SU(d)-Symmetric Random Unitaries: Quantum Scrambling, Error Correction, and Machine Learning

TL;DR

It is shown that SU(d)-symmetric unitaries can be used to construct asymptotically optimal unitaries and derived an overpartameterization threshold via the quantum neural tangent kernel (QNTK) required for exponential convergence guarantee of generic ansatz for geometric quantum machine learning, which reveals that the number of parameters required scales only with the dimension of desired subspaces rather than that of the entire Hilbert space.

Abstract

Quantum information processing in the presence of continuous symmetry is of wide importance and exhibits many novel physical and mathematical phenomena. SU(d) is a continuous group of particular interest since it represents a fundamental type of non-Abelian symmetry and also plays a vital role in quantum computation. Here, we explicate three particularly interesting applications of symmetric random unitaries in diverse contexts ranging from physics to quantum computing: information scrambling with non-Abelian conserved quantities, covariant quantum error correcting random codes, and geometric quantum machine learning. First, we show that, in the presence of SU(d) symmetry, the local conserved quantities would exhibit residual values even at $t \rightarrow \infty$ which decays as $Ω(1/n^{3/2})$ under local Pauli basis for qubits and $Ω(1/n^{(d+2)^2/2})$ under symmetric basis for general qudits with respect to the system size, in contrast to O(1/n) decay for U(1) case and the exponential decay for no-symmetry case in the sense of out-of-time ordered correlator. Second, we show that SU(d)-symmetric unitaries can be used to construct asymptotically optimal (in the sense of saturating the fundamental limits on the code error, or the approximate Eastin--Knill theorems) SU(d)-covariant codes that encode any constant number of logical qudits, extending [Kong & Liu; PRXQ 3, 020314 (2022)]. Finally, we derive an overpartameterization threshold via the quantum neural tangent kernel required for exponential convergence guarantee of generic ansatz for geometric quantum machine learning, which reveals that the number of parameters required scales only with the dimension of desired subspaces rather than the entire Hilbert space. Our work invites further research on quantum information with continuous symmetries, where the mathematical tools developed in this work are expected to be useful.

Figures (5)

• Figure 1: Hayden-Preskill decoupling as a general set-up where $R$ and $A$ form a maximally entangled pair and $\Psi$ is a generic density matrix in $\bar{A}$ which is purified with the memory register $\operatorname{MEM}$. In the case that $\Psi$ itself is a pure state, we omit the MEM register for the case of SU$(d)$-covariant codes.
• Figure 2: The log-log plot Purified distance between the decoupled states between $R$ and $\bar{B}$ and $\rho_{\operatorname{avg}}(R \cup \bar{B})$ with three logical qubits. The residual entanglement displayed is a consequence of the Eastin--Knill theorems and imperfect information recovery due to conservation law faist20Tajima2021Tajima2022Nakata2023
• Figure 3: Tensor network diagram illustrating the deduction from Eq.\ref{['eq:Tr-rho-20']} to \ref{['eq:Tr-rho-2']}.
• Figure 4: Tensor network diagram illustrating the deduction from Eq.\ref{['eq:rho_avg^2_1']} to \ref{['eq:rho_avg^2_3']}.
• Figure 5: Tensor network diagram illustrating how to obtain Eq.\ref{['eq:Code1']} when $\rho_{\bar{A}}$ is initialized as a pure state represented as dots connected by dashed line in the diagram.

Theorems & Definitions (25)

• Definition A.1
• Definition A.2
• Definition A.3
• Definition A.4
• Proposition A.5
• Theorem A.6
• Proposition A.7
• Definition A.8
• Lemma A.9
• proof