Qubit-efficient entanglement spectroscopy using qubit resets

Justin Yirka1 and Yiğit Subaşı2

1Department of Computer Science, The University of Texas at Austin, Austin, TX 78712, USA
2Computer, Computational, and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

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One strategy to fit larger problems on NISQ devices is to exploit a tradeoff between circuit width and circuit depth. Unfortunately, this tradeoff still limits the size of tractable problems since the increased depth is often not realizable before noise dominates. Here, we develop $\textit{qubit-efficient}$ quantum algorithms for entanglement spectroscopy which avoid this tradeoff. In particular, we develop algorithms for computing the trace of the $n$-th power of the density operator of a quantum system, $Tr(\rho^n)$, (related to the Rényi entropy of order $n$) that use fewer qubits than any previous efficient algorithm while achieving similar performance in the presence of noise, thus enabling spectroscopy of larger quantum systems on NISQ devices. Our algorithms, which require a number of qubits independent of $n$, are variants of previous algorithms with width proportional to $n$, an asymptotic difference. The crucial ingredient in these new algorithms is the ability to measure and reinitialize subsets of qubits in the course of the computation, allowing us to reuse qubits and increase the circuit depth without suffering the usual noisy consequences. We also introduce the notion of $\textit{effective circuit depth}$ as a generalization of standard circuit depth suitable for circuits with qubit resets. This tool helps explain the noise-resilience of our qubit-efficient algorithms and should aid in designing future algorithms. We perform numerical simulations to compare our algorithms to the original variants and show they perform similarly when subjected to noise. Additionally, we experimentally implement one of our qubit-efficient algorithms on the Honeywell System Model H0, estimating $Tr(\rho^n)$ for larger n than possible with previous algorithms.

In the near- to medium-term, during the so-called Noisy Intermediate-Scale Quantum (NISQ) era, quantum devices don’t have many qubits and can’t stay coherent for long. As a result, only quantum circuits of limited width (the number of qubits) and depth (similar to the runtime) can be successfully executed. For many applications, larger problem size corresponds to larger circuits. Therefore, quantum computers will remain limited to small problems for a significant time unless new solutions are found.

One strategy to fit larger problems on NISQ devices is to exploit a tradeoff between circuit width and circuit depth. Unfortunately, since both of these resources are limited, this tradeoff still limits size of tractable problems.

Here, we develop qubit-efficient quantum algorithms for the task of Entanglement Spectroscopy which avoid this tradeoff.

In this application, the goal is to learn the structure of the entanglement between subsystems of a larger quantum system. Entanglement spectroscopy will be important in order to process the output of one of the most promising applications of quantum computers, quantum simulation of many-body systems.

In particular, we develop algorithms for Entanglement Spectroscopy that use asymptotically fewer qubits than any previous (efficient) algorithms while achieving similar performance in the presence of noise. Thus, our algorithms avoid the usual tradeoff and enable spectroscopy of larger quantum systems on NISQ devices than previously possible.

The crucial ingredient in these new algorithms is the ability to measure and reinitialize subsets of qubits in the course of the computation. By carefully arranging these qubits resets, we are able to reuse qubits and increase the circuit depth without suffering the usual noisy consequences.

We perform experiments to test our algorithms on both simulated and real quantum hardware. We experimentally implement one of our qubit-efficient algorithms on the 6-qubit Honeywell System Model H0. In these experiments, we perform spectroscopy for larger parameters than possible with any previous algorithm on this size device. We also run numerical simulations using IBM’s Qiskit to test the noise-resilience of our algorithms and compare them to previous algorithms.

Finally, we introduce the notion of effective circuit depth as a generalization of standard circuit depth that is suitable for circuits with qubit resets. This tool helps explain the noise-resilience of our qubit-efficient algorithms, while standard circuit depth does not, and should aid in designing future algorithms.

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[2] Kun Wang, Yu-Ao Chen, and Xin Wang, "Measurement Error Mitigation via Truncated Neumann Series", arXiv:2103.13856.

[3] Michael Foss-Feig, Stephen Ragole, Andrew Potter, Joan Dreiling, Caroline Figgatt, John Gaebler, Alex Hall, Steven Moses, Juan Pino, Ben Spaun, Brian Neyenhuis, and David Hayes, "Entanglement from tensor networks on a trapped-ion QCCD quantum computer", arXiv:2104.11235.

[4] Kok Chuan Tan and Tyler Volkoff, "Variational quantum algorithms to estimate rank, quantum entropies, fidelity, and Fisher information via purity minimization", Physical Review Research 3 3, 033251 (2021).

[5] Maria Kieferova, Ortiz Marrero Carlos, and Nathan Wiebe, "Quantum Generative Training Using Rényi Divergences", arXiv:2106.09567.

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