary operations relevant for numerous applications in quantum information and quantum computing. In this work we study quantitative connections between these two notions. Specifically, we prove that, for$d$dimensional Hilbert space, unitaries constituting$\delta $-approximate$t$-expanders form$\epsilon $-nets for$t\simeq \frac {d^{5/2}}{ \epsilon }$and$\delta \simeq \left ({\frac { \epsilon ^{3/2}}{d}}\right)^{d^{2}}$. We also show that for arbitrary$t$,$\epsilon $-nets can be used to construct$\delta $-approximate unitary$t$-designs for$\delta \simeq \epsilon t$, where the notion of approximation is based on the diamond norm. Finally, we prove that the degree of an exact unitary$t$design necessary to obtain an$\epsilon $-net must grow at least as fast as$\frac {1}{ \epsilon }$(for fixed dimension) and not slower than$d^{2}$(for fixed$\epsilon $). This shows near optimality of our result connecting$t$-designs and$\epsilon $-nets. We apply our findings in the context of quantum computing. First, we show that that approximate t-designs can be generated by shallow random circuits formed from a set of universal two-qudit gates in the parallel and sequential local architectures considered in (Brandão et al., 2016). Importantly, our gate sets need not to be symmetric (i.e., contains gates together with their inverses) or consist of gates with algebraic entries. Second, we consider compilation of quantum gates and prove a non-constructive Solovay-Kitaev theorem for general universal gate sets. Our main technical contribution is a new construction of efficient polynomial approximations to the Dirac delta in the space of quantum channels, which can be of independent interest.

DOI
10.1109/tit.2021.3128110

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Epsilon-nets and approximate unitaryttt-designs are natural notions that capture properties of unitary operations relevant for numerous applications in quantum information and quantum computing. In this work we study quantitative connections between these two notions. Specifically, we prove that, forddddimensional Hilbert space, unitaries constitutingδ\delta δ-approximatettt-expanders formϵ\epsilon ϵ-nets fort≃d5/2ϵt\simeq \frac {d^{5/2}}{ \epsilon }t≃ϵd5/2​andδ≃(ϵ3/2d)d2\delta \simeq \left ({\frac { \epsilon ^{3/2}}{d}}\right)^{d^{2}}δ≃(dϵ3/2​)d2. We also show that for arbitraryttt,ϵ\epsilon ϵ-nets can be used to constructδ\delta δ-approximate unitaryttt-designs forδ≃ϵt\delta \simeq \epsilon tδ≃ϵt, where the notion of approximation is based on the diamond norm. Finally, we prove that the degree of an exact unitarytttdesign necessary to obtain anϵ\epsilon ϵ-net must grow at least as fast as1ϵ\frac {1}{ \epsilon }ϵ1​(for fixed dimension) and not slower thand2d^{2}d2(for fixedϵ\epsilon ϵ). This shows near optimality of our result connectingttt-designs andϵ\epsilon ϵ-nets. We apply our findings in the context of quantum computing. First, we show that that approximate t-designs can be generated by shallow random circuits formed from a set of universal two-qudit gates in the parallel and sequential local architectures considered in (Brandão et al., 2016). Importantly, our gate sets need not to be symmetric (i.e., contains gates together with their inverses) or consist of gates with algebraic entries. Second, we consider compilation of quantum gates and prove a non-constructive Solovay-Kitaev theorem for general universal gate sets. Our main technical contribution is a new construction of efficient polynomial approximations to the Dirac delta in the space of quantum channels, which can be of independent interest.
The capacity to randomly pick a unitary across the whole unitary group is a powerful tool across physics and quantum information. A unitary t-design is designed to tackle this challenge in an efficient way, yet constructions to date rely on heavy constraints. In particular, they are composed of ensembles of unitaries which, for technical reasons, must contain inverses and whose entries are algebraic. In this work, we reduce the requirements for generating an ε -approximate unitary t-design. To do so, we first construct a specific n-qubit random quantum circuit composed of a sequence of randomly chosen 2-qubit gates, chosen from a set of unitaries which is approximately universal on U ( 4 ) , yet need not contain unitaries and their inverses nor are in general composed of unitaries whose entries are algebraic; dubbed r e l a x e d seed. We then show that this relaxed seed, when used as a basis for our construction, gives rise to an ε -approximate unitary t-design efficiently, where the depth of our random circuit scales as p o l y ( n , t , l o g ( 1 / ε ) ) , thereby overcoming the two requirements which limited previous constructions. We suspect the result found here is not optimal and can be improved; particularly because the number of gates in the relaxed seeds introduced here grows with n and t. We conjecture that constant sized seeds such as those which are usually present in the literature are sufficient.
Abstract The concepts of ε -nets and unitary ( δ -approximate) t -designs are important and ubiquitous across quantum computation and information. Both notions are closely related and the quantitative relations between t , δ and ε find applications in areas such as (non-constructive) inverse-free Solovay–Kitaev like theorems and random quantum circuits. In recent work, quantitative relations have revealed the close connection between the two constructions, with ε -nets functioning as unitary δ -approximate t -designs and vice-versa, for appropriate choice of parameters. In this work we improve these results, significantly increasing the bound on the δ required for a δ -approximate t -design to form an ε -net from δ ≃ ( ϵ 3 / 2 / d ) d 2 to δ ≃ ( ϵ / d 1 / 2 ) d 2 . We achieve this by constructing polynomial approximations to the Dirac delta using heat kernels on the projective unitary group PU ( d ) ≅ U ( d ) , whose properties we studied and which may be applicable more broadly. We also outline the possible applications of our results in quantum circuit overheads, quantum complexity and black hole physics.
At its core a ttt-design is a method for sampling from a set of unitaries in a way which mimics sampling randomly from the Haar measure on the unitary group, with applications across quantum information processing and physics. We construct new families of quantum circuits on nnn-qubits giving rise to ε\varepsilonε-approximate unitary ttt-designs efficiently in O(n3t12)O(n^3t^{12})O(n3t12) depth. These quantum circuits are based on a relaxation of technical requirements in previous constructions. In particular, the construction of circuits which give efficient approximate ttt-designs by Brandao, Harrow, and Horodecki (F.G.S.L Brandao, A.W Harrow, and M. Horodecki, Commun. Math. Phys. (2016).) required choosing gates from ensembles which contained inverses for all elements, and that the entries of the unitaries are algebraic. We reduce these requirements, to sets that contain elements without inverses in the set, and non-algebraic entries, which we dub partially invertible universal sets. We then adapt this circuit construction to the framework of measurement based quantum computation(MBQC) and give new explicit examples of nnn-qubit graph states with fixed assignments of measurements (graph gadgets) giving rise to unitary ttt-designs based on partially invertible universal sets, in a natural way. We further show that these graph gadgets demonstrate a quantum speedup, up to standard complexity theoretic conjectures. We provide numerical and analytical evidence that almost any assignment of fixed measurement angles on an nnn-qubit cluster state give efficient ttt-designs and demonstrate a quantum speedup.
Random unitaries are useful in quantum information and related fields but hard to generate with limited resources. An approximate unitary k-design is a measure over an ensemble of unitaries such that the average is close to a Haar (uniformly) random ensemble up to the first k moments. A strong notion of approximation bounds the distance from Haar randomness in relative error: the weighted twirl induced by an approximate design can be written as a convex combination involving that of an exact design and vice versa. The main focus of our work is on efficient constructions of approximate designs, in particular whether relative-error designs in sublinear depth are possible. We give a positive answer to this question as part of our main results: 1. Twirl-Swap-Twirl: Let A and B be systems of the same size. Consider a protocol that locally applies k-design unitaries to A^k and B^k respectively, then exchanges l qudits between each copy of A and B respectively, then again applies local k-design unitaries. This protocol yields an ε-approximate relative k-design when l = O(k log k + log(1/ε)). In particular, this bound is independent of the size of A and B as long as it is sufficiently large compared to k and 1/ε. 2. Twirl-Crosstwirl: Let A_1, … , A_P be subsystems of a multipartite system A. Consider the following protocol for k copies of A: (1) locally apply a k-design unitary to each A_p for p = 1, … , P; (2) apply a "crosstwirl" k-design unitary across a joint system combining l qudits from each A_p. Assuming each A_p’s dimension is sufficiently large compared to other parameters, one can choose l to be of the form 2 (Pk + 1) log_q k + log_q P + log_q(1/ε) + O(1) to achieve an ε-approximate relative k-design. As an intermediate step, we show that this protocol achieves a k-tensor-product-expander, in which the approximation error is in 2 → 2 norm, using communication logarithmic in k. 3. Recursive Crosstwirl: Consider an m-qudit system with connectivity given by a lattice in spatial dimension D. For every D = 1, 2, …, we give a construction of an ε-approximate relative k-design using unitaries of spatially local circuit depth O ((log m + log(1/ε) + k log k ) k polylog(k)). Moreover, across the boundaries of spatially contiguous sub-regions, unitaries used in the design ensemble require only area law communication up to corrections logarithmic in m. Hence they generate only that much entanglement on any product state input. These constructions use the alternating projection method to analyze overlapping Haar twirls, giving a bound on the convergence speed to the full twirl with respect to the 2-norm. Using von Neumann subalgebra indices to replace system dimension, the 2-norm distance converts to relative error without introducing system size. The Recursive Crosstwirl construction answers one variant of [Harrow and Mehraban, 2023, Open Problem 1], showing that with a specific, layered architecture, random circuits produce relative error k-designs in sublinear depth. Moreover, it addresses [Harrow and Mehraban, 2023, Open Problem 7], showing that structured circuits in spatial dimension D of depth << m^{1/D} may achieve approximate k-designs.
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.
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.