Quantum State Designs from Minimally Random Quantum Circuits

TL;DR

This work analyzes quantum-state-design generation using minimally random brickwork circuits in two settings: boundary-only randomness and fully-random single-qudit gates with fixed two-qudit interactions. By tracking the $k$-moment density matrices and frame potentials, the authors prove that almost all choices of two-site gates yield Haar-equivalent $k$-designs at large times, while the approach exhibits a characteristic two-step relaxation absent in fully random circuits. Dual-unitary gates, especially those with high entangling power or perfect-tensor properties, accelerate design generation, achieving Haar-random-like distributions more rapidly than Haar circuits for many $k$, with the fastest scaling $ au_k^{(\epsilon)} \simeq \frac{k}{2}L - \frac{\ln \epsilon}{4\ln d}$ in the DU/perfect-tensor limit. The findings reveal a two-tier universality in minimally random dynamics and point to optimized gate choices as a practical route to efficient quantum-state design and tomography in near-term devices.

Abstract

Random many-body states are both a useful tool to model certain physical systems and an important asset for quantum computation. Realising them, however, generally requires an exponential (in system size) amount of resources. Recent research has presented a way out by showing that one can generate random states, or more precisely a controlled approximation of them, by applying a quantum circuit built in terms of few-body unitary gates. Most of this research, however, has been focussed on the case of quantum circuits composed by completely random unitary gates. Here we consider what happens for circuits that, instead, involve a minimal degree of randomness. Specifically, we concentrate on two different settings: (a) brickwork quantum circuits with a single one-qudit random matrix at a boundary; (b) brickwork quantum circuits with fixed interactions but random one-qudit gates everywhere. We show that, for any given initial state, (a) and (b) produce a distribution of states approaching the Haar distribution in the limit of large circuit depth. More precisely, we show that the moments of the distribution produced by our circuits can approximate the ones of the Haar distribution in a depth proportional to the system size. Interestingly we find that in both Cases (a) and (b) the relaxation to the Haar distribution occurs in two steps - this is in contrast with what happens in fully random circuits. Moreover, we show that choosing appropriately the fixed interactions, for example taking the local gate to be a dual-unitary gate with high enough entangling power, minimally random circuits produce a Haar random distribution more rapidly than fully random circuits. In particular, dual-unitary circuits with maximal entangling power - i.e. perfect tensors - appear to provide the optimal quantum state design preparation for any design number.

Figures (11)

• Figure 1: Calculating $\Delta_2^{(k)}(t)$ with sampled averaging for $L = 4$ (8 qubits) for various $k$. Here we take random Pauli's on the boundary and evaluate the expression in Eq. \ref{['eq:framepot']} for all configurations for $t\leq 9$. For $t>9$ we take $10^{9}$ samples. We set $\delta = 0$ in the top figure and $\delta = 0.1$ in the bottom figure.
• Figure 2: Calculating $\Delta_2^{(1)}(t)$ with explicit averaging for $L = 8$ (16 qubits) for various $\delta$. Curve captures the behaviour for both random Pauli and Haar averaging on the boundary.
• Figure 3: $r_1^{(1)}$ for various $\delta$ extracted from the $L = 8$ (16 qubits) data presented in Fig. \ref{['fig:explicitk1']}.
• Figure 4: Calculating $\Delta_2^{(2)}(t)$ with explicit averaging for various $\delta$. All data is from system size $L = 4$ (8 qubits). (Top) Boundary randomness is random Pauli matrices [Case \ref{['CaseA1']}]. (Bottom) Boundary randomness is with one site Haar random unitaries [Case \ref{['CaseA2']}].
• Figure 5: $r_1^{(2)},r_2^{(2)}$ for various $\delta$ from the $L = 4$ data presented in Fig. \ref{['fig:explicitk2']}. The full Haar value is calculated for $d=2$ in Eq. \ref{['eq:haardecay']}.