Jun 4, 2023
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CSX/vol-1/no-03/Probing critical states of matter on a digital quantum computer
PTL PREMIUM
Received 19 January 2023
Reviewed 28 February 2023
Accepted 25 April 2023
Probing critical states of matter on a digital quantum computer
Reza Haghshenas
1Quantinuum, 303 S. Technology Ct., Broomfield, Colorado 80021, USA
Eli Chertkov
Matthew DeCross
Thomas M. Gatterman
Justin A. Gerber,1 Kevin Gilmore,1 Dan Gresh,1 Nathan Hewitt,1 Chandler V. Horst,1 Mitchell
Matheny,1 Tanner Mengle,1 Brian Neyenhuis,1 David Hayes,1 and Michael Foss-Feig1, †
Canadian Science Letters X
2023 ° 03(05) ° 01-03
https://www.wikipt.org/csx-home
DOI: 10.1490/6576500.983csx
Funding Agent Details
Not Applicable.
Abstract
Although quantum mechanics underpins the microscopic behaviour of all materials, its effects are often obscured at the macroscopic level by thermal fluctuations. A notable exception is a zerotemperature phase transition, where scaling laws emerge entirely due to quantum correlations over a diverging length scale. The accurate description of such transitions is challenging for classical simulation methods of quantum systems, and is a natural application space for quantum simulation. These quantum simulations are, however, not without their own challenges — representing quantum critical states on a quantum computer requires encoding entanglement of a large number of degrees of freedom, placing strict demands on the coherence and fidelity of the computer’s operations. Using Quantinuum’s H1-1 quantum computer, we address these challenges by employing hierarchical quantum tensor-network techniques, creating the ground state of the critical transverse-field Ising chain on 128-sites with sufficient fidelity to extract accurate critical properties of the model. Our results suggest a viable path to quantum-assisted tensor network contraction beyond the limits of classical methods.
Introduction
Simulating quantum systems is a natural task for quantum computers, and is widely considered amongst their most important and most feasible near-term applications. Despite this consensus, it would be wrong to infer that the path to outperforming the best classical methods for simulating quantum systems is easy, or even clearly laid out. One challenge is that quantum states with low entanglement can be accurately and efficiently represented using classical tensor-network (TN) techniques [1], i.e., low-entanglement problems are not hard for classical computers. On the other hand, quantum states with high entanglement are generally difficult to accurately produce on existing quantum processors due to hardware imperfections. An important situation in which classical tensor network methods reveal their vulnerability to entanglement is the study of quantum critical points. In one spatial dimension, matrix product states (MPS) cannot accurately describe critical systems in the large systemsize limit unless the bond-dimension (and therefore classical simulation overhead) grows polynomially with system size [2]. In practice, the polynomial overheads are tame enough to allow accurate MPS calculations for 1D critical systems, though with considerably more difficulty than calculations away from criticality. In dimensions d > 1 or for systems out of equilibrium, the growth of entanglement with either system size or evolution time remains a significant obstacle to performing accurate TN calculations classically.
Conclusion
We employ a form of zero-noise extrapolation (ZNE) [14], which works for our native arbitrary-angle two-qubit gate UZZ(θ) = exp(−i θ 2 Z ⊗ Z), to mitigate the errors in our experimentally measured observables. The expectation value of an observable E(p) computed from the quantum computer depends on the amount of noise caused by noisy operations like two-qubit gates, with p setting the scale of the noise. For small enough p, we expect E(p) ≈ E0 + pE1 to be well approximated by a first-order Taylor expansion, with E(p = 0) = E0 corresponding to the ideal zero-noise expectation value. By performing two experiments, one at p and one at mp for small enough m, we can use a simple linear extrapolation to estimate the zero-noise value as
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