34 research outputs found
Learning t-doped stabilizer states
In this paper, we present a learning algorithm aimed at learning states
obtained from computational basis states by Clifford circuits doped with a
finite number t of non-Clifford gates. To tackle this problem, we introduce a
novel algebraic framework for t-doped stabilizer states by utilizing tools from
stabilizer entropy. Leveraging this new structure, we develop an algorithm that
uses sampling from the distribution obtained by squaring expectation values of
Pauli operators that can be obtained by Bell sampling on the state and its
conjugate in the computational basis. The algorithm requires resources of
complexity O(\exp(t)\poly(n)) and exhibits an exponentially small probability
of failure.Comment: L.L. and S.O. contributed equally to this wor
Isospectral twirling and quantum chaos
We show that the most important measures of quantum chaos, such as frame potentials, scrambling, Loschmidt echo and out-of-time-order correlators (OTOCs), can be described by the unified framework of the isospectral twirling, namely the Haar average of a k-fold unitary channel. We show that such measures can then always be cast in the form of an expectation value of the isospectral twirling. In literature, quantum chaos is investigated sometimes through the spectrum and some other times through the eigenvectors of the Hamiltonian generating the dynamics. We show that thanks to this technique, we can interpolate smoothly between integrable Hamiltonians and quantum chaotic Hamiltonians. The isospectral twirling of Hamiltonians with eigenvector stabilizer states does not possess chaotic features, unlike those Hamiltonians whose eigenvectors are taken from the Haar measure. As an example, OTOCs obtained with Clifford resources decay to higher values compared with universal resources. By doping Hamiltonians with non-Clifford resources, we show a crossover in the OTOC behavior between a class of integrable models and quantum chaos. Moreover, exploiting random matrix theory, we show that these measures of quantum chaos clearly distinguish the finite time behavior of probes to quantum chaos corresponding to chaotic spectra given by the Gaussian Unitary Ensemble (GUE) from the integrable spectra given by Poisson distribution and the Gaussian Diagonal Ensemble (GDE)
Isospectral twirling and quantum chaos
We show that the most important measures of quantum chaos like frame
potentials, scrambling, Loschmidt echo, and out-of-time-order correlators
(OTOCs) can be described by the unified framework of the isospectral twirling,
namely the Haar average of a -fold unitary channel. We show that such
measures can then be always cast in the form of an expectation value of the
isospectral twirling. In literature, quantum chaos is investigated sometimes
through the spectrum and some other times through the eigenvectors of the
Hamiltonian generating the dynamics. We show that, by exploiting random matrix
theory, these measures of quantum chaos clearly distinguish the finite time
profiles of probes to quantum chaos corresponding to chaotic spectra given by
the Gaussian Unitary Ensemble (GUE) from the integrable spectra given by
Poisson distribution and the Gaussian Diagonal Ensemble (GDE). On the other
hand, we show that the asymptotic values do depend on the eigenvectors of the
Hamiltonian. We see that the isospectral twirling of Hamiltonians with
eigenvectors stabilizer states does not possess chaotic features, unlike those
Hamiltonians whose eigenvectors are taken from the Haar measure. As an example,
OTOCs obtained with Clifford resources decay to higher values compared with
universal resources. Finally, we show a crossover in the OTOC behavior between
a class of integrable models and quantum chaos.Comment: Updated version with several new result
Transitions in entanglement complexity in random quantum circuits by measurements
Random Clifford circuits doped with non Clifford gates exhibit transitions to
universal entanglement spectrum statistics[1] and quantum chaotic behavior. In
[2] we proved that the injection of non Clifford gates into a -qubit
Clifford circuit drives the transition towards the universal value of the
purity fluctuations. In this paper, we show that doping a Clifford circuit with
single qubit non Clifford measurements is both necessary and sufficient
to drive the transition to universal fluctuations of the purity
Transitions in entanglement complexity in random quantum circuits by measurements
Random Clifford circuits doped with non Clifford gates exhibit transitions to universal entanglement spectrum statistics [1] and quantum chaotic behavior. In [2] we proved that the injection of Ω(n) non Clifford gates into a n-qubit Clifford circuit drives the transition towards the universal value of the purity fluctuations. In this paper, we show that doping a Clifford circuit with Ω(n) single qubit non Clifford measurements is both necessary and sufficient to drive the transition to universal fluctuations of the purity
Phase transition in Stabilizer Entropy and efficient purity estimation
Stabilizer Entropy (SE) quantifies the spread of a state in the basis of
Pauli operators. It is a computationally tractable measure of
non-stabilizerness and thus a useful resource for quantum computation. SE can
be moved around a quantum system, effectively purifying a subsystem from its
complex features. We show that there is a phase transition in the residual
subsystem SE as a function of the density of non-Clifford resources. This phase
transition has important operational consequences: it marks the onset of a
subsystem purity estimation protocol that requires
many queries to a circuit containing
non-Clifford gates that prepares the state from a stabilizer state. Thus, for
, it estimates the purity with polynomial resources and, for
highly entangled states, attains an exponential speed-up over the known
state-of-the-art algorithms
Quantum chaos is quantum
It is well known that a quantum circuit on N qubits composed of Clifford gates with the addition of k non Clifford gates can be simulated on a classical computer by an algorithm scaling as poly(N)exp (k)[1]. We show that, for a quantum circuit to simulate quantum chaotic behavior, it is both necessary and sufficient that k = Θ(N). This result implies the impossibility of simulating quantum chaos on a classical computer
On the practical usefulness of the Hardware Efficient Ansatz
Variational Quantum Algorithms (VQAs) and Quantum Machine Learning (QML)
models train a parametrized quantum circuit to solve a given learning task. The
success of these algorithms greatly hinges on appropriately choosing an ansatz
for the quantum circuit. Perhaps one of the most famous ansatzes is the
one-dimensional layered Hardware Efficient Ansatz (HEA), which seeks to
minimize the effect of hardware noise by using native gates and connectives.
The use of this HEA has generated a certain ambivalence arising from the fact
that while it suffers from barren plateaus at long depths, it can also avoid
them at shallow ones. In this work, we attempt to determine whether one should,
or should not, use a HEA. We rigorously identify scenarios where shallow HEAs
should likely be avoided (e.g., VQA or QML tasks with data satisfying a volume
law of entanglement). More importantly, we identify a Goldilocks scenario where
shallow HEAs could achieve a quantum speedup: QML tasks with data satisfying an
area law of entanglement. We provide examples for such scenario (such as
Gaussian diagonal ensemble random Hamiltonian discrimination), and we show that
in these cases a shallow HEA is always trainable and that there exists an
anti-concentration of loss function values. Our work highlights the crucial
role that input states play in the trainability of a parametrized quantum
circuit, a phenomenon that is verified in our numerics
Stability of topological purity under random local unitaries
In this work, we provide an analytical proof of the robustness of topological
entanglement under a model of random local perturbations. We define a notion of
average topological subsystem purity and show that, in the context of quantum
double models, this quantity does detect topological order and is robust under
the action of a random quantum circuit of shallow depth.Comment: Added new reference
Random Matrix Theory of the Isospectral twirling
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
of quantum many-body systems: Frame potentials, Loschmidt Echos, OTOCs,
Entanglement, Tripartite mutual information, coherence, distance to equilibrium
states, work in quantum batteries and extension to CP-maps. Moreover, we
perform averages in these ensembles by random matrix theory and show how these
quantities clearly separate chaotic quantum dynamics from non chaotic ones.Comment: Submission to SciPos