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Convergence rates for arbitrary statistical moments of random quantum circuits
Winton G Brown1, Lorenza Viola
1Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, New Hampshire 03755, USA.
Physical Review Letters
|September 28, 2010
Summary
Random quantum circuits efficiently create approximate unitary designs by converging quickly to Haar measure averages. This convergence is demonstrated through a novel mapping to a Lipkin-Meshkov-Glick Hamiltonian, showing spectral gap scaling with qubit number.
Area of Science:
- Quantum Information Science
- Quantum Computing
- Statistical Mechanics
Background:
- Random quantum circuits are crucial for quantum information processing.
- Understanding the convergence of circuit elements to Haar measure is key for applications like quantum error correction and simulation.
- Characterizing the dynamics of quantum systems requires analyzing moments of unitary evolution.
Purpose of the Study:
- To determine the rate at which averages of polynomials in random quantum circuit elements converge to Haar measure averages.
- To establish the connection between random quantum circuits and statistical mechanics models.
- To demonstrate the efficiency of random quantum circuits in implementing approximate unitary designs.
Main Methods:
- Analysis of a class of random quantum circuits with gates applied to random qubit pairs.
- Mapping the superoperator for t-order moments on n qubits to a multilevel SU(4^t) Lipkin-Meshkov-Glick Hamiltonian.
- Investigating the spectral gap of the Hamiltonian in the thermodynamic limit (n -> infinity).
Main Results:
- The ground-state manifold of the mapped Hamiltonian is spanned by factorized eigenstates.
- Under a mean-field ansatz, the spectral gap scales as 1/n.
- This scaling implies rapid convergence to Haar measure averages for arbitrary fixed t.
Conclusions:
- Random quantum circuits provide an efficient method for implementing epsilon-approximate unitary t-designs.
- The theoretical framework connects quantum circuit complexity to solvable models in statistical mechanics.
- The findings have implications for the design and analysis of quantum algorithms and protocols.
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