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arXiv:2312.08447 · Quantum

Entanglement spectrum of matchgate circuits with universal and non-universal resources

Andrew M. Projansky, Joshuah T. Heath, James Daniel Whitfield

Published Aug 7, 2024• 3 citations
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Abstract

The entanglement level statistics of a quantum state have recently been proposed to be a signature of universality in the underlying quantum circuit. This is a consequence of level repulsion in the entanglement spectra being tied to the integrability of entanglement generated. However, such studies of the level-spacing statistics in the entanglement spectrum have thus far been limited to the output states of Clifford and Haar random circuits on product state inputs. In this work, we provide the first example of a circuit which is composed of a simulable gate set, yet has a Wigner-Dyson distributed entanglement level spectrum without any perturbing universal element. We first show that, for matchgate circuits acting on random product states, Wigner-Dyson statistics emerge by virtue of a single SWAP gate, in direct analog to previous studies on Clifford circuits. We then examine the entanglement spectrum of matchgate circuits with varied input states, and find a sharp jump in the complexity of entanglement as we go from two- to three-qubit entangled inputs. Studying Clifford and matchgate hybrid circuits, we find examples of classically simulable circuits whose output states exhibit Wigner-Dyson entanglement level statistics in the absence of universal quantum gate elements. Our study thus provides strong evidence that entanglement spectrum is not strongly connected to notions of simulability in any given quantum circuit.

DOI
10.22331/q-2024-08-07-1432

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The entanglement level statistics of a quantum state have recently been proposed to be a signature of universality in the underlying quantum circuit. This is a consequence of level repulsion in the entanglement spectra being tied to the integrability of entanglement generated. However, such studies of the level-spacing statistics in the entanglement spectrum have thus far been limited to the output states of Clifford and Haar random circuits on product state inputs. In this work, we provide the first example of a circuit which is composed of a simulable gate set, yet has a Wigner-Dyson distributed entanglement level spectrum without any perturbing universal element. We first show that, for matchgate circuits acting on random product states, Wigner-Dyson statistics emerge by virtue of a single SWAP gate, in direct analog to previous studies on Clifford circuits. We then examine the entanglement spectrum of matchgate circuits with varied input states, and find a sharp jump in the complexity of entanglement as we go from two- to three-qubit entangled inputs. Studying Clifford and matchgate hybrid circuits, we find examples of classically simulable circuits whose output states exhibit Wigner-Dyson entanglement level statistics in the absence of universal quantum gate elements. Our study thus provides strong evidence that entanglement spectrum is not strongly connected to notions of simulability in any given quantum circuit.
Clifford circuits are insufficient for universal quantum computation or creating t t -designs with t\ge 4 t ≥ 4 . While the entanglement entropy is not a telltale of this insufficiency, the entanglement spectrum of a time evolved random product state is: the entanglement levels are Poisson-distributed for circuits restricted to the Clifford gate-set, while the levels follow Wigner-Dyson statistics when universal gates are used. In this paper we show, using finite-size scaling analysis of different measures of level spacing statistics, that in the thermodynamic limit, inserting a single T (\pi/8) ( π / 8 ) gate in the middle of a random Clifford circuit is sufficient to alter the entanglement spectrum from a Poisson to a Wigner-Dyson distribution.
We study the classical simulation complexity, in both the weak and strong senses, of matchgate (MG) computations supplemented with all combinations of settings involving inclusion of intermediate adaptive or nonadaptive computational basis measurements, product state or magic and general entangled state inputs, and single- or multiple-line outputs. We find a striking parallel to known results for Clifford circuits, after some rebranding of resources. We also give bounds on the amount of classical simulation effort required in the case of limited access to intermediate measurements and entangled inputs. In further settings we show that adaptive MG circuits remain classically efficiently simulable if arbitrary two-qubit entangled input states on consecutive lines are allowed, but become quantum universal for three or more lines. And if adaptive measurements in noncomputational bases are allowed, even with just computational basis inputs, we get quantum universal power again.
We show that in a quantum system evolving unitarily under a stochastic quantum circuit the notions of irreversibility, universality of computation, and entanglement are closely related. As the state evolves from an initial product state, it gets asymptotically maximally entangled. We define irreversibility as the failure of searching for a disentangling circuit using a Metropolis-like algorithm. We show that irreversibility corresponds to Wigner–Dyson statistics in the level spacing of the entanglement eigenvalues, and that this is obtained from a quantum circuit made from a set of universal gates for quantum computation. If, on the other hand, the system is evolved with a non-universal set of gates, the statistics of the entanglement level spacing deviates from Wigner–Dyson and the disentangling algorithm succeeds. These results open a new way to characterize irreversibility in quantum systems.
Universal features of chaotic quantum dynamics underlie our understanding of thermalization in closed quantum systems and the complexity of quantum computations. Reversible automaton circuits, comprised of classical logic gates, have emerged as a tractable means to study such dynamics. Despite generating no entanglement in the computational basis, these circuits nevertheless capture many features expected from fully quantum evolutions. In this work, we demonstrate that the differences between automaton dynamics and fully quantum dynamics are revealed by the operator entanglement spectrum, much like the entanglement spectrum of a quantum state distinguishes between the dynamics of states under Clifford and Haar random circuits. While the operator entanglement spectrum under random unitary dynamics is governed by the eigenvalue statistics of random Gaussian matrices, we show evidence that under random automaton dynamics it is described by the statistics of Bernoulli random matrices, whose entries are random variables taking values 0 or 1 . We study the crossover between automaton and generic unitary operator dynamics as the automaton circuit is doped with gates that introduce superpositions, namely Hadamard or R x = e − i π 4 X gates. We find that a constant number of superposition-generating gates is sufficient to drive the operator dynamics to the random-circuit universality class, similar to earlier results on Clifford circuits doped with T gates. This establishes the operator entanglement spectrum as a useful tool for probing the chaoticity and universality class of quantum dynamics.
We study the level-spacing statistics in the entanglement spectrum of output states of random universal quantum circuits where qubits are subject to a finite probability of projection to the computational basis at each time step. 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. We identify a second transition within the area law phase by repartioning the system randomly into two subsystems and probing the entanglement level statistics. This second transition separates a pure Poisson level statistics phase at large projective measurement rates from a regime of residual level repulsion in the entanglement spectrum, characterized by nonuniversal level spacing statistics that interpolates between the Wigner-Dyson and Poisson distributions. By applying a tensor network contraction algorithm introduced in [Z.-C. Yang et al., Phys. Rev. E 97, 033303 (2018)] to the circuit spacetime, we identify this second projective-measurement-driven transition as a percolation transition of entangled bonds. The same behavior is observed in both circuits of random two-qubit unitaries and circuits of universal gate sets, including the set implemented by Google in its Sycamore circuits.
Abstract Though Cliffords and matchgates are both examples of classically simulable circuits, they are considered simulable for different reasons. The celebrated Gottesman–Knill explains the simulability Cliffords, and the efficient simulability of matchgates is understood via Pfaffians of antisymmetric matrices. We take the perspective that by studying Clifford-matchgate hybrid circuits, we expand the set of known simulable circuits and reach a better understanding of what unifies these two circuit families. While the simulability of Clifford conjugated matchgate circuits for single qubit outputs has been briefly considered, the simulability of Clifford and matchgate hybrid circuits has not been generalized up to this point. In this paper we extend that work, studying simulability of marginals as well as Pauli expectation values of Clifford and matchgate hybrid circuits. We describe a hierarchy of Clifford circuits, and find that as we consider more general Cliffords, we lose some amount of simulability of bitstring outputs. We then show that the known simulability of Pauli expectation values of Clifford circuits acting on product states can be generalized to Clifford circuits acting after any matchgate circuit. We conclude with general discussion about the relationship between Cliffords and matchgates, and show that both circuit families can be understood as being Gaussian.