[Paper Review] Unitary k-designs from random number-conserving quantum circuits

Sumner N. Hearth, Michael O. Flynn+2 more|arXiv (Cornell University)|Jun 1, 2023

Mathematical Approximation and Integration4 citations

TL;DR

This paper establishes that random number-conserving quantum circuits generate unitary k-designs on the group of number-conserving unitaries, with convergence depth scaling as $\tau \gtrsim kL^{d+2}$ in $d$ spatial dimensions. Using a $k$-fold replicated statistical mechanics model, it shows that gapless Goldstone modes from spontaneously broken $U(1)$ symmetries control the slow convergence, and proves that finite moments up to $k_c = O(L^d)$ are indistinguishable from the Haar ensemble.

ABSTRACT

Local random circuits scramble efficiently and accordingly have a range of applications in quantum information and quantum dynamics. With a global $U(1)$ charge however, the scrambling ability is reduced; for example, such random circuits do not generate the entire group of number-conserving unitaries. We establish two results using the statistical mechanics of $k$-fold replicated circuits. First, we show that finite moments cannot distinguish the ensemble that local random circuits generate from the Haar ensemble on the entire group of number-conserving unitaries. Specifically, the circuits form a $k_c$-design with $k_c = O(L^d)$ for a system in $d$ spatial dimensions with linear dimension $L$. Second, for $k < k_c$, we derive bounds on the depth $τ$ required for the circuit to converge to an approximate $k$-design. The depth is lower bounded by diffusion $k L^2 \ln(L) \lesssim τ$. In contrast, without number conservation $τ\sim ext{poly}(k) L$. The convergence of the circuit ensemble is controlled by the low-energy properties of a frustration-free quantum statistical model which spontaneously breaks $k$ $U(1)$ symmetries. We conjecture that the associated Goldstone modes set the spectral gap for arbitrary spatial and qudit dimensions, leading to an upper bound $τ\lesssim k L^{d+2}$.

Motivation & Objective

• To determine the depth at which random number-conserving quantum circuits converge to a $k$-design on the group of number-conserving unitaries.
• To understand how $U(1)$ conservation and diffusive charge transport affect the convergence rate of random circuits to approximate unitary designs.
• To analyze the role of low-energy excitations—specifically gapless Goldstone modes—arising from spontaneously broken $k$-fold $U(1)$ symmetries in the replicated circuit model.
• To derive rigorous lower bounds on the convergence depth $\tau$ for $k$-designs using variational and statistical mechanics techniques.
• To establish that finite moments of the circuit ensemble match those of the Haar measure on the full number-conserving unitary group up to $k_c = O(L^d)$

Proposed method

• Construct a $k$-fold replicated circuit model to compute the $k$-th moment operator $\hat{T}_k^{U_t}$, encoding all $(k,k)$-order moments of the circuit unitary.
• Map the problem to a frustration-free quantum statistical model with $k$ copies of the system and $k$ conjugate $*$-replicas, each carrying a conserved $U(1)$ charge.
• Identify the low-energy spectrum of the replicated Hamiltonian, showing that it supports gapless Goldstone modes due to spontaneous $U(1)^k$ symmetry breaking.
• Use variational bounds on the overlap between the circuit ensemble and the Haar measure to derive a lower bound on the convergence depth $\tau$.
• Apply a partition function approach to sum over excited states of the Goldstone modes, showing that higher-occupation and higher-momentum modes are exponentially suppressed.
• Derive the asymptotic bound $\tau \gtrsim kL^{d+2}$ for convergence to an $\varepsilon$-approximate $k$-design, consistent with the contribution of the lowest-energy Goldstone modes alone.

Experimental results

Research questions

• RQ1 What is the minimum depth $\tau$ required for a random number-conserving quantum circuit to form an $\varepsilon$-approximate $k$-design on the group of number-conserving unitaries?
• RQ2 How does $U(1)$ conservation alter the convergence rate of random circuits compared to unconstrained unitary circuits?
• RQ3 To what extent do gapless Goldstone modes from spontaneously broken $U(1)^k$ symmetries govern the convergence dynamics of the circuit ensemble?
• RQ4 Are finite moments of the circuit ensemble indistinguishable from the Haar measure on the full number-conserving unitary group for $k < k_c$?
• RQ5 Does the leading-order contribution to convergence come from the lowest-energy Goldstone modes, or are higher-momentum or higher-occupation modes significant?

Key findings

• The circuit ensemble forms a $k$-design on the group of number-conserving unitaries up to $k_c = O(L^d)$, meaning all finite moments up to this $k$ are indistinguishable from the Haar measure.
• The convergence depth scales as $\tau \gtrsim kL^{d+2}$, which is slower than the $\tau \gtrsim kL^d$ scaling in the absence of number conservation.
• The slow convergence is governed by gapless Goldstone modes arising from spontaneous $U(1)^k$ symmetry breaking in the $k$-fold replicated model.
• The leading contribution to the convergence rate comes from the lowest-energy Goldstone modes; higher-momentum and higher-occupation modes are exponentially suppressed.
• The derived lower bound on $\tau$ matches the asymptotic behavior predicted by the lowest-energy modes alone, confirming their dominance in the convergence dynamics.
• The results are derived using variational bounds and a partition function approach in the replicated statistical model, and are conjectured to be tight for arbitrary spatial and qudit dimensions.