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arXiv:2408.14463

Unitary Designs from Random Symmetric Quantum Circuits

Liu, Hanqing, Austin Hulse, Iman Marvian

Published Aug 26, 2024• 0 citations
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Abstract

In this work, we study distributions of unitaries generated by random quantum circuits containing only symmetry-respecting gates. We develop a unified approach applicable to all symmetry groups and obtain an equation that determines the exact design properties of such distributions. It has been recently shown that the locality of gates imposes various constraints on realizable unitaries, which in general, significantly depend on the symmetry under consideration. These constraints typically include restrictions on the relative phases between sectors with inequivalent irreducible representations of the symmetry. We call a set of symmetric gates semi-universal if they realize all unitaries that respect the symmetry, up to such restrictions. For instance, while 2-qubit gates are semi-universal for $\mathbb{Z}_2$, U(1), and SU(2) symmetries in qubit systems, SU(d) symmetry with $d\ge 3$ requires 3-qudit gates for semi-universality. Failure of semi-universality precludes the distribution generated by the random circuits from being even a 2-design for the Haar distribution over symmetry-respecting unitaries. On the other hand, when semi-universality holds, under mild conditions, satisfied by U(1) and SU(2) for example, the distribution becomes a $t$-design for $t$ growing polynomially with the number of qudits, where the degree is determined by the locality of gates. More generally, we present a simple linear equation that determines the maximum integer $t_{\max}$ for which the uniform distribution of unitaries generated by the circuits is a $t$-design for all $t\leq t_{\max}$. Notably, for U(1), SU(2) and cyclic groups, we determine the exact value of $t_{\max}$ as a function of the number of qubits and locality of the gates, and for SU(d), we determine the exact value of $t_{\max}$ for up to $4$-qudit gates.

DOI
10.48550/arxiv.2408.14463

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In this work, we study distributions of unitaries generated by random quantum circuits containing only symmetry-respecting gates. We develop a unified approach applicable to all symmetry groups and obtain an equation that determines the exact design properties of such distributions. It has been recently shown that the locality of gates imposes various constraints on realizable unitaries, which in general, significantly depend on the symmetry under consideration. These constraints typically include restrictions on the relative phases between sectors with inequivalent irreducible representations of the symmetry. We call a set of symmetric gates semi-universal if they realize all unitaries that respect the symmetry, up to such restrictions. For instance, while 2-qubit gates are semi-universal for Z2\mathbb{Z}_2Z2​, U(1), and SU(2) symmetries in qubit systems, SU(d) symmetry with d≥3d\ge 3d≥3 requires 3-qudit gates for semi-universality. Failure of semi-universality precludes the distribution generated by the random circuits from being even a 2-design for the Haar distribution over symmetry-respecting unitaries. On the other hand, when semi-universality holds, under mild conditions, satisfied by U(1) and SU(2) for example, the distribution becomes a ttt-design for ttt growing polynomially with the number of qudits, where the degree is determined by the locality of gates. More generally, we present a simple linear equation that determines the maximum integer tmax⁡t_{\max}tmax​ for which the uniform distribution of unitaries generated by the circuits is a ttt-design for all t≤tmax⁡t\leq t_{\max}t≤tmax​. Notably, for U(1), SU(2) and cyclic groups, we determine the exact value of tmax⁡t_{\max}tmax​ as a function of the number of qubits and locality of the gates, and for SU(d), we determine the exact value of tmax⁡t_{\max}tmax​ for up to 444-qudit gates.
Local symmetric quantum circuits are those that contain only local unitaries that respect a certain symmetry. These circuits provide a simple framework to study the dynamics and phases of complex quantum systems with conserved charges; however, some of their basic properties have not yet been understood. Recently, it has been shown that such quantum circuits only generate a restricted subset of symmetric unitary transformations. In this paper, we consider circuits with 2-local SU(d)-invariant unitaries acting on qudits, i.e., d-dimensional quantum systems. Our results reveal a significant distinction between the cases of d=2 and d>=3. For qubits with SU(2) symmetry, arbitrary global rotationally-invariant unitaries can be generated with 2-local ones, up to relative phases between the subspaces corresponding to inequivalent irreducible representations (irreps) of the symmetry, i.e., sectors with different angular momenta. On the other hand, for d>=3, in addition to similar constraints on relative phases between the irreps, locality also restricts the generated unitaries inside these conserved subspaces. In particular, for certain irreps of SU(d), the dynamics under 2-local SU(d)-invariant unitaries can be mapped to the dynamics of a non-interacting (free) fermionic system, whereas for general 3-local ones, the corresponding fermionic model is interacting. Using this correspondence, we obtain new conservation laws for dynamics under 2-local SU(d)-invariant unitaries. Furthermore, we identify a Z2 symmetry related to the parity of permutations which imposes additional conservation laws for systems with n =3. Our results imply that the distribution of unitaries generated by random 2-local SU(d)-invariant unitaries does not converge to the Haar measure over the group of all SU(d)-invariant unitaries, and in fact, for d>=3, is not even a 2-design for the Haar distribution.
Quantum circuits with symmetry-respecting gates have attracted broad interest in quantum information science. While recent work has developed a theory for circuits with Abelian symmetries, revealing important distinctions between Abelian and non-Abelian cases, a comprehensive framework for non-Abelian symmetries has been lacking. In this work, we develop novel techniques and a powerful framework that is particularly useful for understanding circuits with non-Abelian symmetries. Using this framework we settle an open question on quantum circuits with SU(d) symmetry. We show that 3-qudit SU(d)-invariant gates are semi-universal, i.e., generate all SU(d)-invariant unitaries, up to certain constraints on the relative phases between sectors with inequivalent representation of symmetry. Furthermore, we prove that these gates achieve full universality when supplemented with 3 ancilla qudits. Interestingly, we find that studying circuits with 3-qudit gates is also useful for a better understanding of circuits with 2-qudit gates. In particular, we establish that even though 2-qudit SU(d)-invariant gates are not themselves semi-universal, they become universal with at most 11 ancilla qudits. Additionally, we investigate the statistical properties of circuits composed of random SU(d)-invariant gates. Our findings reveal that while circuits with 2-qudit gates do not form a 2-design for the Haar measure over SU(d)-invariant unitaries, circuits with 3-qudit gates generate a t-design, with t that is quadratic in the number of qudits.
We have established the method of characterizing the unitary design generated by a symmetric local random circuit. Concretely, we have shown that the necessary and sufficient condition for the circuit asymptotically forming a t-design is given by simple integer optimization for general symmetry and locality. By using the result, we explicitly give the maximal order of unitary design under the Z_{2}, U(1), and SU(2) symmetries for general locality. This work reveals the relation between the fundamental notions of symmetry and locality in terms of randomness.
We define the unitary design strength of a finite unitary ensemble to be the maximal integer ttt for which its ttt-th statistical moment is identical to that of the unitary group. In the presence of physical symmetry, not all operators of an ensemble are symmetry-compatible. This motivates us to define the symmetric design strength of a finite unitary ensemble to be the maximal integer ttt for which the ttt-th moments of its symmetry-compatible sub-ensemble match those of the group of symmetry-compatible unitaries. We show that in sufficiently high dimension and for a broad family of symmetries, which includes the global on-site U(1)\mathrm{U}(1)U(1) and SU(d)\mathrm{SU}(d)SU(d) symmetries, there exist finite unitary ensembles of arbitrarily high design strength whose symmetric design strength is strictly higher. Ensembles with this property may be more versatile under symmetry constraints for protocols that rely on unitary randomization than they are without symmetry. Our results extend the previous work [Mitsuhashi and Yoshioka, PRX Quantum 4.4 (Nov. 2023)] showing that the symmetric design strength of the Clifford group is strictly upper bounded by its unitary design strength for symmetries that do not trivialize the symmetry-compatible unitaries. Due to the special significance of unitary designs with a group structure, we also derive a variety of bounds on the unitary and symmetric design strengths of finite unitary ensembles whose operators form a group. In particular, we provide a simple proof that an arbitrary ensemble of operators forming a finite group must have a symmetric design strength of at most two under the global on-site U(1)\mathrm{U}(1)U(1) and SU(d)\mathrm{SU}(d)SU(d) symmetries. Along the way, we show that there exist uniformly weighted groups of arbitrarily high symmetric design strength in arbitrary dimension and analyze their structure.
In the accompanying paper [Y. Mitsuhashi, et al., Phys. Rev. Lett. 134, 180404 (2025)], we established the method of characterizing the maximal order of asymptotic unitary designs generated by symmetric local random circuits, and explicitly specified the order in the cases of Z2,U(1){\mathbb{Z}}_{2}, \mathrm{U}(1)Z2​,U(1), and SU(2)\mathrm{SU}(2)SU(2) symmetries. Here, we provide full details on the derivation of the main theorems for general symmetry and for concrete symmetries. Furthermore, we consider a general framework where we have access to a finite set of connected compact unitary subgroups, which includes symmetric local unitary gate sets.
Local symmetric quantum circuits provide a simple framework to study the dynamics and phases of complex quantum systems with conserved charges. However, some of their basic properties have not yet been understood. Recently, it has been shown that such quantum circuits only generate a restricted subset of symmetric unitary transformations [I. Marvian, Nature Physics, 2022]. In this paper, we consider circuits with 2-local SU(d)-invariant unitaries acting on qudits, i.e., d-dimensional quantum systems. Our results reveal a significant distinction between the cases of d = 2 and d>2. For qubits with SU(2) symmetry, arbitrary global rotationally-invariant unitaries can be generated with 2-local ones, up to relative phases between the subspaces corresponding to inequivalent irreducible representations (irreps) of the symmetry, i.e., sectors with different angular momenta. On the other hand, for d>2, in addition to similar constraints on the relative phases between the irreps, locality also restricts the generated unitaries inside these conserved subspaces. These constraints impose conservation laws that hold for dynamics under 2-local SU(d)-invariant unitaries, but are violated under general SU(d)-invariant unitaries. Based on this result, we show that the distribution of unitaries generated by random 2-local SU(d)-invariant unitaries does not converge to the Haar measure over the group of all SU(d)-invariant unitaries, and in fact, for d>2, is not even a 2-design for the Haar distribution.