Random Unitary Operators Research Articles

Comparison of quantum advantage experiments using random circuit sampling

Random circuit sampling, the task of sampling bit strings from a random unitary operator, has been implemented to demonstrate quantum advantage on the Sycamore quantum processor with 53 qubits and on the Zuchongzhi quantum processor with 56 and 61 qubits. Recently, it was claimed that classical computers using tensor network simulation could catch on to current noisy quantum processors for random circuit sampling. While the linear cross-entropy benchmark fidelity was used to certify all these claims, it may not capture statistical properties of outputs in detail. Here, we compare the bit strings sampled from classical computers using tensor network simulation by Pan et al. [F. Pan, K. Chen, and P. Zhang, Phys. Rev. Lett. 129, 090502 (2022)] and by Kalachev et al. [G. Kalachev, P. Panteleev, P. Zhou, and M.-H. Yung, arXiv:2112.15083] with the bit strings from the Sycamore quantum processor. It is shown that all of Kalachev et al.'s samples passed the NIST random number tests. The heat maps of bit strings show that Pan et al.'s and Kalachev et al.'s samples are quite different from the Sycamore or Zuchongzhi samples. The analysis with the Marchenko-Pastur distribution and the Wasssertein distances demonstrates that Kalachev et al.'s samples are statistically closer to the Sycamore samples than Pan et al.'s while the three datasets have similar values for the linear cross-entropy fidelity. Our finding implies that further study is needed to certify or beat the claims of quantum advantage using random circuit sampling.

Faithfulness and Sensitivity for Ancilla-Assisted Process Tomography.

A system-ancilla bipartite state capable of containing the complete information of an unknown quantum channel acting on the system is called faithful. In this work, we extend the applicability and generality of faithfulness significantly by introducing its local variant and examining their relationship when applied to various classes of quantum channels. In doing so, we discovered that, in the original proof by D'Ariano and Presti, only sufficiency was shown, not the full equivalence between faithfulness of state and invertibility of the corresponding Jamiołkowski map. We complete the proof by showing necessity and examine how far this characterization of faithfulness can be generalized by applying it to various classes of quantum channels. We also explore a more general notion we call sensitivity, the property of quantum state being altered by any nontrivial action of quantum channel. We study their relationship by characterizing both properties for important classes of quantum channels such as unital channels, random unitary operations, and unitary operations. Unexpected (non)equivalence results among them shed light on the structure of quantum channels by showing that we need only two classes of quantum states for characterizing quantum states faithful or sensitive to various subclasses of quantum channels.

Hilbert space fragmentation produces an effective attraction between fractons

Fracton systems exhibit restricted mobility of their excitations due to the presence of higher-order conservation laws. Here we study the time evolution of a one-dimensional fracton system with charge and dipole moment conservation using a random unitary circuit description. Previous work has shown that when the random unitary operators act on four or more sites, an arbitrary initial state eventually thermalizes via a universal subdiffusive dynamics. In contrast, a system evolving under three-site gates fails to thermalize due to strong ``fragmentation'' of the Hilbert space. Here we show that three-site gate dynamics causes a given initial state to evolve toward a highly nonthermal state on a timescale consistent with Brownian diffusion. Strikingly, the dynamics produces an effective attraction between isolated fractons or between a single fracton and the boundaries of the system, as in the Casimir effect of quantum electrodynamics. We show how this attraction can be understood by exact mapping to a simple classical statistical mechanics problem, which we solve exactly for the case of an initial state with either one or two fractons.

Hamiltonian tomography by the quantum quench protocol with random noise

In this article, we introduce a framework for Hamiltonian tomography of multi-qubit systems with random noise. We adopt the quantum quench protocol to reconstruct a many-body Hamiltonian by local measurements that are distorted by random unitary operators and time uncertainty. In particular, we consider a transverse field Ising Hamiltonians describing interactions of two spins $1/2$ and three-qubit Hamiltonians of a heteronuclear system within the radio-frequency field. For a sample of random Hamiltonians, we report the fidelity of reconstruction versus the amount of noise quantified by two parameters. Furthermore, we discuss the correlation between the accuracy of Hamiltonian tomography and the number of pairs of quantum states involved in the framework. The results provide valuable insight into the robustness of the protocol against random noise.

Quantum State Tomography of Four-Level Systems with Noisy Measurements

In this article, we investigate the problem of state reconstruction of four-level quantum systems. A realistic scenario is considered with measurement results distorted by random unitary operators. Two frames which define injective measurements are applied and compared. By introducing arbitrary rotations, we can test the performance of the framework versus the amount of experimental noise. The results of numerical simulations are depicted on graphs and discussed. In particular, a class of entangled states is reconstructed. The concurrence is used as a figure of merit in order to quantify how well entanglement is preserved through noisy measurements.

Characterizing Correlations via Skew Information

The skew information has important applications in quantum theory. In this paper, a measure for correlations in terms of the skew information is introduced and discussed. Its fundamental properties are obtained, which include the measure is locally unitary invariant and convex on the set of states with the fixed marginal state, moreover, it decreases under local random unitary operation. Lastly, we propose an entanglement criterion in terms of the quantum skew information and illustrate the significance of this criterion by some examples.

The Cost of Randomness for Converting a Tripartite Quantum State to be Approximately Recoverable

We introduce and analyze a task in which a tripartite quantum state is transformed to an approximately recoverable state by a randomizing operation on one of the three subsystems. We consider cases where the initial state is a tensor product of $n$ copies of a tripartite state $\rho^{ABC}$, and is transformed by a random unitary operation on $A^n$ to another state which is approximately recoverable from its reduced state on $A^nB^n$ (Case 1) or $B^nC^n$ (Case 2). We analyze the minimum cost of randomness per copy required for the task in an asymptotic limit of infinite copies and vanishingly small error of recovery, mainly focusing on the case of pure states. We prove that the minimum cost in Case 1 is equal to the Markovianizing cost of the state, for which a single-letter formula is known. With an additional requirement on the convergence speed of the recovery error, we prove that the minimum cost in Case 2 is also equal to the Markovianizing cost. Our results have an application for distributed quantum computation.

Symmetrizing cost of quantum states

We introduce and analyze a task that we call symmetrization, in which a state of a quantum system, associated with a symmetry group, is transformed by a random unitary operation to a symmetric state. Each element of the unitary ensemble is required to be symmetry preserving, in the sense that it keeps the set of symmetric states invariant. We consider an asymptotic limit of infinitely many copies and vanishingly small error, and analyze the symmetrizing cost, that is, the minimum cost of randomness per copy required for symmetrization. We prove that the symmetrizing cost of an arbitrary quantum state is equal to the relative entropy of frameness, thereby providing it with a direct operational meaning.

Operator entanglement entropy of the time evolution operator in chaotic systems

We study the growth of the operator entanglement entropy (EE) of the time evolution operator in chaotic, many-body localized and Floquet systems. In the random field Heisenberg model we find a universal power law growth of the operator EE at weak disorder, a logarithmic growth at strong disorder, and extensive saturation values in both cases. In a Floquet spin model, the saturation value after an initial linear growth is identical to the value of a random unitary operator (the Page value). We understand these properties by mapping the operator EE to a global quench problem evolved with a similar parent-Hamiltonian in an enlarged Hilbert space with the same chaotic, MBL and Floquet properties as the original Hamiltonian. The scaling and saturation properties reflect the spreading of the state EE of the corresponding time evolution. We conclude that the EE of the evolution operator should characterize the propagation of information in these systems.

Markovianizing Cost of Tripartite Quantum States

We introduce and analyze a task that we call Markovianization, in which a tripartite quantum state is transformed to a quantum Markov chain by a randomizing operation on one of the three subsystems. We consider cases where the initial state is the tensor product of $n$ copies of a tripartite state $\rho^{ABC}$, and is transformed to a quantum Markov chain conditioned by $B^n$ with a small error, using a random unitary operation on $A^n$. In an asymptotic limit of infinite copies and vanishingly small error, we analyze the Markovianizing cost, that is, the minimum cost of randomness per copy required for Markovianization. For tripartite pure states, we derive a single-letter formula for the Markovianizing costs. Counterintuitively, the Markovianizing cost is not a continuous function of states, and can be arbitrarily large even if the state is close to a quantum Markov chain. Our results have an application in analyzing the cost of resources for simulating a bipartite unitary gate by local operations and classical communication.

Passive states as optimal inputs for single-jump lossy quantum channels

The passive states of a quantum system minimize the average energy among all the states with a given spectrum. We prove that passive states are the optimal inputs of single-jump lossy quantum channels. These channels arise from a weak interaction of the quantum system of interest with a large Markovian bath in its ground state, such that the interaction Hamiltonian couples only consecutive energy eigenstates of the system. We prove that the output generated by any input state $\rho$ majorizes the output generated by the passive input state $\rho_0$ with the same spectrum of $\rho$. Then, the output generated by $\rho$ can be obtained applying a random unitary operation to the output generated by $\rho_0$. This is an extension of De Palma et al., IEEE Trans. Inf. Theory 62, 2895 (2016), where the same result is proved for one-mode bosonic Gaussian channels. We also prove that for finite temperature this optimality property can fail already in a two-level system, where the best input is a coherent superposition of the two energy eigenstates.

Quantifying Correlations via the Wigner-Yanase-Dyson Skew Information

In this paper, based on a discussion about the Wigner-Yanase-Dyson (WYD) skew information, the measure Fa,α(ρab) for correlations in terms of the WYD skew information is introduced and discussed. The following conclusions are obtained. For a classical-quantum state ρab, Fa,α(ρab)=0 if and only if ρab is a product state; Fa,α(ρab) is locally unitary invariant and convex on the set of states with the fixed marginal ρa; Fa,α(ρab) decreases under local random unitary operation on Hb; For a quantum-classical state ρab, Fa,α(ρab) decreases under local operation on Hb; Lastly, Fa,α(ρab) is computed for the pure states and the Bell-diagonal states, respectively.

Random unitary evolution model of quantum Darwinism with pure decoherence

We study the behavior of Quantum Darwinism [W.H. Zurek, Nat. Phys. 5, 181 (2009)] within the iterative, random unitary operations qubit-model of pure decoherence [J. Novotný, G. Alber, I. Jex, New J. Phys. 13, 053052 (2011)]. We conclude that Quantum Darwinism, which describes the quantum mechanical evolution of an open system S from the point of view of its environment E, is not a generic phenomenon, but depends on the specific form of input states and on the type of S-E-interactions. Furthermore, we show that within the random unitary model the concept of Quantum Darwinism enables one to explicitly construct and specify artificial input states of environment E that allow to store information about an open system S of interest with maximal efficiency.

Spectral Transition for Random Quantum Walks on Trees

We define and analyze random quantum walks on homogeneous trees of degree $q\geq 3$. Such walks describe the discrete time evolution of a quantum particle with internal degree of freedom in $\C^q$ hopping on the neighboring sites of the tree in presence of static disorder. The one time step random unitary evolution operator of the particle depends on a unitary matrix $C\in U(q)$ which monitors the strength of the disorder. We prove for any $q$ that there exist open sets of matrices in $U(q)$ for which the random evolution has either pure point spectrum almost surely or purely absolutely continuous spectrum, thereby showing the existence of a spectral transition driven by $C\in U(q)$. For $q\in\{3,4\}$, we establish properties of the spectral diagram which provide a description of the spectral transition.

Percolation induced effects in two-dimensional coined quantum walks: analytic asymptotic solutions

Quantum walks on graphs can model physical processes and serve as efficient tools in quantum information theory. Once we admit random variations in the connectivity of the underlying graph, we arrive at the problem of percolation, where the long-time behaviour appears untreatable with direct numerical methods. We develop novel analytic methods based on the theory of random unitary operations which help us to determine explicitly the asymptotic dynamics of quantum walks on two-dimensional finite integer lattices with percolation. Based on this theory, we find new unexpected features of percolated walks like asymptotic position inhomogeneity or special directional symmetry breaking.

Irreversibility of a quantum walk induced by controllable decoherence employing random unitary operations

Quantum walk is different from random walk in reversibility and interference. Observation of the reduced reversibility in a realistic quantum walk is of scientific interest in understanding the unique quantum behavior. We propose an idea to experimentally investigate the decoherence-induced irreversibility of quantum walks with trapped ions in phase space via the average fidelity decay. By introducing two controllable decoherence sources, i.e., the phase damping channel (i.e., dephasing) and the high temperature amplitude reservoir (i.e., dissipation), in the intervals between the steps of quantum walk, we find that the high temperature amplitude reservoir shows more detrimental effects than the phase damping channel on quantum walks. Our study also shows that the average fidelity decay works better than the position variance for characterizing the transition from quantum walks to random walk. Experimental feasibility to monitor the irreversibility is justified using currently available techniques.

Stone's representation theorem of a group of random unitary operators on complete complex random inner product modules

Let [ a,b ] be a finite real interval and ( S ,‖·‖) a complete random normed module endowed with the ( e , λ )-topology. We first introduce the Riemann integral for abstract-valued functions from [ a,b ] to S and give a sufficient condition for such a continuous function with the almost surely bounded range to be Riemannintegrable. Then we investigate the relation between random spectral measures and ordinary random measures. Finally, based on the above two preliminaries, we establish the Stone’s representation theorem of a group of random unitary operators on complete complex random inner product modules.

Disordered quantum walks in one lattice dimension

We study a spin-$\frac{1}{2}$12-particle moving on a one-dimensional lattice subject to disorder induced by a random, space-dependent quantum coin. The discrete time evolution is given by a family of random unitary quantum walk operators, where the shift operation is assumed to be deterministic. Each coin is an independent identically distributed random variable with values in the group of two-dimensional unitary matrices. We derive sufficient conditions on the probability distribution of the coins such that the system exhibits dynamical localization. Put differently, the tunneling probability between two lattice sites decays rapidly for almost all choices of random coins and after arbitrary many time steps with increasing distance. Our findings imply that this effect takes place if the coin is chosen at random from the Haar measure, or some measure continuous with respect to it, but also for a class of discrete probability measures which support consists of two coins, one of them being the Hadamard coin.

Asymptotic evolution of quantum walks on theN-cycle subject to decoherence on both the coin and position degrees of freedom

Consider a discrete-time quantum walk on the $N$-cycle subject to decoherence both on the coin and the position degrees of freedom. By examining the evolution of the density matrix of the system, we derive some new conclusions about the asymptotic behavior of the system. When $N$ is odd, the density matrix of the system tends, in the long run, to the maximally mixed state, independent of the initial state. When $N$ is even, although the behavior of the system is not necessarily asymptotically stationary, in this case too an explicit formulation is obtained of the asymptotic dynamics of the system. Moreover, this approach enables us to specify the limiting behavior of the mutual information, viewed as a measure of quantum entanglement between subsystems (coin and walker). In particular, our results provide efficient theoretical confirmation of the findings of previous authors, who arrived at their results through extensive numerical simulations. Our results can be attributed to an important theorem which, for a generalized random unitary operation, explicitly identifies the structure of all of its eigenspaces corresponding to eigenvalues of unit modulus.

Asymptotic evolution of random unitary operations

AbstractWe analyze the asymptotic dynamics of quantum systems resulting from large numbers of iterations of random unitary operations. Although, in general, these quantum operations cannot be diagonalized it is shown that their resulting asymptotic dynamics is described by a diagonalizable superoperator. We prove that this asymptotic dynamics takes place in a typically low dimensional attractor space which is independent of the probability distribution of the unitary operations applied. This vector space is spanned by all eigenvectors of the unitary operations involved which are associated with eigenvalues of unit modulus. Implications for possible asymptotic dynamics of iterated random unitary operations are presented and exemplified in an example involving random controlled-not operations acting on two qubits.