Cross Entropy Benchmark for Measurement-Induced Phase Transitions

Yaodong Li, Yijian Zou, Paolo Glorioso, Ehud Altman, and Matthew P. A. Fisher

Phys. Rev. Lett. 130, 220404 – Published 1 June 2023

Abstract

We investigate prospects of employing the linear cross entropy to experimentally access measurement-induced phase transitions without requiring any postselection of quantum trajectories. For two random circuits that are identical in the bulk but with different initial states, the linear cross entropy $χ$ between the bulk measurement outcome distributions in the two circuits acts as an order parameter, and can be used to distinguish the volume law from area law phases. In the volume law phase (and in the thermodynamic limit) the bulk measurements cannot distinguish between the two different initial states, and $χ = 1$ . In the area law phase $χ < 1$ . For circuits with Clifford gates, we provide numerical evidence that $χ$ can be sampled to accuracy $ϵ$ from $O (1 / ϵ^{2})$ trajectories, by running the first circuit on a quantum simulator without postselection, aided by a classical simulation of the second. We also find that for weak depolarizing noise the signature of the measurement-induced phase transitions is still present for intermediate system sizes. In our protocol we have the freedom of choosing initial states such that the “classical” side can be simulated efficiently, while simulating the “quantum” side is still classically hard.

Received 9 January 2023
Accepted 16 May 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.220404

Physics Subject Headings (PhySH)

Entropy Open quantum systems Phase transitions Quantum entanglement Quantum measurements Quantum simulation

Quantum Information, Science & TechnologyStatistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Yaodong Li ^1,2, Yijian Zou², Paolo Glorioso ², Ehud Altman³, and Matthew P. A. Fisher ¹

¹Department of Physics, University of California, Santa Barbara, California 93106, USA
²Department of Physics, Stanford University, Stanford, California 94305, USA
³Department of Physics, University of California, Berkeley, California 94720, USA

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Vol. 130, Iss. 22 — 2 June 2023

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Images

Figure 1
The layout of the hybrid circuit considered in this Letter. Different from the usual setup [9], we have an additional “encoding” stage before the hybrid evolution for time $t_{encoding} = 2 L$ , following Ref. [5]. We call the evolution after the encoding stage the “circuit bulk,” which lasts for another $t_{bulk} = 2 L$ . The total circuit time is $T = t_{encoding} + t_{bulk} = 4 L$ . We will compare two different initial states $ρ$ and $σ$ (left unspecified for the moment) undergoing the same circuit evolution.
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Figure 2
(a) Numerical results for $χ_{C}$ when averaged over 300 Clifford circuits in the bulk (denoted by $E_{C}$ ), with the initial states $ρ = (1 / 2^{L}) 1$ and $σ = (| 0 ⟩ ⟨ 0 |)^{\otimes L}$ . Here, for each $C$ , the calculation is exact, and $M$ can be thought of as infinity in Eq. (11). Inset: collapsing the data to a scaling form, with parameters $p_{c}$ and $ν$ close to those found near the MIPT in entanglement entropy [3, 9]. (b),(c) The behavior of $χ$ when depolarizing noise is present in the $ρ$ circuit. As we see, at noise rate $q = 0.1 %$ (b), there is still evidence for a phase transition, although the location of the transition has shifted from $p_{c} \approx 0.16$ to $p_{c} \approx 0.14$ . At noise rate 1% (c), there is no crossing, and any signature of the phase transition is completely washed out. (d),(e) The convergence of the sample average $μ_{M_{C}} = (M_{C})^{- 1} \sum_{j = 1}^{M_{C}} χ_{C_{j}}$ to $χ = E_{C} χ_{C}$ with increasing number of circuit samples $M_{C}$ . For each of $L \in {64, 128}$ , we plot $σ_{M_{C}}^{2} = E [{(μ_{M_{C}} - χ)}^{2}]$ for $M_{C} \leq 500$ , whereas $χ$ is estimated using $M_{C} = 2000$ circuits. The results are consistent with the central limit theorem; see Eq. (12). From this plot we see that sample variance is suppressed by large $L$ when $p < p_{c}$ , and is independent of $L$ when $p \geq p_{c}$ . This justifies our choice of a relatively small $M_{C}$ that is independent of the system size.
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Figure 3
(a) Numerical results of $χ$ for initial states $ρ = \otimes_{i = 1}^{L / 2} (| 0 ⟩ ⟨ 0 |_{2 i - 1} \otimes | T ⟩ ⟨ T |_{2 i})$ and $σ = (| 0 ⟩ ⟨ 0 |)^{\otimes L}$ obtained from random Clifford circuits. Here, $M_{C} = 300$ circuit realizations are taken for each $L$ , and for each circuit, we use $M = 100$ runs to estimate $χ_{C}$ , following Eq. (11). Compared to Fig. 2, the results are qualitatively similar, despite a different choice of initial state and smaller system sizes. (b) Numerical results of $χ$ for initial states $ρ = (| + ⟩ ⟨ + |)^{\otimes L}$ and $σ = (| 0 ⟩ ⟨ 0 |)^{\otimes L}$ obtained from random Haar circuits. Here, $M_{C} = 150$ circuit realizations are taken for each $L$ , and for each circuit we estimate Eqs. (8) and (9) separately, using $M = 3200$ runs each.
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