Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Annual Symposium on Foundations of Computer Science

Explicit orthogonal and unitary designs

Ryan O’Donnell, Rocco A. Servedio, Pedro Paredes

Published Nov 6, 2023• 8 citations
Publisher

Abstract

We give a strongly explicit construction of ϵ approximate k-designs for the orthogonal group O(N) and the unitary group U(N), for $N=2^{n}$. Our designs are of cardinality $\operatorname{poly}(N^{k}/\epsilon)$ (equivalently, they have seed length $O(nk+\log(1/\epsilon)))$; up to the polynomial, this matches the number of design elements used by the construction consisting of completely random matrices.

DOI
10.1109/focs57990.2023.00073

Related papers

Explicit orthogonal and unitary designs

Ryan O’Donnell, Rocco A. Servedio, Pedro Paredes

We give a strongly explicit construction of $\varepsilon$-approximate $k$-designs for the orthogonal group $\mathrm{O}(N)$ and the unitary group $\mathrm{U}(N)$, for $N=2^n$. Our designs are of cardinality $\mathrm{poly}(N^k/\varepsilon)$ (equivalently, they have seed length $O(nk + \log(1/\varepsilon)))$; up to the polynomial, this matches the number of design elements used by the construction consisting of completely random matrices.

arXiv:2310.13597•Oct 20, 2023•0 citations

On Unitary t-Designs from Relaxed Seeds

Rawad Mezher, Joe Ghalbouni, Joseph Dgheim +1 more

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.

arXiv:1911.03704•Entropy•Jan 12, 2020•6 citations

Efficient unitary designs and pseudorandom unitaries from permutations

Chi-Fang Chen, Adam Bouland, Fernando G. S. L. Brandão +3 more

In this work we give an efficient construction of unitary $k$-designs using $\tilde{O}(k\cdot poly(n))$ 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 $S(N)$ to random unitaries over $U(N)$ for $N=2^n$. In particular, we show that products of exponentiated sums of $S(N)$ permutations with random phases approximately match the first $2^{Ω(n)}$ moments of the Haar measure. By substituting either $\tilde{O}(k)$-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-$N$) expansion in random matrix theory and the polynomial method, which allows us to prove query lower bounds at finite-$N$ by interpolating from the much simpler large-$N$ limit. The key technical 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 $N$.

arXiv:2404.16751•Apr 25, 2024•4 citations

On the explicit constructions of certain unitary t -designs

Eiichi Bannai, Mikio Nakahara, Zhao Da +1 more

Abstract 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.

arXiv:1906.04583•Journal of Physics A Mathematical and Theoretical•Oct 22, 2019•12 citations

Arts & crafts: Strong random unitaries and geometric locality

Marten Folkertsma, Lorenzo Grevink, Jonas Helsen +1 more

We study the problem of constructing strong approximate unitary $k$-designs on $D$-dimensional grids (and more generally on Cartesian products of graphs), building on the work of Schuster et al. arXiv:2509.26310 which establishes strong unitary designs in 1D and in all-to-all connectivity. We provide two constructions. The first construction leverages the existing all-to-all connectivity result with general routing theory to provide flexible (but slightly suboptimal) strong $k$-designs in arbitrary connectivities. The second construction is more direct, requires no auxiliaries and has provably optimal depth (in the number of qubits $n$) for $D$-dimensional grids with constant dimension. Combining these techniques also allows us to construct strong pseudorandom unitaries on $D$-dimensional grids with provably optimal depth.

quant-pharXiv:2605.03023•May 6, 2026•0 citations

Efficient Unitary Designs from Random Sums and Permutations

Chi-Fang Chen, Jordan Docter, Michelle Xu +3 more

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$N$.

Annual Symposium on Foundations of Computer Science•Oct 27, 2024•13 citations
We give a strongly explicit construction of ϵ approximate k-designs for the orthogonal group O(N) and the unitary group U(N), for N=2nN=2^{n}N=2n. Our designs are of cardinality poly⁡(Nk/ϵ)\operatorname{poly}(N^{k}/\epsilon)poly(Nk/ϵ) (equivalently, they have seed length O(nk+log⁡(1/ϵ)))O(nk+\log(1/\epsilon)))O(nk+log(1/ϵ))); up to the polynomial, this matches the number of design elements used by the construction consisting of completely random matrices.
We give a strongly explicit construction of ε\varepsilonε-approximate kkk-designs for the orthogonal group O(N)\mathrm{O}(N)O(N) and the unitary group U(N)\mathrm{U}(N)U(N), for N=2nN=2^nN=2n. Our designs are of cardinality poly(Nk/ε)\mathrm{poly}(N^k/\varepsilon)poly(Nk/ε) (equivalently, they have seed length O(nk+log⁡(1/ε)))O(nk + \log(1/\varepsilon)))O(nk+log(1/ε))); up to the polynomial, this matches the number of design elements used by the construction consisting of completely random matrices.
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.
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.
Abstract 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.
We study the problem of constructing strong approximate unitary kkk-designs on DDD-dimensional grids (and more generally on Cartesian products of graphs), building on the work of Schuster et al. arXiv:2509.26310 which establishes strong unitary designs in 1D and in all-to-all connectivity. We provide two constructions. The first construction leverages the existing all-to-all connectivity result with general routing theory to provide flexible (but slightly suboptimal) strong kkk-designs in arbitrary connectivities. The second construction is more direct, requires no auxiliaries and has provably optimal depth (in the number of qubits nnn) for DDD-dimensional grids with constant dimension. Combining these techniques also allows us to construct strong pseudorandom unitaries on DDD-dimensional grids with provably optimal depth.
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.