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Exact Replica Mechanics and Born-Record Large Deviations in Monitored Quantum Circuits

Amin Abaee

Published Oct 14, 2026• 0 citations
Publisher

Abstract

We develop an exact finite-dimensional theory of the replica mechanics underlying measurement-induced entanglement transitions in monitored quantum circuits, separating three structures that are often conflated: linear outcome-summed replica channels, normalized conditional trajectories, and asymptotic large deviations. For fixed-basis projective measurements we determine the minimal invariant unital algebra of the one-wire feature space and prove that independently Haar-averaged two-site gates on a perfect matching close the \( n \)-replica permutation space exactly for every replica number, with an explicit Weingarten bond channel. The resulting one-period transfer operators have real, positive, semisimple nonzero spectra that compress to bond-label space; at \( n=2 \) the model is exactly the anisotropic triangular-lattice Ising model of Bao, Choi and Altman, with a closed-form critical point and a closed free-fermion spectrum. Carrying the same computation through \( n=3,4,5 \) locates the annealed critical points \( p_c^{(2)}=0.2338 \) <\( p_c^{(3)}=0.305(3) \)<\( p_c^{(4)}\approx0.383 \)<\( p_c^{(5)}\approx0.47 \) with the Potts-class exponents at \( n=3,4 \), and supports, at the three-size level, the first-order prediction of the q=n-Potts identification at n=5: an exact three-size test (L=4,6,8) on a symmetry-reduced orbit basis of dimension 3,865 over the 2.1×108 bond-label space shows the two-phase gap closing faster than the continuous baseline at every size, with the scaled gap falling below the continuous envelope at the matched sizes and dual effective closing rates ordered consistently with the continuous, marginal and first-order behaviours at matched sizes---a directional trend, with the registered conditional that completes it: if the closing is genuinely exponential, a fourth size (L=10) will show the effective closing rate stabilizing at a positive interface tension, and if the asymptotic is a slowly decaying power law, that rate will continue to decay---and at local dimension d=3 the same three-size test keeps the gap level below the d=2 values at every size. We further prove a no-freezing theorem for finite-reachable record processes, establish the worst-case complexity of conditional-trajectory decision, derive the exact replica interpolation identity with its measurable self-averaging criterion, and give the first numerical quenched record scaled cumulant generating functions for Haar circuits, which exhibit genuine record multifractality; the Clifford record-count disorder statistics are closed exactly through a verified conjugation 3-design.dummy

DOI
10.20944/preprints202610.1401.v1

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We develop an exact finite-dimensional theory of the replica mechanics underlying measurement-induced entanglement transitions in monitored quantum circuits, separating three structures that are often conflated: linear outcome-summed replica channels, normalized conditional trajectories, and asymptotic large deviations. For fixed-basis projective measurements we determine the minimal invariant unital algebra of the one-wire feature space and prove that independently Haar-averaged two-site gates on a perfect matching close the ( n )-replica permutation space exactly for every replica number, with an explicit Weingarten bond channel. The resulting one-period transfer operators have real, positive, semisimple nonzero spectra that compress to bond-label space; at ( n=2 ) the model is exactly the anisotropic triangular-lattice Ising model of Bao, Choi and Altman, with a closed-form critical point and a closed free-fermion spectrum. Carrying the same computation through ( n=3,4,5 ) locates the annealed critical points ( p_c^{(2)}=0.2338 ) <( p_c^{(3)}=0.305(3) )<( p_c^{(4)}\approx0.383 )<( p_c^{(5)}\approx0.47 ) with the Potts-class exponents at ( n=3,4 ), and supports, at the three-size level, the first-order prediction of the q=n-Potts identification at n=5: an exact three-size test (L=4,6,8) on a symmetry-reduced orbit basis of dimension 3,865 over the 2.1×108 bond-label space shows the two-phase gap closing faster than the continuous baseline at every size, with the scaled gap falling below the continuous envelope at the matched sizes and dual effective closing rates ordered consistently with the continuous, marginal and first-order behaviours at matched sizes---a directional trend, with the registered conditional that completes it: if the closing is genuinely exponential, a fourth size (L=10) will show the effective closing rate stabilizing at a positive interface tension, and if the asymptotic is a slowly decaying power law, that rate will continue to decay---and at local dimension d=3 the same three-size test keeps the gap level below the d=2 values at every size. We further prove a no-freezing theorem for finite-reachable record processes, establish the worst-case complexity of conditional-trajectory decision, derive the exact replica interpolation identity with its measurable self-averaging criterion, and give the first numerical quenched record scaled cumulant generating functions for Haar circuits, which exhibit genuine record multifractality; the Clifford record-count disorder statistics are closed exactly through a verified conjugation 3-design.dummy
We study the critical properties of random quantum circuits with a U(1)U(1)U(1) symmetry subject to local projective measurements that explicitly break this symmetry. We find that, at the measurement-induced phase transition, symmetry-breaking measurements act as a relevant perturbation at large scales, leading to the same universal critical properties as the corresponding monitored random circuit with non-symmetric unitary dynamics. In particular, we consider monitored U(1)U(1)U(1)-symmetric Haar-random circuits in the limit of large local Hilbert-space dimension, where the trajectory-averaged entanglement entropy can be exactly obtained in terms of a classical statistical mechanics model. In this model, the charge associated with the conservation law follows a symmetric simple exclusion process, in which symmetry-breaking measurements correspond to disordered defects that create and destroy charges. We prove that the charge correlation length remains finite for any measurement rate, ruling out a charge-sharpening transition, in contrast to the case of symmetry-preserving measurements. We further support our predictions at finite local Hilbert-space dimension through numerical finite-size scaling analyses of the entanglement transition in monitored U(1)U(1)U(1)-symmetric Haar and stabilizer random circuits.
We study the critical properties of random quantum circuits with a U(1)U(1)U(1) symmetry subject to local projective measurements that explicitly break this symmetry. We find that, at the measurement-induced phase transition, symmetry-breaking measurements act as a relevant perturbation at large scales, leading to the same universal critical properties as the corresponding monitored random circuit with non-symmetric unitary dynamics. In particular, we consider monitored U(1)U(1)U(1)-symmetric Haar-random circuits in the limit of large local Hilbert-space dimension, where the trajectory-averaged entanglement entropy can be exactly obtained in terms of a classical statistical mechanics model. In this model, the charge associated with the conservation law follows a symmetric simple exclusion process, in which symmetry-breaking measurements correspond to disordered defects that create and destroy charges. We prove that the charge correlation length remains finite for any measurement rate, ruling out a charge-sharpening transition, in contrast to the case of symmetry-preserving measurements. We further support our predictions at finite local Hilbert-space dimension through numerical finite-size scaling analyses of the entanglement transition in monitored U(1)U(1)U(1)-symmetric Haar and stabilizer random circuits.
Monitored quantum circuits combine scrambling with measurement-conditioned state updates, while non-Clifford perturbations inject magic into otherwise stabilizer-compatible dynamics. Rigorous results on stationary magic and its weak-injection asymptotics remain limited even for finite-dimensional Clifford-based monitored models. We study an NNN-qudit process of prime local dimension ddd. Each cycle draws a fresh uniform global Clifford unitary, applies a local weak non-Clifford rotation and a projective measurement on one qudit, and then returns to the inverse Clifford frame. For every fixed injection strength, we prove that the induced pure-state Markov chain has a unique invariant probability measure and attracts every initial law geometrically in Wasserstein distance. At zero injection, the invariant law is supported on the finite stabilizer layer. After rescaling transverse deviations from this layer, the resulting blown-up stationary laws converge weakly to the invariant law of an affine tangent recursion. Combining this tangent law with Poisson representations and the first nonzero local resource germs determines the sharp vanishing rates of stationary magic. For every fixed N≥2N\ge2N≥2, odd-prime Gross--Wigner mana admits a linear expansion with a strictly positive coefficient, whereas qubit 222-stabilizer Rényi entropy admits a quadratic expansion with a strictly positive coefficient. For N=1N=1N=1, both stationary resource averages vanish identically for all injection strengths. The distinct orders arise from the different local resource geometries, together with quadratic-order zero-reference branch contributions in qubit case.
The authors explore critical properties of entanglement phase transitions in random tensor networks and monitored quantum circuits with Clifford tensors/gates for on-site Hilbert space dimensions which are powers of a prime number ppp. Exact mappings to replica spin models and characterizations of their symmetry groups predict that all universal properties will depend only on ppp. These predictions are confirmed with extensive numerical simulations. The authors also 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.
On finite system sizes, monitored one-dimensional free-fermion chains display a broad crossover in entanglement scaling as the measurement rate increases, from sub-extensive behavior at weak monitoring toward an area law at strong monitoring. Analytical field theory and recent large-scale simulations indicate that this apparent change is not a finite-rate transition in the thermodynamic limit. I test, using trajectory-resolved correlation-matrix simulations (chains up to L=96L=96L=96 and up to 128 Born-rule trajectories per parameter point), whether two quantities built from the single-particle entanglement spectrum are useful finite-size diagnostics: the bipartite particle-number fluctuation FA=∑kνk(1−νk)F_A=\sum_kν_k(1-ν_k)FA​=∑k​νk​(1−νk​) and a participation-style effective number of entangling modes, MA=exp⁡[−∑kwkln⁡wk]\mathcal{M}_A=\exp[-\sum_k w_k\ln w_k]MA​=exp[−∑k​wk​lnwk​], with wk∝νk(1−νk)w_k\proptoν_k(1-ν_k)wk​∝νk​(1−νk​). Both quantities track the crossover. However, for trajectory- and time-averaged steady-state values, M\mathcal{M}M is nearly a deterministic function of FFF: a pooled curve explains 99.7\% of its variance across all rates, so M\mathcal{M}M carries little independent information. Its main advantage is statistical: its trajectory-to-trajectory coefficient of variation is up to a factor of ∼2\sim2∼2 smaller than that of the entropy or FFF under strong monitoring. Subsystem-scaling comparisons at L=48L=48L=48--969696 also show that a pure logarithmic law is not statistically preferred under weak monitoring; a small residual quasi-extensive component and a drifting logarithmic coefficient are instead consistent with a crossover rather than a critical phase. Thus, FFF remains the natural experimentally motivated companion to the entropy, whereas M\mathcal{M}M is best viewed as a variance-reduced numerical summary of essentially the same information.
In this thesis, I describe research results on three topics : (i) a phase transition in the area-law regime of quantum circuits driven by projection measurements; (ii) ultra slow dynamics in two dimensional spin circuits; and (iii) tensor network methods applied to boolean satisfiability problems. (i) Nonuniversal entanglement level statistics in projection-driven quantum circuits; Non-thermalized closed quantum many-body systems have drawn considerable attention, due to their relevance to experimentally controllable quantum systems. In the first part of the thesis, we study the level-spacing statistics in the entanglement spectrum of output states of random universal quantum circuits where, at each time step, qubits are subject to a finite probability of projection onto states of the computational basis. We encounter two phase transitions with increasing projection rate: The first is the volume-to-area law transition observed in quantum circuits with projective measurements; The second separates the pure Poisson level statistics phase at large projective measurement rates from a regime of residual level repulsion in the entanglement spectrum within the area-law phase, characterized by non-universal level spacing statistics that interpolates between the Wigner-Dyson and Poisson distributions. The same behavior is observed in both circuits of random two-qubit unitaries and circuits of universal gates, including the set implemented by Google in its Sycamore circuits. (ii) Ultra-slow dynamics in a translationally invariant spin model for multiplication and factorization; Slow relaxation of glassy systems in the absence of disorder remains one of the most intriguing problems in condensed matter physics. In the second part of the thesis we investigate slow relaxation in a classical model of short-range interacting Ising spins on a translationally invariant two-dimensional lattice that mimics a reversible circuit that, depending on the choice of boundary conditions, either multiplies or factorizes integers. We prove that, for open boundary conditions, the model exhibits no finite-temperature phase transition. Yet we find that it displays glassy dynamics with astronomically slow relaxation times, numerically consistent with a double exponential dependence on the inverse temperature. The slowness of the dynamics arises due to errors that occur during thermal annealing that cost little energy but flip an extensive number of spins. We argue that the energy barrier that needs to be overcome in order to heal such defects scales linearly with the correlation length, which diverges exponentially with inverse temperature, thus yielding the double exponential behavior of the relaxation time. (iii) Reversible circuit embedding on tensor networks for Boolean satisfiability; Finally, in the third part of the thesis we present an embedding of Boolean satisfiability (SAT) problems on a two-dimensional tensor network. The embedding uses reversible circuits encoded into the tensor network whose trace counts the number of solutions of the satisfiability problem. We specifically present the formulation of #2SAT, #3SAT, and #3XORSAT formulas into planar tensor networks. We use a compression-decimation algorithm introduced by us to propagate constraints in the network before coarse-graining the boundary tensors. Iterations of these two steps gradually collapse the network while slowing down the growth of bond dimensions. For the case of #3XORSAT, we show numerically that this procedure recognizes, at least partially, the simplicity of XOR constraints for which it achieves subexponential time to solution. For a #P-complete subset of #2SAT we find that our algorithm scales with size in the same way as state-of-the-art #SAT counters, albeit with a larger prefactor. We find that the compression step performs less efficiently for #3SAT than for #2SAT.