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Cutoff phenomenon and entropic uncertainty for random quantum circuits

Sangchul Oh, Sabre Kais

TL;DR

It is found that random quantum states, as stationary states of random walks on a unitary group, are invariant under the quantum Fourier transform (QFT), thus the entropic uncertainty ofrandom quantum states has balanced Shannon entropies for the computational basis and the QFT basis.

Abstract

How fast a state of a system converges to a stationary state is one of the fundamental questions in science. Some Markov chains and random walks on finite groups are known to exhibit the non-asymptotic convergence to a stationary distribution, called the cutoff phenomenon. Here, we examine how quickly a random quantum circuit could transform a quantum state to a Haar-measure random quantum state. We find that random quantum states, as stationary states of random walks on a unitary group, are invariant under the quantum Fourier transform. Thus the entropic uncertainty of random quantum states has balanced Shannon entropies for the computational bases and the quantum Fourier transform bases. By calculating the Shannon entropy for random quantum states and the Wasserstein distances for the eigenvalues of random quantum circuits, we show that the cutoff phenomenon occurs for the random quantum circuit. It is also demonstrated that the Dyson-Brownian motion for the eigenvalues of a random unitary matrix as a continuous random walk exhibits the cutoff phenomenon. The results here imply that random quantum states could be generated with shallow random circuits.

Cutoff phenomenon and entropic uncertainty for random quantum circuits

TL;DR

It is found that random quantum states, as stationary states of random walks on a unitary group, are invariant under the quantum Fourier transform (QFT), thus the entropic uncertainty ofrandom quantum states has balanced Shannon entropies for the computational basis and the QFT basis.

Abstract

How fast a state of a system converges to a stationary state is one of the fundamental questions in science. Some Markov chains and random walks on finite groups are known to exhibit the non-asymptotic convergence to a stationary distribution, called the cutoff phenomenon. Here, we examine how quickly a random quantum circuit could transform a quantum state to a Haar-measure random quantum state. We find that random quantum states, as stationary states of random walks on a unitary group, are invariant under the quantum Fourier transform. Thus the entropic uncertainty of random quantum states has balanced Shannon entropies for the computational bases and the quantum Fourier transform bases. By calculating the Shannon entropy for random quantum states and the Wasserstein distances for the eigenvalues of random quantum circuits, we show that the cutoff phenomenon occurs for the random quantum circuit. It is also demonstrated that the Dyson-Brownian motion for the eigenvalues of a random unitary matrix as a continuous random walk exhibits the cutoff phenomenon. The results here imply that random quantum states could be generated with shallow random circuits.
Paper Structure (4 sections, 10 equations, 3 figures)

This paper contains 4 sections, 10 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Cutoff Phenomenon for Random Quantum Circuits
  3. Dyson-Brownian Motions on a Unitary Group
  4. Summary

Figures (3)

  • Figure 1: The distributions of $p_i$ for (a) a uniform superposition of all basis states $N=20$, (c) a random quantum state generated by a random unitary matrix ${\rm U}(N)$ with $N=20$, and (e) a random quantum state of $n=14$ qubits generated by a random quantum circuit implemented on the Sycamore processor Arute2019Martinis2022. Plots (b), (d), and (f) are the distributions of $p_k$ after the quantum Fourier transform of (a), (b), and (c), respectively. The red lines indicate $p=1/N$ and the Shannon entropy $H$ for each state is labeled by $H$.
  • Figure 2: For a random quantum circuit for $n=14$ qubits and up to $m=14$ cycles Arute2019, (a) Shannon entropy of the average of quantum states and (b) Wasserstein distance between the eigenvalues of random circuits and the Haar random unitary operator are plotted as a function of the number of gates applied, i.e., the depth of a quantum circuit.
  • Figure 3: For the Dyson-Brownian random walk, (a) trajectories of eigenvalues of a random unitary operator and (b) the Shannon entropy $H(\left| {\psi} \right\rangle)$ of a quantum state are plotted. Here, we take $20\times 20$ random Hamiltonian matrices, i.e., $N = 20$ and the time step $\delta t = 0.01$. In (b) the red dotted horizontal line represents the Shannon entropy of a random quantum state, $\ln N -1 + \gamma \approx 2.5572$ for $N=20$.