Computing exact moments of local random quantum circuits via tensor networks

Abstract

A basic primitive in quantum information is the computation of the moments ${\mathbb{E}}_U[\mathrm{Tr}[U\rho U^{†}O{]}^t]$. These describe the distribution of expectation values obtained by sending a state $\rho$ through a random unitary $U$, sampled from some distribution, and measuring the observable $O$. While the exact calculation of these moments is generally hard, if $U$ is composed of local random gates, one can estimate ${\mathbb{E}}_U[\mathrm{Tr}[U\rho U^{†}O{]}^t]$ by performing Monte Carlo simulations of a Markov chain-like process. However, this approach can require a prohibitively large number of samples, or suffer from the sign problem. In this work, we instead propose to estimate the moments via tensor networks, where the local gates moment operators are mapped to small dimensional tensors acting on their local commutant bases. By leveraging representation theoretical tools, we study the local tensor dimension and we provide bounds for the bond dimension of the matrix product states arising from deep circuits. We compare our techniques against Monte Carlo simulations, showing that we can significantly out-perform them. Then, we showcase how tensor networks can exactly compute the second moment when $U$ is a quantum neural network acting on thousands of qubits and having thousands of gates. To finish, we numerically study the anticoncentration phenomena of circuits with orthogonal random gates, a task which cannot be studied via Monte Carlo due to sign problems.

arXiv comment15 + 8 pages, 9 figures, updated to published version

References in corpus (42)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets
• Barren plateaus in quantum neural network training landscapes
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Quantum Convolutional Neural Networks
• Black holes as mirrors: quantum information in random subsystems
• Characterizing Quantum Supremacy in Near-Term Devices
• Strong quantum computational advantage using a superconducting quantum processor
• Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
• Fast Scramblers
• The ITensor Software Library for Tensor Network Calculations
• Chaos in quantum channels
• Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws
• Theory of the phase transition in random unitary circuits with measurements
• Measurement-induced criticality in random quantum circuits
• Random Quantum Circuits
• Holographic duality from random tensor networks
• Towards the fast scrambling conjecture
• Random Quantum Circuits are Approximate 2-designs
• Local random quantum circuits are approximate polynomial-designs
• Information Scrambling in Computationally Complex Quantum Circuits
• Provably efficient machine learning for quantum many-body problems
• Beyond Barren Plateaus: Quantum Variational Algorithms Are Swamped With Traps
• Diagnosing Barren Plateaus with Tools from Quantum Optimal Control
• Measurement and entanglement phase transitions in all-to-all quantum circuits, on quantum trees, and in Landau-Ginsburg theory
• Matchgates and classical simulation of quantum circuits
• A Lie Algebraic Theory of Barren Plateaus for Deep Parameterized Quantum Circuits
• Introduction to Haar Measure Tools in Quantum Information: A Beginner's Tutorial
• Approximate unitary $t$-designs by short random quantum circuits using nearest-neighbor and long-range gates
• Universal Quantum Information Compression
• Random quantum circuits are approximate unitary $t$-designs in depth $O\left(n{t}^{5+o(1)}\right)$
• Symmetry Principles in Quantum Systems Theory
• Fixed Depth Hamiltonian Simulation via Cartan Decomposition
• Anticoncentration theorems for schemes showing a quantum speedup
• Algebraic Compression of Quantum Circuits for Hamiltonian Evolution
• Fermion Sampling: a robust quantum computational advantage scheme using fermionic linear optics and magic input states
• A Dynamic Systems Approach to Fermions and Their Relation to Spins
• Tight and Efficient Gradient Bounds for Parameterized Quantum Circuits
• Sign Problem in Quantum Monte Carlo Simulation
• RTNI - A symbolic integrator for Haar-random tensor networks
• Efficient unitary designs and pseudorandom unitaries from permutations
• Efficient Unitary T-designs from Random Sums

Cited by in corpus (13)

• Barren Plateaus in Variational Quantum Computing
• Does provable absence of barren plateaus imply classical simulability?
• Magic spreading in random quantum circuits
• Variational quantum simulation: a case study for understanding warm starts
• Classically estimating observables of noiseless quantum circuits
• Quantum Convolutional Neural Networks are Effectively Classically Simulable
• Fermionic Magic Resources of Quantum Many-Body Systems
• Exact spectral gaps of random one-dimensional quantum circuits
• Architectures and random properties of symplectic quantum circuits
• Anticoncentration in Clifford Circuits and Beyond: From Random Tensor Networks to Pseudo-Magic States
• Experimental measurement and a physical interpretation of quantum shadow enumerators
• Detecting entanglement from few partial transpose moments and their decay via weight enumerators
• Moments of Quantum Channel Ensembles