Statistical mechanics model for Clifford random tensor networks and monitored quantum circuits

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Statistical mechanics model for Clifford random tensor networks and monitored quantum circuits

Yaodong Li, Romain Vasseur, Matthew P. A. Fisher, and Andreas W. W. Ludwig

Phys. Rev. B 109, 174307 – Published 13 May 2024

Abstract

We study entanglement transitions in Clifford (stabilizer) random tensor networks (RTNs) and monitored quantum circuits, by introducing an exact mapping onto a (replica) statistical mechanics model. For RTNs and monitored quantum circuits with random Haar unitary gates, entanglement properties can be computed using statistical mechanics models whose fundamental degrees of freedom (“spins”) are permutations, because all operators commuting with the action of the unitaries on a tensor product Hilbert space are (linear combinations of) permutations of the tensor factors (“Schur-Weyl duality”). When the unitary gates are restricted to the smaller group of Clifford unitaries, the set of all operators commuting with this action, called the commutant, will be larger, and no longer form a group. We use the recent full characterization of this commutant by Gross et al. [Commun. Math. Phys. 385, 1325 (2021)] to construct statistical mechanics models for both Clifford RTNs and monitored quantum circuits, for on-site Hilbert-space dimensions which are powers of a prime number $p$ . The elements of the commutant form the spin degrees of freedom of these statistical mechanics models, and we show that the Boltzmann weights are invariant under a symmetry group involving orthogonal matrices with entries in the finite number field $F_{p}$ (“Galois field”) with $p$ elements. This implies that the symmetry group and consequently all universal properties of entanglement transitions in Clifford circuits and RTNs will, respectively, in general depend on and only on the prime $p$ . We show that Clifford monitored circuits with on-site Hilbert-space dimension $d = p^{M}$ are described by percolation in the limits $d \to \infty$ at (a) $p =$ fixed but $M \to \infty$ , and at (b) $M = 1$ but $p \to \infty$ . In the limit (a) we calculate the effective central charge, and in the limit (b) we derive the following universal minimal cut entanglement entropy $S_{A} = (\sqrt{3} / π) ln p ln L_{A}$ for $d = p$ large at the transition. We verify those predictions numerically, and present extensive numerical results for critical exponents at the transition in monitored Clifford circuits for prime number on-site Hilbert-space dimension $d = p$ for a variety of different values of $p$ , finding that projective and forced measurement schemes yield the same critical exponents and that they approach percolation values at large $p$ . We clearly establish multifractal scaling of the purity, reflected in a continuous spectrum of critical exponents, while the typical exponent is the prefactor of the logarithm in the entanglement entropy. As a technical result, we generalize the notion of the Weingarten function, previously known for averages involving the Haar measure, to averages over the Clifford group.

Received 10 October 2023
Revised 4 April 2024
Accepted 5 April 2024

DOI:https://doi.org/10.1103/PhysRevB.109.174307

Physics Subject Headings (PhySH)

Phase transitions Quantum circuits Quantum entanglement

Statistical Physics & ThermodynamicsQuantum Information, Science & Technology

Authors & Affiliations

Yaodong Li ¹, Romain Vasseur², Matthew P. A. Fisher¹, and Andreas W. W. Ludwig¹

¹Department of Physics, University of California, Santa Barbara, California 93106, USA
²Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA

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Vol. 109, Iss. 17 — 1 May 2024

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Images

Figure 1
(a) A tensor $T_{v}$ at vertex $v$ with coordination number $z = 4$ . (b) Arranging the tensors in a network (taken to be square lattice here), we obtain a “tensor network state” $ρ [T]$ [see Eq. (4)] by contracting indices on all bulk edges, as implemented by projecting onto the maximally entangled state $| I_{e} 〉$ . The state $ρ [T]$ is supported on the boundary qudits (highlighted by purple squares) and is in general unnormalized.
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Figure 2
The architecture of the monitored quantum circuit model. Each blue block represents a random unitary gate, and each hollowed circle represents a rank-1 projective measurement, occurring randomly with probability $p_{0}$ . The dynamical state $ρ (t)$ at circuit depth $t$ is supported on qudits at the final time step (highlighted with purple squares) and is labeled by the measurement record $m$ , see Eq. (33). We consider two cases, depending on whether the random unitaries are sampled from the Haar measure on $U (D = 2 d)$ (the “random Haar circuit”), or from the uniform distribution the Clifford group $Cliff (2 M, p) \subset U (D)$ (the “random Clifford circuit”).
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Figure 3
Statistical mechanics model for monitored quantum circuits. (a) The average over replicated unitary gates leads to degrees of freedom belonging to the commutant, defined on vertices; Weingarten weights ${Wg}_{D}$ , represented by the dashed line; and link weights $W$ , corresponding to purple solid lines. (b) The statistical mechanics model is defined on an anisotropic honeycomb lattice.
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Figure 4
Numerical results for $d = p^{M = 1}$ . The fits for the critical exponents are summarized in Table 1. (a) We plot $S (A = [z_{1}, z_{2}])$ against $ln w_{12}$ [see Eq. (56)] at different values of $d = p$ , where we see a clear dependence of the slope ( $= 2 h_{a | b}$ ) on $p$ . With increasing $p$ , the value of $h_{a | b}$ approaches that of minimal cut percolation [see Eq. (54)]. (b) The mutual negativity $N (A, B)$ between two disjoint subregions $A = [z_{1}, z_{2}]$ and $B = [z_{3}, z_{4}]$ against the cross ratio $η = (w_{12} w_{34}) / (w_{13} w_{24})$ . The power law at small $η$ defines a critical exponent $h_{MN}$ , which has a clear dependence on $p$ , and approaches minimal cut percolation at large $p$ . (c) A “localizable entanglement” between $A$ and $B$ . We perform a single-qudit projective measurements on each qudit outside $A \cup B$ , and calculate the mutual information $I (A, B)$ subsequently. The power law at small $η$ is another critical exponent $h_{f | f}^{(1)}$ , which should in general take different values at different $p$ , as we illustrate.
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Figure 5
Numerical results for $d = p^{M}$ , where we take $p \in {2, 3}$ and $M \in {1, 2}$ . Comparing Fig. 4 and Eq. (56), we see that $h_{a | b}$ depends on the choice of $p$ but not on $M$ .
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Figure 6
Numerical results for the first three cumulants of the second Rényi entanglement entropy $S_{A}^{(2)} = S_{A = | x_{1}, x_{2} |}$ : Averaged entanglement entropy (blue) from (61), its second cumulant (red) from (62), and its third cumulant (green) from (63), versus the chord distance, here denoted by $R_{12}$ . The slopes of the plots yield the exponent $x^{(1)} = h_{a | b} \approx 0.53, - x^{(2)} = h_{a | b}^{(2)} \approx 0.32$ , and $x^{(3)} = h_{a | b}^{(3)} \approx 0.15$ . (The notations $h_{a | b}, h_{a | b}^{(2)}$ , and $h_{a | b}^{(3)}$ serve as a reminder that these are the exponents associated with the boundary-condition changing (bcc) operator [21, 25, 56].)
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