Operator relaxation and the optimal depth of classical shadows

Abstract

Classical shadows are a powerful method for learning many properties of quantum states in a sample-efficient manner, by making use of randomized measurements. Here we study the sample complexity of learning the expectation value of Pauli operators via ``shallow shadows'', a recently-proposed version of classical shadows in which the randomization step is effected by a local unitary circuit of variable depth $t$. We show that the shadow norm (the quantity controlling the sample complexity) is expressed in terms of properties of the Heisenberg time evolution of operators under the randomizing (``twirling'') circuit -- namely the evolution of the weight distribution characterizing the number of sites on which an operator acts nontrivially. For spatially-contiguous Pauli operators of weight $k$, this entails a competition between two processes: operator spreading (whereby the support of an operator grows over time, increasing its weight) and operator relaxation (whereby the bulk of the operator develops an equilibrium density of identity operators, decreasing its weight). From this simple picture we derive (i) an upper bound on the shadow norm which, for depth $t\sim \log (k)$, guarantees an exponential gain in sample complexity over the $t=0$ protocol in any spatial dimension, and (ii) quantitative results in one dimension within a mean-field approximation, including a universal subleading correction to the optimal depth, found to be in excellent agreement with infinite matrix product state numerical simulations. Our work connects fundamental ideas in quantum many-body dynamics to applications in quantum information science, and paves the way to highly-optimized protocols for learning different properties of quantum states.

arXiv comment(4+eps pages, 4 figures) main text + (11 pages, 5 figures) supplementary material. v2: switched from state-averaged shadow norm to regular shadow norm, added references. v3: added supplemental section S6 on non-contiguous operators. Accepted version

References in corpus (44)

• Quantum Computing in the NISQ era and beyond
• Fault-tolerant quantum computation by anyons
• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• The density-matrix renormalization group in the age of matrix product states
• A bound on chaos
• Predicting Many Properties of a Quantum System from Very Few Measurements
• Strong quantum computational advantage using a superconducting quantum processor
• Suppressing quantum errors by scaling a surface code logical qubit
• Quantum Phases of Matter on a 256-Atom Programmable Quantum Simulator
• Probing Topological Spin Liquids on a Programmable Quantum Simulator
• Operator Spreading in Random Unitary Circuits
• Local stabilizer codes in three dimensions without string logical operators
• Chaos in quantum channels
• Quantum Entanglement Growth Under Random Unitary Dynamics
• Probing entanglement entropy via randomized measurements
• Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws
• Quantum Simulators: Architectures and Opportunities
• Fractons
• Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws
• Measuring the scrambling of quantum information
• Towards the fast scrambling conjecture
• The randomized measurement toolbox
• Fracton Phases of Matter
• Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation
• Information Scrambling in Computationally Complex Quantum Circuits
• Provably efficient machine learning for quantum many-body problems
• Sample-optimal tomography of quantum states
• Onset of Random Matrix Behavior in Scrambling Systems
• Statistical correlations between locally randomized measurements: a toolbox for probing entanglement in many-body quantum states
• Fermionic partial tomography via classical shadows
• Robust shadow estimation
• Classical Shadows With Noise
• Experimental Estimation of Quantum State Properties from Classical Shadows
• An order parameter for symmetry-protected phases in one dimension
• Classical Shadow Tomography with Locally Scrambled Quantum Dynamics
• A Random Unitary Circuit Model for Black Hole Evaporation
• Operator Dynamics in Brownian Quantum Circuit
• Standard Form of Qudit Stabilizer Groups
• Shadow process tomography of quantum channels
• Sorting topological stabilizer models in three dimensions
• Measuring Arbitrary Physical Properties in Analog Quantum Simulation
• Markovian Entanglement Dynamics under Locally Scrambled Quantum Evolution
• Scrambling and quantum chaos indicators from long-time properties of operator distributions
• On Classical and Hybrid Shadows of Quantum States

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