For an isolated generic quantum system out of equilibrium, the long time average of observables is given by the diagonal ensemble, i.e. the mixed state with the same probability for energy eigenstates as the initial state but without coherences between different energies. In this work we present a method to approximate the diagonal ensemble using tensor networks. Instead of simulating the real time evolution, we adapt a filtering scheme introduced earlier in [Phys. Rev. B 101, 144305 (2020)] to this problem. We analyze the performance of the method on a non-integrable spin chain, for which we observe that local observables converge towards thermal values polynomially with the inverse width of the filter.
We propose a scheme for identifying an unknown Bell diagonal state. In our scheme the measurements are performed on the probe qubits instead of the Bell diagonal state. The distinguished advantage is that the quantum state of the evolved Bell diagonal state ensemble plus probe states will still collapse on the original Bell diagonal state ensemble after the measurement on probe states, i.e. our identification is quantum-state nondestructive. It is also shown finally how to realize our scheme in the framework of cavity electrodynamics.
The diagonal ensemble is the infinite time average of a quantum state following unitary dynamics. In analogy to the time average of a classical phase space dynamics, it is intimately related to the ergodic properties of the quantum system giving information on the spreading of the initial state in the eigenstates of the Hamiltonian. In this work we apply a concept from quantum information, known as total correlations, to the diagonal ensemble. Forming an upper-bound on the multipartite entanglement, it quantifies the combination of both classical and quantum correlations in a mixed state. We g…
The intriguing phenomenon of many-body localization (MBL) has attracted significant interest recently, but a complete characterization is still lacking. In this work, we introduce the total correlations, a concept from quantum information theory capturing multi-partite correlations, to the study of this phenomenon. We demonstrate that the total correlations of the diagonal ensemble provides a meaningful diagnostic tool to pin-down, probe, and better understand the MBL transition and ergodicity breaking in quantum systems. In particular, we show that the total correlations has sub-linear depend…
A density operator, $ρ= {P}_{α} |α> <α| + {P}_{β} |β> <β|$, with ${P}_{α}$ and ${P}_{β}$ linearly independent normalized wave functions, must be traced normalized, so ${P}_{β} = 1 - {P}_{α}$. However, unless $<α|β> = 0$, ${P}_{α}$ and ${P}_{β}$ cannot be interpreted as probabilities of finding $|α>$ and $|β>$ respectively. We show that a density matrix comprised of two (${P}_{α}$ and ${P}_{β}$ nonzero) non-orthogonal projectors have unique spectral decomposition into diagonal form with orthogonal projectors. Only then, according to axioms of Von Neumann and Fock, can we have probability interp…
We study quantum tomography from a continuous measurement record obtained by measuring expectation values of a set of Hermitian operators obtained from unitary evolution of an initial observable. For this purpose, we consider the application of a random unitary, diagonal in a fixed basis at each time step and quantify the information gain in tomography using Fisher information of the measurement record and the Shannon entropy associated with the eigenvalues of covariance matrix of the estimation. Surprisingly, very high fidelity of reconstruction is obtained using random unitaries diagonal in …
Nonlocal properties of an ensemble of diagonal random unitary matrices of order $N^2$ are investigated. The average Schmidt strength of such a bipartite diagonal quantum gate is shown to scale as $\log N$, in contrast to the $\log N^2$ behavior characteristic to random unitary gates. Entangling power of a diagonal gate $U$ is related to the von Neumann entropy of an auxiliary quantum state $ρ=AA^{\dagger}/N^2$, where the square matrix $A$ is obtained by reshaping the vector of diagonal elements of $U$ of length $N^2$ into a square matrix of order $N$. This fact provides a motivation to study t…
A new scheme is proposed which will permit electron spin resonance pulse techniques to be used to realize a quantum computer with a 100 qbits, or more. The computation is performed on effective pure states which correspond to off-diagonal blocks of the density matrix. Described is a scheme which very efficiently performs the preparation stage and which permits ``pseudo-projective measurement'' to be made on the output. With such measurements all members of the ensemble remain coherent.
Pure quantum states are often approximately encoded as classical bit strings such as those representing probability amplitudes and those describing circuits that generate the quantum states. The crucial quantity is the minimum length of classical bit strings from which the original pure states are approximately reconstructible. We derive asymptotically tight bounds on the minimum bit length required for probabilistic encodings with which one can approximately reconstruct the original pure state as an ensemble of the quantum states encoded in classical strings. We also show that such a probabil…
We study quantum quenches in the XXZ spin-$1/2$ Heisenberg chain from families of ferromagnetic and antiferromagnetic initial states. Using Bethe ansatz techniques, we compute short-range correlators in the complete generalized Gibbs ensemble (GGE), which takes into account all local and quasi-local conservation laws. We compare our results to exact diagonalization and numerical linked cluster expansion calculations for the diagonal ensemble finding excellent agreement and thus providing a very accurate test for the validity of the complete GGE. Furthermore, we compute the diagonal entropy in …
We study the fidelity decay of the $k$-body embedded ensembles of random matrices for bosons distributed over two single-particle states. Fidelity is defined in terms of a reference Hamiltonian, which is a purely diagonal matrix consisting of a fixed one-body term and includes the diagonal of the perturbing $k$-body embedded ensemble matrix, and the perturbed Hamiltonian which includes the residual off-diagonal elements of the $k$-body interaction. This choice mimics the typical mean-field basis used in many calculations. We study separately the cases $k=2$ and $3$. We compute the ensemble-ave…
Diagonal quantum circuits are quantum circuits comprising only diagonal gates in the computational basis. In spite of a classical feature of diagonal quantum circuits in the sense of commutativity of all gates, their computational power is highly likely to outperform classical one and they are exploited for applications in quantum informational tasks. We review computational power of diagonal quantum circuits and their applications. We focus on the computational power of instantaneous quantum polynomial-time (IQP) circuits, which are a special type of diagonal quantum circuits. We then review …
A simple method is proposed to prepare conveniently the effective pure state |00...0><0...00| with any number of qubits in a quantum ensemble. The preparation is based on the temporal averaging (Knill, Chuang, and Laflamme, Phys.Rev.A 57, 3348(1998)). The quantum circuit to prepare the effective pure state is designed in a unified and systematical form and is explicitly decomposed completely into a product of a series of one-qubit quantum gates and the two-qubit diagonal quantum gates. The preparation could be programmed and implemented conveniently on an NMR quantum computer.
We present an efficient algorithm for twirling a multi-qudit quantum state. The algorithm can be used for approximating the twirling operation in an ensemble of physical systems in which the systems cannot be individually accessed. It can also be used for computing the twirled density matrix on a classical computer. The method is based on a simple non-unitary operation involving a random unitary. When applying this basic building block iteratively, the mean squared error of the approximation decays exponentially. In contrast, when averaging over random unitary matrices the error decreases only…
Ensemble Engineering to Overcome Destructive Cancellation in Quantum Measurements
Purdue University（普渡大学）；North Carolina State University（北卡罗来纳州立大学）；National Center for Computational Sciences, Oak Ridge National Laboratory（橡树岭国家实验室国家计算科学中心）
Energy level statistics of Hermitian random matrices $\hat H$ with Gaussian independent random entries $H_{i\geq j}$ is studied for a generic ensemble of almost diagonal random matrices with $ <|H_{ii}|^{2} > \sim 1$ and $<|H_{i\neq j}|^{2} >= b {\cal F}(|i-j|) \ll 1$. We perform a regular expansion of the spectral form-factor $K(τ) = 1 + b K_{1}(τ) + b^{2} K_{2}(τ) + ... $ in powers of $b \ll 1$ with the coefficients $K_{m}(τ)$ that take into account interaction of (m+1) energy levels. To calculate $K_{m}(τ)$, we develop a diagrammatic technique which is based on the Trotter formula and on th…
Geometry of quantum states in random matrix ensembles and the chaos-integrability transition
Indian Institute of Technology, Kanpur（坎普尔印度理工学院）；Gwangju Institute of Science and Technology（光州科学技术院）；Asia Pacific Center for Theoretical Physics（亚太理论物理中心）
We investigate protocols for generating a state $t$-design by using a fixed separable initial state and a diagonal-unitary $t$-design in the computational basis, which is a $t$-design of an ensemble of diagonal unitary matrices with random phases as their eigenvalues. We first show that a diagonal-unitary $t$-design generates a $O(1/2^N)$-approximate state $t$-design, where $N$ is the number of qubits. We then discuss a way of improving the degree of approximation by exploiting non-diagonal gates after applying a diagonal-unitary $t$-design. We also show that it is necessary and sufficient to …
We study the density of states (DOS) for disordered systems whose spectral statistics can be described by a Gaussian ensemble of almost diagonal Hermitian random matrices. The matrices have independent random entries $ H_{i \geq j} $ with small off-diagonal elements: $ <|H_{i \neq j}|^{2} > \ll <|H_{ii}|^{2} > \sim 1 $. Using the recently suggested method of a {\it virial expansion in the number of interacting energy levels} (Journ.Phys.A {\bf 36}, 8265), we calculate the leading correction to the Poissonian DOS in the cases of the Gaussian Orthogonal and Unitary Ensembles. We apply the genera…
Motivated by studies of typical properties of quantum states in statistical mechanics, we introduce phase-random states, an ensemble of pure states with fixed amplitudes and uniformly distributed phases in a fixed basis. We first show that canonical states typically appear in subsystems of phase-random states. We then investigate the simulatability of phase-random states, which is directly related to that of time evolution in closed systems, by studying their entanglement properties. We find that starting from a separable state, time evolutions under Hamiltonians composed of only separable eig…
We study efficient generations of random diagonal-unitary matrices, an ensemble of unitary matrices diagonal in a given basis with randomly distributed phases for their eigenvalues. Despite the simple algebraic structure, they cannot be achieved by quantum circuits composed of a few-qubit diagonal gates. We introduce diagonal-unitary $t$-designs and present two quantum circuits that implement diagonal-unitary $2$-designs with the computational basis in $N$-qubit systems. One is composed of single-qubit diagonal gates and controlled-phase gates with randomized phases, which achieves an exact di…
We present a systematic construction of probes into the dynamics of isospectral ensembles of Hamiltonians by the notion of Isospectral twirling, expanding the scopes and methods of ref.[1]. The relevant ensembles of Hamiltonians are those defined by salient spectral probability distributions. The Gaussian Unitary Ensembles (GUE) describes a class of quantum chaotic Hamiltonians, while spectra corresponding to the Poisson and Gaussian Diagonal Ensemble (GDE) describe non chaotic, integrable dynamics. We compute the Isospectral twirling of several classes of important quantities in the analysis …
We propose a quantum-classical hybrid algorithm of the power method, here dubbed as quantum power method, to evaluate $\hat{\cal H}^n |ψ\rangle$ with quantum computers, where $n$ is a nonnegative integer, $\hat{\cal H}$ is a time-independent Hamiltonian of interest, and $|ψ\rangle$ is a quantum state. We show that the number of gates required for approximating $\hat{\cal H}^n$ scales linearly in the power and the number of qubits, making it a promising application for near term quantum computers. Using numerical simulation, we show that the power method can control systematic errors in approxi…
The unitary dynamics of isolated quantum systems does not allow a pure state to thermalize. Because of that, if an isolated quantum system equilibrates, it will do so to the predictions of the so-called "diagonal ensemble" $ρ_{DE}$. Building on the intuition provided by Jaynes' maximum entropy principle, in this paper we present a novel technique to generate progressively better approximations to $ρ_{DE}$. As an example, we write down a hierarchical set of ensembles which can be used to describe the equilibrium physics of small isolated quantum systems, going beyond the "thermal ansatz" of Gib…
We investigate the Joule expansion of nonintegrable quantum systems that contain bosons or spinless fermions in one-dimensional lattices. A barrier initially confines the particles to be in half of the system in a thermal state described by the canonical ensemble and is removed at time $t = 0$. We investigate the properties of the time-evolved density matrix, the diagonal ensemble density matrix and the corresponding canonical ensemble density matrix with an effective temperature determined by the total energy conservation using exact diagonalization. The weights for the diagonal ensemble and …
We numerically investigate statistical ensembles for the occupations of eigenstates of an isolated quantum system emerging as a result of quantum quenches. The systems investigated are sparse random matrix Hamiltonians and disordered lattices. In the former case, the quench consists of sudden switching-on the off-diagonal elements of the Hamiltonian. In the latter case, it is sudden switching-on of the hopping between adjacent lattice sites. The quench-induced ensembles are compared with the so-called "quantum micro-canonical" (QMC) ensemble describing quantum superpositions with fixed energy …
Spontaneous symmetry breaking originats in quantum mechanical measurement of the relevant observable defining the physical situation, order parameter is the average of this observable. A modification is made on the random-phase postulate validating the ensemble description. Off-diagonal long-range order, macroscopic wavefunction and interference effects in many-particle systems present when there is a so-called nucleation of quantum state, which is proposed to be the origin of spontaneous gauge symmetry breaking, for which nonconservation of particle number N is not essential. The approach bas…
We introduce structured random matrix ensembles, constructed to model many-body quantum systems with local interactions. These ensembles are employed to study equilibration of isolated many-body quantum systems, showing that rather complex matrix structures, well beyond Wigner's full or banded random matrices, are required to faithfully model equilibration times. Viewing the random matrices as connectivities of graphs, we analyse the resulting network of classical oscillators in Hilbert space with tools from network theory. One of these tools, called the maximum flow value, is found to be an e…
