Unitary T-designs Research Articles

Approximate t -Designs in Generic Circuit Architectures

Unitary t-designs are distributions on the unitary group whose first t moments appear maximally random. Previous work has established several upper bounds on the depths at which certain specific random quantum circuit ensembles approximate t-designs. Here we show that these bounds can be extended to any fixed architecture of Haar-random two-site gates. This is accomplished by relating the spectral gaps of such architectures to those of one-dimensional brickwork architectures. Our bound depends on the details of the architecture only via the typical number of layers needed for a block of the circuit to form a connected graph over the sites. When this quantity is bounded, the circuit forms an approximate t-design in at most linear depth. We give numerical evidence for a stronger bound that depends only on the number of connected blocks into which the architecture can be divided. We also give an implicit bound for nondeterministic architectures in terms of properties of the corresponding distribution over fixed architectures. Published by the American Physical Society 2024

Continuous-Variable Quantum State Designs: Theory and Applications

We generalize the notion of quantum state designs to infinite-dimensional spaces. We first prove that, under the definition of continuous-variable (CV) state t-designs from [Blume-Kohout , ], no state designs exist for t≥2. Similarly, we prove that no CV unitary t-designs exist for t≥2. We propose an alternative definition for CV state designs, which we call rigged t-designs, and provide explicit constructions for t=2. As an application of rigged designs, we develop a design-based shadow-tomography protocol for CV states. Using energy-constrained versions of rigged designs, we define an average fidelity for CV quantum channels and relate this fidelity to the CV entanglement fidelity. As an additional result of independent interest, we establish a connection between torus 2-designs and complete sets of mutually unbiased bases. Published by the American Physical Society 2024

Approximate Unitary t-Designs by Short Random Quantum Circuits Using Nearest-Neighbor and Long-Range Gates

We prove that {{,textrm{poly},}}(t) cdot n^{1/D}-depth local random quantum circuits with two qudit nearest-neighbor gates on a D-dimensional lattice with n qudits are approximate t-designs in various measures. These include the “monomial” measure, meaning that the monomials of a random circuit from this family have expectation close to the value that would result from the Haar measure. Previously, the best bound was {{,textrm{poly},}}(t)cdot n due to Brandão–Harrow–Horodecki (Commun Math Phys 346(2):397–434, 2016) for D=1. We also improve the “scrambling” and “decoupling” bounds for spatially local random circuits due to Brown and Fawzi (Scrambling speed of random quantum circuits, 2012). One consequence of our result is that assuming the polynomial hierarchy ({{,mathrm{textsf{PH}},}}) is infinite and that certain counting problems are #{textsf{P}}-hard “on average”, sampling within total variation distance from these circuits is hard for classical computers. Previously, exact sampling from the outputs of even constant-depth quantum circuits was known to be hard for classical computers under these assumptions. However the standard strategy for extending this hardness result to approximate sampling requires the quantum circuits to have a property called “anti-concentration”, meaning roughly that the output has near-maximal entropy. Unitary 2-designs have the desired anti-concentration property. Our result improves the required depth for this level of anti-concentration from linear depth to a sub-linear value, depending on the geometry of the interactions. This is relevant to a recent experiment by the Google Quantum AI group to perform such a sampling task with 53 qubits on a two-dimensional lattice (Arute in Nature 574(7779):505–510, 2019; Boixo et al. in Nate Phys 14(6):595–600, 2018) (and related experiments by USTC), and confirms their conjecture that O(sqrt{n}) depth suffices for anti-concentration. The proof is based on a previous construction of t-designs by Brandão et al. (2016), an analysis of how approximate designs behave under composition, and an extension of the quasi-orthogonality of permutation operators developed by Brandão et al. (2016). Different versions of the approximate design condition correspond to different norms, and part of our contribution is to introduce the norm corresponding to anti-concentration and to establish equivalence between these various norms for low-depth circuits. For random circuits with long-range gates, we use different methods to show that anti-concentration happens at circuit size O(nln ^2 n) corresponding to depth O(ln ^3 n). We also show a lower bound of Omega (n ln n) for the size of such circuit in this case. We also prove that anti-concentration is possible in depth O(ln n ln ln n) (size O(n ln n ln ln n)) using a different model.

Efficient Unitary Designs with a System-Size Independent Number of Non-Clifford Gates

Many quantum information protocols require the implementation of random unitaries. Because it takes exponential resources to produce Haar-random unitaries drawn from the full n-qubit group, one often resorts to t-designs. Unitary t-designs mimic the Haar-measure up to t-th moments. It is known that Clifford operations can implement at most 3-designs. In this work, we quantify the non-Clifford resources required to break this barrier. We find that it suffices to inject O(t^{4}log ^{2}(t)log (1/varepsilon )) many non-Clifford gates into a polynomial-depth random Clifford circuit to obtain an varepsilon -approximate t-design. Strikingly, the number of non-Clifford gates required is independent of the system size – asymptotically, the density of non-Clifford gates is allowed to tend to zero. We also derive novel bounds on the convergence time of random Clifford circuits to the t-th moment of the uniform distribution on the Clifford group. Our proofs exploit a recently developed variant of Schur-Weyl duality for the Clifford group, as well as bounds on restricted spectral gaps of averaging operators.

Random quantum circuits are approximate unitary t-designs in depth O(nt5+o(1))

The applications of random quantum circuits range from quantum computing and quantum many-body systems to the physics of black holes. Many of these applications are related to the generation of quantum pseudorandomness: Random quantum circuits are known to approximate unitary t-designs. Unitary t-designs are probability distributions that mimic Haar randomness up to tth moments. In a seminal paper, Brandão, Harrow and Horodecki prove that random quantum circuits on qubits in a brickwork architecture of depth O(nt10.5) are approximate unitary t-designs. In this work, we revisit this argument, which lower bounds the spectral gap of moment operators for local random quantum circuits by Ω(n−1t−9.5). We improve this lower bound to Ω(n−1t−4−o(1)), where the o(1) term goes to 0 as t→∞. A direct consequence of this scaling is that random quantum circuits generate approximate unitary t-designs in depth O(nt5+o(1)). Our techniques involve Gao's quantum union bound and the unreasonable effectiveness of the Clifford group. As an auxiliary result, we prove fast convergence to the Haar measure for random Clifford unitaries interleaved with Haar random single qubit unitaries.

Explicit construction of exact unitary designs

In this paper, we first provide an inductive construction of designs on compact groups G from those on subgroups K when (G,K) is a Gelfand pair. We then use the construction to provide explicit constructions of unitary t-designs in the unitary group U(d) for all t and d. To the best of our knowledge, the explicit constructions were so far known only for very special cases. Here, explicit construction means that the entries of the unitary matrices are given by the values of elementary functions at the root of given polynomials. We will compare our constructions with previously known ones in the case of unitary 4-designs in U(4). We finally remark that this method can be applied to the orthogonal groups O(d) and thus provides another explicit construction of spherical t-design on the sphere Sd−1 in the d dimensional Euclidean space by the induction on d.

Weak approximate unitary designs and applications to quantum encryption

Unitary t-designs are the bread and butter of quantum information theory and beyond. An important issue in practice is that of efficiently constructing good approximations of such unitary t-designs. Building on results by Aubrun (Comm. Math. Phys. 2009), we prove that sampling dtpoly(t,log⁡d,1/ϵ) unitaries from an exact t-design provides with positive probability an ϵ-approximate t-design, if the error is measured in one-to-one norm. As an application, we give a randomized construction of a quantum encryption scheme that has roughly the same key size and security as the quantum one-time pad, but possesses the additional property of being non-malleable against adversaries without quantum side information.

On Unitary t-Designs from Relaxed Seeds.

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 -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 , yet need not contain unitaries and their inverses nor are in general composed of unitaries whose entries are algebraic; dubbed seed. We then show that this relaxed seed, when used as a basis for our construction, gives rise to an -approximate unitary t-design efficiently, where the depth of our random circuit scales as , 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.

On the explicit constructions of certain unitary t-designs

Unitary t-designs are ‘good’ finite subsets of the unitary group that approximate the whole unitary group well. Unitary t-designs have been applied in randomized benchmarking, tomography, quantum cryptography and many other areas of quantum information science. If a unitary t-design itself is a group then it is called a unitary t-group. Although it is known that unitary t-designs in exist for any t and d, the unitary t-groups do not exist for if , as it is shown by Guralnick–Tiep (Guralnick and Tiep 2005 Represent. Theory 9 138–208) and Bannai–Navarro–Rizo–Tiep (Bannai et al 2018 J. Math. Soc. Japan (accepted)). Explicit constructions of exact unitary t-designs in are not easy in general. In particular, explicit constructions of unitary 4-designs in have been an open problem in quantum information theory. We prove that some exact unitary -designs in the unitary group are constructed from unitary t-groups in that satisfy certain specific conditions. Based on this result, we specifically construct exact unitary 3-designs in from the unitary 2-group in and also unitary 4-designs in from the unitary 3-group in numerically. We also discuss some related problems and a few applications to physics problems.

Estimating average fidelity error based on local random quantum circuits

This paper proposes a general method to estimate the practical average fidelity error using an ϵ-approximate unitary 2t-design. We then build a specific model of local random circuits to construct the ϵ-approximate unitary t-design under a tighter bound on the diamond norm of the corresponding error. For a twirled t-tensor product noisy channel constructed by local random circuits, we extend the estimation of average fidelity using approximate unitary 2t-designs to an -fold noise scenario. Compared to previous randomized benchmarking, the average fidelity estimated by the method of local random circuits requires less experiments but longer circuit lengths for large n quantum system.

Local Random Quantum Circuits are Approximate Polynomial-Designs

We prove that local random quantum circuits acting on n qubits composed of O(t^{10} n^2) many nearest neighbor two-qubit gates form an approximate unitary t-design. Previously it was unknown whether random quantum circuits were a t-design for any t > 3. The proof is based on an interplay of techniques from quantum many-body theory, representation theory, and the theory of Markov chains. In particular we employ a result of Nachtergaele for lower bounding the spectral gap of frustration-free quantum local Hamiltonians; a quasi-orthogonality property of permutation matrices; a result of Oliveira which extends to the unitary group the path-coupling method for bounding the mixing time of random walks; and a result of Bourgain and Gamburd showing that dense subgroups of the special unitary group, composed of elements with algebraic entries, are infty-copy tensor-product expanders. We also consider pseudo-randomness properties of local random quantum circuits of small depth and prove that circuits of depth O(t^{10}n) constitute a quantum t-copy tensor-product expander. The proof also rests on techniques from quantum many-body theory, in particular on the detectability lemma of Aharonov, Arad, Landau, and Vazirani. We give applications of the results to cryptography, equilibration of closed quantum dynamics, and the generation of topological order. In particular we show the following pseudo-randomness property of generic quantum circuits: Almost every circuit U of size O(n^k) on n qubits cannot be distinguished from a Haar uniform unitary by circuits of size O(n^{(k-9)/11}) that are given oracle access to U.

Identification of a reversible quantum gate: assessing the resources

We assess the resources needed to identify a reversible quantum gate among a finite set of alternatives, including in our analysis both deterministic and probabilistic strategies. Among the probabilistic strategies, we consider unambiguous gate discrimination—where errors are not tolerated but inconclusive outcomes are allowed—and we prove that parallel strategies are sufficient to unambiguously identify the unknown gate with minimum number of queries. This result is used to provide upper and lower bounds on the query complexity and on the minimum ancilla dimension. In addition, we introduce the notion of generalized t-designs, which includes unitary t-designs and group representations as special cases. For gates forming a generalized t-design we give an explicit expression for the maximum probability of correct gate identification and we prove that there is no gap between the performances of deterministic strategies and those of probabilistic strategies. Hence, evaluating of the query complexity of perfect deterministic discrimination is reduced to the easier problem of evaluating the query complexity of unambiguous discrimination. Finally, we consider discrimination strategies where the use of ancillas is forbidden, providing upper bounds on the number of additional queries needed to make up for the lack of entanglement with the ancillas.

Convergence Rates for Arbitrary Statistical Moments of Random Quantum Circuits

We consider a class of random quantum circuits where at each step a gate from a universal set is applied to a random pair of qubits, and determine how quickly averages of arbitrary finite-degree polynomials in the matrix elements of the resulting unitary converge to Haar measure averages. This is accomplished by mapping the superoperator that describes t order moments on n qubits to a multilevel SU(4^{t}) Lipkin-Meshkov-Glick Hamiltonian. We show that, for arbitrary fixed t, the ground-state manifold is exactly spanned by factorized eigenstates and, under the assumption that a mean-field ansatz accurately describes the low-lying excitations, the spectral gap scales as 1/n in the thermodynamic limit. Our results imply that random quantum circuits yield an efficient implementation of ϵapproximate unitary tdesigns.

Exact and approximate unitary 2-designs and their application to fidelity estimation

We develop the concept of a unitary t-design as a means of expressing operationally useful subsets of the stochastic properties of the uniform (Haar) measure on the unitary group U(2^n) on n qubits. In particular, sets of unitaries forming 2-designs have wide applicability to quantum information protocols. We devise an O(n)-size in-place circuit construction for an approximate unitary 2-design. We then show that this can be used to construct an efficient protocol for experimentally characterizing the fidelity of a quantum process on n qubits with quantum circuits of size O(n) without requiring any ancilla qubits, thereby improving upon previous approaches.

Unitary designs and codes

A unitary design is a collection of unitary matrices that approximate the entire unitary group, much like a spherical design approximates the entire unit sphere. In this paper, we use irreducible representations of the unitary group to find a general lower bound on the size of a unitary t-design in U(d), for any d and t. We also introduce the notion of a unitary code--a subset of U(d) in which the trace inner product of any pair of matrices is restricted to only a small number of distinct absolute values--and give an upper bound for the size of a code with s inner product values in U(d), for any d and s. These bounds can be strengthened when the particular inner product values that occur in the code or design are known. Finally, we describe some constructions of designs: we give an upper bound on the size of the smallest weighted unitary t-design in U(d), and we catalogue some t-designs that arise from finite groups.