Unitary Circuits Research Articles

Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus

In this paper, we propose a general scheme to analyze the gradient vanishing phenomenon, also known as the barren plateau phenomenon, in training quantum neural networks with the ZX-calculus. More precisely, we extend the barren plateaus theorem from unitary 2-design circuits to any parameterized quantum circuits under certain reasonable assumptions. The main technical contribution of this paper is representing certain integrations as ZX-diagrams and computing them with the ZX-calculus. The method is used to analyze four concrete quantum neural networks with different structures. It is shown that, for the hardware efficient ansatz and the MPS-inspired ansatz, there exist barren plateaus, while for the QCNN ansatz and the tree tensor network ansatz, there exists no barren plateau.

Optical Implementation of 2 × 2 Universal Unitary Matrix Transformations

AbstractUnitary operations are a specific class of linear transformations that have become an essential ingredient for the realization of classical and quantum information processing. The ability of implementing any n‐dimensional unitary signal transformation by using a reconfigurable optical hardware has recently led to the pioneering concept of programmable linear optical processor, whose basic building block (BB) must be correctly designed to guarantee that the whole system is able to perform universal (i.e., arbitrary) unitary matrix transformations. Here, it is demonstrated that the present architectures of the BB do not fulfil the universal unitary property (at least) in optical processors, limiting the number of unitary matrix transformations that may be generated. Aiming to solve this fundamental constraint, the theoretical tools required to analyze and design universal unitary optical circuits and their corresponding BBs are presented. The consequences of this mathematical framework are explored, obtaining a simple route to implement different BB architectures, all of them guaranteeing a true universal unitary functionality in the resulting optical processors. These findings may pave the way to revisit the design of high‐dimensional unitary optical processors, unleashing the potential of programmable integrated photonics technology.

Self-organized error correction in random unitary circuits with measurement

Quantum systems evolving under unitary dynamics with a low rate of projective measurements can realize a highly entangled steady state. However, the stability of this phase against the disentangling action of measurements has not been fully understood. Here, the authors quantify the ability of this highly entangled phase to hide local quantum information, thereby relating its stability to its behavior as an emergent quantum error-correcting code. The authors estimate the tolerable measurement rate, above which this quantum code becomes unstable.

Optimization schemes for unitary tensor-network circuit

We discuss the variational optimization of a unitary tensor-network circuit with different network structures. The ansatz is performed based on a generalization of well-developed multi-scale entanglement renormalization algorithm and also the conjugate-gradient method with an effective line search. We present the benchmarking calculations for different network structures by studying the Heisenberg model in a strongly disordered magnetic field and a tensor-network $QR$-decomposition.

Consequences of preserving reversibility in quantum superchannels

Similarly to quantum states, quantum operations can also be transformed by means of quantum superchannels, also known as process matrices. Quantum superchannels with multiple slots are deterministic transformations which take independent quantum operations as inputs. While they are enforced to respect the laws of quantum mechanics, the use of input operations may lack a definite causal order, and characterizations of general superchannels in terms of quantum objects with a physical implementation have been missing. In this paper, we provide a mathematical characterization for pure superchannels with two slots (also known as bipartite pure processes), which are superchannels preserving the reversibility of quantum operations. We show that the reversibility preserving condition restricts all pure superchannels with two slots to be either a quantum circuit only consisting of unitary operations or a coherent superposition of two unitary quantum circuits where the two input operations are differently ordered. The latter may be seen as a generalization of the quantum switch, allowing a physical interpretation for pure two-slot superchannels. An immediate corollary is that purifiable bipartite processes cannot violate device-independent causal inequalities.

Quantum circuit cutting with maximum-likelihood tomography

We introduce maximum-likelihood fragment tomography (MLFT) as an improved circuit cutting technique for running clustered quantum circuits on quantum devices with a limited number of qubits. In addition to minimizing the classical computing overhead of circuit cutting methods, MLFT finds the most likely probability distribution for the output of a quantum circuit, given the measurement data obtained from the circuit’s fragments. We demonstrate the benefits of MLFT for accurately estimating the output of a fragmented quantum circuit with numerical experiments on random unitary circuits. Finally, we show that circuit cutting can estimate the output of a clustered circuit with higher fidelity than full circuit execution, thereby motivating the use of circuit cutting as a standard tool for running clustered circuits on quantum hardware.

Scrambling and Lyapunov exponent in spatially extended systems

Scrambling of information in a quantum many-body system, quantified by the out-of-time-ordered correlator (OTOC), is a key manifestation of quantum chaos. A regime of exponential growth in the OTOC, characterized by a Lyapunov exponent, has so far mostly been observed in systems with a high-dimensional local Hilbert space and in weakly-coupled systems. Here, we propose a general criterion for the existence of a well-defined regime of exponential growth of the OTOC in spatially extended systems with local interactions. In such systems, we show that a parametrically long period of exponential growth requires the butterfly velocity to be much larger than the Lyapunov exponent times a microscopic length scale, such as the lattice spacing. As an explicit example, we study a random unitary circuit with tunable interactions. In this model, we show that in the weakly interacting limit the above criterion is satisfied, and there is a prolonged window of exponential growth. Our results are based on numerical simulations of both Clifford and universal random circuits supported by an analytical treatment.

Void formation in operator growth, entanglement, and unitarity

The structure of the Heisenberg evolution of operators plays a key role in explaining diverse processes in quantum many-body systems. In this paper, we discuss a new universal feature of operator evolution: an operator can develop a void during its evolution, where its nontrivial parts become separated by a region of identity operators. Such processes are present in both integrable and chaotic systems, and are required by unitarity. We show that void formation has important implications for unitarity of entanglement growth and generation of mutual information and multipartite entanglement. We study explicitly the probability distributions of void formation in a number of unitary circuit models, and conjecture that in a quantum chaotic system the distribution is given by the one we find in random unitary circuits, which we refer to as the random void distribution. We also show that random unitary circuits lead to the same pattern of entanglement growth for multiple intervals as in (1 + 1)-dimensional holographic CFTs after a global quench, which can be used to argue that the random void distribution leads to maximal entanglement growth.

Multipole conservation laws and subdiffusion in any dimension.

Subdiffusion is a generic feature of chaotic many-body dynamics with multipole conservation laws and subsystem symmetries. We numerically study this subdiffusive dynamics, using quantum automaton random unitary circuits, in a broad range of models including one-dimensional models with dipole and quadrupole conservation, two-dimensional models with dipole conservation, and two-dimensional models with subsystem symmetry on the triangular lattice. Our results are in complete agreement with recent hydrodynamic predictions for such theories.

Entanglement Phase Transitions in Measurement-Only Dynamics

Unitary circuits subject to repeated projective measurements can undergo an entanglement phase transition (EPT) as a function of the measurement rate. This transition is generally understood in terms of a competition between the scrambling effects of unitary dynamics and the disentangling effects of measurements. We find that, surprisingly, EPTs are possible even in the absence of scrambling unitary dynamics, where they are best understood as arising from measurements alone. This motivates us to introduce \emph{measurement-only models}, in which the "scrambling" and "un-scrambling" effects driving the EPT are fundamentally intertwined and cannot be attributed to physically distinct processes. This represents a novel form of an EPT, conceptually distinct from that in hybrid unitary-projective circuits. We explore the entanglement phase diagrams, critical points, and quantum code properties of some of these measurement-only models. We find that the principle driving the EPTs in these models is \emph{frustration}, or mutual incompatibility, of the measurements. Suprisingly, an entangling (volume-law) phase is the generic outcome when measuring sufficiently long but still local ($\gtrsim 3$-body) operators. We identify a class of exceptions to this behavior ("bipartite ensembles") which cannot sustain an entangling phase, but display dual area-law phases, possibly with different kinds of quantum order, separated by self-dual critical points. Finally, we introduce a measure of information spreading in dynamics with measurements and use it to demonstrate the emergence of a statistical light-cone, despite the non-locality inherent to quantum measurements.

Postselection-Free Entanglement Dynamics via Spacetime Duality.

The dynamics of entanglement in "hybrid" nonunitary circuits (for example, involving both unitary gates and quantum measurements) has recently become an object of intense study. A major hurdle toward experimentally realizing this physics is the need to apply postselection on random measurement outcomes in order to repeatedly prepare a given output state, resulting in an exponential overhead. We propose a method to sidestep this issue in a wide class of nonunitary circuits by taking advantage of spacetime duality. This method maps the purification dynamics of a mixed state under nonunitary evolution onto a particular correlation function in an associated unitary circuit. This translates to an operational protocol which could be straightforwardly implemented on a digital quantum simulator. We discuss the signatures of different entanglement phases, and demonstrate examples via numerical simulations. With minor modifications, the proposed protocol allows measurement of the purity of arbitrary subsystems, which could shed light on the properties of the quantum error correcting code formed by the mixed phase in this class of hybrid dynamics.

Solving quantum statistical mechanics with variational autoregressive networks and quantum circuits

We extend the ability of an unitary quantum circuit by interfacing it with a classical autoregressive neural network. The combined model parametrizes a variational density matrix as a classical mixture of quantum pure states, where the autoregressive network generates bitstring samples as input states to the quantum circuit. We devise an efficient variational algorithm to jointly optimize the classical neural network and the quantum circuit to solve quantum statistical mechanics problems. One can obtain thermal observables such as the variational free energy, entropy, and specific heat. As a byproduct, the algorithm also gives access to low energy excitation states. We demonstrate applications of the approach to thermal properties and excitation spectra of the quantum Ising model with resources that are feasible on near-term quantum computers.

Growth of genuine multipartite entanglement in random unitary circuits

We study the growth of genuine multipartite entanglement in random quantum circuit models, which include random unitary circuit models and the random Clifford circuit. We find that for the random Clifford circuit, the growth of multipartite entanglement remains slower in comparison to the random unitary case. However, the final saturation value of multipartite entanglement is almost the same in both cases. The behavior is then compared to the genuine multipartite entanglement obtained in random matrix product states with a moderately high bond dimension. We then relate the behavior of multipartite entanglement to other global properties of the system, viz. the delocalization of the many-body wavefunctions in Hilbert space. Along with this, we analyze the robustness of such highly entangled quantum states obtained through random unitary dynamics under weak measurements.

Variational quantum algorithm for nonequilibrium steady states

We propose a quantum-classical hybrid algorithm to simulate the non-equilibrium steady state of an open quantum many-body system, named the dissipative-system Variational Quantum Eigensolver (dVQE). To employ the variational optimization technique for a unitary quantum circuit, we map a mixed state into a pure state with a doubled number of qubits and design the unitary quantum circuit to fulfill the requirements for a density matrix. This allows us to define a cost function that consists of the time evolution generator of the quantum master equation. Evaluation of physical observables is, in turn, carried out by a quantum circuit with the original number of qubits. We demonstrate our dVQE scheme by both numerical simulation on a classical computer and actual quantum simulation that makes use of the device provided in Rigetti Quantum Cloud Service.

Nonlinear entanglement growth in inhomogeneous space-times

Entanglement has become central for the characterization of quantum matter both in and out of equilibrium. In a dynamical context entanglement exhibits universal linear temporal growth in generic systems, which stems from the underlying linear light cones as they occur in planar geometries. Inhomogeneous spacetimes can lead, however, to strongly bent trajectories. While such bent trajectories crucially impact correlation spreading and therefore the light-cone structure, it has remained elusive how this influences the entanglement dynamics. In this work we investigate the real-time evolution of the entanglement entropy in one-dimensional quantum systems after quenches which change the underlying spacetime background of the Hamiltonian. Concretely, we focus on the Rindler space describing the spacetime in close vicinity to a black hole. As a main result we find that entanglement grows sublinearly in a generic fashion both for interacting and noninteracting quantum matter. We further observe that the asymptotic relaxation becomes exponential, as opposed to algebraic for planar Minkowski spacetimes, and that in the vicinity of the black hole the relaxation time for large subsystems becomes independent of the subsystem size. We study entanglement dynamics both for the case of noninteracting fermions, allowing for exact numerical solutions, and for random unitary circuits representing a paradigmatic class of ergodic systems.

Entanglement Membrane in Chaotic Many-Body Systems

In certain analytically-tractable quantum chaotic systems, the calculation of out-of-time-order correlation functions, entanglement entropies after a quench, and other related dynamical observables, reduces to an effective theory of an ``entanglement membrane'' in spacetime. These tractable systems involve an average over random local unitaries defining the dynamical evolution. We show here how to make sense of this membrane in more realistic models, which do not involve an average over random unitaries. Our approach relies on introducing effective pairing degrees of freedom in spacetime, describing a pairing of forward and backward Feynman trajectories, inspired by the structure emerging in random unitary circuits. This provides a framework for applying ideas of coarse-graining to dynamical quantities in chaotic systems. We apply the approach to some translationally invariant Floquet spin chains studied in the literature. We show that a consistent line tension may be defined for the entanglement membrane, and that there are qualitative differences in this tension between generic models and ``dual-unitary'' circuits. These results allow scaling pictures for out-of-time-order correlators and for entanglement to be taken over from random circuits to non-random Floquet models. We also provide an efficient numerical algorithm for determining the entanglement line tension in 1+1D.

Hybrid quantum variational algorithm for simulating open quantum systems with near-term devices

Hybrid quantum-classical (HQC) algorithms make it possible to use near-term quantum devices supported by classical computational resources by useful control schemes. In this paper, we develop an HQC algorithm using an efficient variational optimization approach to simulate open system dynamics under the Noisy-Intermediate Scale Quantum computer. Using the time-dependent variational principle (TDVP) method and extending it to McLachlan TDVP for density matrix which involves minimization of Frobenius norm of the error, we apply the unitary quantum circuit to obtain the time evolution of the open quantum system in the Lindblad formalism. Finally, we illustrate the use of our methods with detailed examples which are in good agreement with analytical calculations.

Measurement-induced quantum criticality under continuous monitoring

We investigate entanglement phase transitions from volume-law to area-law entanglement in a quantum many-body state under continuous position measurement on the basis of the quantum trajectory approach. We find the signatures of the transitions as peak structures in the mutual information as a function of measurement strength, as previously reported for random unitary circuits with projective measurements. At the transition points, the entanglement entropy scales logarithmically and various physical quantities scale algebraically, implying emergent conformal criticality, for both integrable and nonintegrable one-dimensional interacting Hamiltonians; however, such transitions have been argued to be absent in noninteracting regimes in some previous studies. With the aid of $U(1)$ symmetry in our model, the measurement-induced criticality exhibits a spectral signature resembling a Tomonaga-Luttinger liquid theory from symmetry-resolved entanglement. These intriguing critical phenomena are unique to steady-state regimes of the conditional dynamics at the single-trajectory level, and are absent in the unconditional dynamics obeying the Lindblad master equation, in which the system ends up with the featureless, infinite-temperature mixed state. We also propose a possible experimental setup to test the predicted entanglement transition based on the subsystem particle-number fluctuations. This quantity should readily be measured by the current techniques of quantum gas microscopy and is in practice easier to obtain than the entanglement entropy itself.

Scrambling in random unitary circuits: Exact results

We study the scrambling of quantum information in local random unitary circuits by focusing on the tripartite information proposed by Hosur et al. We provide exact results for the averaged R\'enyi-$2$ tripartite information in two cases: (i) the local gates are Haar random and (ii) the local gates are dual-unitary and randomly sampled from a single-site Haar-invariant measure. We show that the latter case defines a one-parameter family of circuits, and prove that for a "maximally chaotic" subset of this family quantum information is scrambled faster than in the Haar-random case. Our approach is based on a standard mapping onto an averaged folded tensor network, that can be studied by means of appropriate recurrence relations. By means of the same method, we also revisit the computation of out-of-time-ordered correlation functions, re-deriving known formulae for Haar-random unitary circuits, and presenting an exact result for maximally chaotic random dual-unitary gates.

Instantaneous braids and Dehn twists in topologically ordered states

A defining feature of topologically ordered states of matter is the existence of locally indistinguishable states on spaces with non-trivial topology. These degenerate states form a representation of the mapping class group (MCG) of the space, which is generated by braids of defects or anyons, and by Dehn twists along non-contractible cycles. These operations can be viewed as fault-tolerant logical gates in the context of topological quantum error correcting codes and topological quantum computation. Here we show that braids and Dehn twists can in general be implemented by a constant depth quantum circuit, with a depth that is independent of code distance $d$ and system size. The circuit consists of a constant depth local quantum circuit (LQC) implementing a local geometry deformation of the quantum state, followed by a permutation on (relabelling of) the qubits. The permutation requires permuting qubits that are separated by a distance of order $d$; it can be implemented by collective classical motion of mobile qubits or as a constant depth circuit using long-range SWAP operations (with a range set by $d$) on immobile qubits. Applying these results to certain non-Abelian quantum error correcting codes demonstrates how universal logical gate sets can be implemented on encoded qubits using only constant depth unitary circuits.