We have generalized the well-known statement that the Clifford group is a unitary 3-design into symmetric cases by extending the notion of unitary design.Concretely, we have proven that a symmetric Clifford group is a symmetric unitary 3-design if and only if the symmetry constraint is described by some Pauli subgroup.We have also found a complete and unique construction method of symmetric Clifford groups with simple quantum gates for Pauli symmetries.For the overall understanding, we have also considered physically relevant U(1) and SU(2) symmetry constraints, which cannot be described by a Pauli subgroup, and have proven that the symmetric Clifford group is a symmetric unitary 1-design but not a 2-design under those symmetries.Our findings are numerically verified by computing the frame potentials, which measure the difference in randomness between the uniform ensemble on the symmetric group of interest and the symmetric unitary group.This work will open a new perspective into quantum information processing, such as randomized benchmarking, and give a deep understanding to many-body systems, such as monitored random circuits.We have generalized the well-known statement that the Clifford group is a unitary 3-design into symmetric cases by extending the notion of unitary design.Concretely, we have proven that a symmetric Clifford group is a symmetric unitary 3-design if and only if the symmetry constraint is described by some Pauli subgroup.We have also found a complete and unique construction method of symmetric Clifford groups with simple quantum gates for Pauli symmetries.For the overall understanding, we have also considered physically relevant U(1) and SU(2) symmetry constraints, which cannot be described by a Pauli subgroup, and have proven that the symmetric Clifford group is a symmetric unitary 1-design but not a 2-design under those symmetries.Our findings are numerically verified by computing the frame potentials, which measure the difference in randomness between the uniform ensemble on the symmetric group of interest and the symmetric unitary group.This work will open a new perspective into quantum information processing, such as randomized benchmarking, and give a deep understanding to many-body systems, such as monitored random circuits.

arXiv:2306.17559PRX QuantumNov 27, 202312 citations

Explicit orthogonal and unitary designs

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

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.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.

Annual Symposium on Foundations of Computer ScienceNov 6, 20238 citations

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.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:2310.13597Oct 20, 20230 citations

Generalized group designs: constructing novel unitary 2-, 3- and 4-designs

Ágoston Kaposi, Zoltán Kolarovszki, Adrian Solymos +1 more

Unitary designs are essential tools in several quantum information protocols. Similarly to other design concepts, unitary designs are mainly used to facilitate averaging over a relevant space, in this case, the unitary group U ( d ) . While it is known that exact unitary t -designs exist for any degree t and dimension d , the most appealing type of designs, group designs (in which the elements of the design form a group), can provide at most 3 -designs. Moreover, even group 2 -designs can exist only in limited dimensions. In this paper, we present novel construction methods for creating exact generalized group designs based on the representation theory of the unitary group and its finite subgroups that overcome the 4 -design-barrier of unitary group designs. Furthermore, a construction is presented for creating generalized group 2 -designs in arbitrary dimensions.Unitary designs are essential tools in several quantum information protocols. Similarly to other design concepts, unitary designs are mainly used to facilitate averaging over a relevant space, in this case, the unitary group U ( d ) . While it is known that exact unitary t -designs exist for any degree t and dimension d , the most appealing type of designs, group designs (in which the elements of the design form a group), can provide at most 3 -designs. Moreover, even group 2 -designs can exist only in limited dimensions. In this paper, we present novel construction methods for creating exact generalized group designs based on the representation theory of the unitary group and its finite subgroups that overcome the 4 -design-barrier of unitary group designs. Furthermore, a construction is presented for creating generalized group 2 -designs in arbitrary dimensions.

arXiv:2405.00919QuantumFeb 23, 20261 citation

The Clifford group forms a unitary 3-design

Zak Webb

Unitary k-designs are finite ensembles of unitary matrices that approximate the Haar distribution over unitary matrices. Several ensembles are known to be 2-designs, including the uniform distribution over the Clifford group, but no family of ensembles was previously known to form a 3-design. We prove that the Clifford group is a 3-design, showing that it is a better approximation to Haar-random unitaries than previously expected. Our proof strategy works for any distribution of unitaries satisfying a property we call Pauli 2-mixing and proceeds without the use of heavy mathematical machinery. We also show that the Clifford group does not form a 4-design, thus characterizing how well random Clifford elements approximate Haar-random unitaries. Additionally, we show that the generalized Clifford group for qudits is not a 3-design unless the dimension of the qudit is a power of 2.Unitary k-designs are finite ensembles of unitary matrices that approximate the Haar distribution over unitary matrices. Several ensembles are known to be 2-designs, including the uniform distribution over the Clifford group, but no family of ensembles was previously known to form a 3-design. We prove that the Clifford group is a 3-design, showing that it is a better approximation to Haar-random unitaries than previously expected. Our proof strategy works for any distribution of unitaries satisfying a property we call Pauli 2-mixing and proceeds without the use of heavy mathematical machinery. We also show that the Clifford group does not form a 4-design, thus characterizing how well random Clifford elements approximate Haar-random unitaries. Additionally, we show that the generalized Clifford group for qudits is not a 3-design unless the dimension of the qudit is a power of 2.

arXiv:1510.02769Quantum Information and ComputationNov 1, 2016159 citations

Third Moments of Qudit Clifford Orbits and 3-Designs Based on Magic Orbits

Huangjun Zhu, Chengsi Mao, Changhao Yi

arXiv:2410.13575Communications in Mathematical PhysicsSep 27, 20260 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=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.

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.