17 research outputs found
Measurement cost of metric-aware variational quantum algorithms
Variational quantum algorithms are promising tools for near-term quantum
computers as their shallow circuits are robust to experimental imperfections.
Their practical applicability, however, strongly depends on how many times
their circuits need to be executed for sufficiently reducing shot-noise. We
consider metric-aware quantum algorithms: variational algorithms that use a
quantum computer to efficiently estimate both a matrix and a vector object. For
example, the recently introduced quantum natural gradient approach uses the
quantum Fisher information matrix as a metric tensor to correct the gradient
vector for the co-dependence of the circuit parameters. We rigorously
characterise and upper bound the number of measurements required to determine
an iteration step to a fixed precision, and propose a general approach for
optimally distributing samples between matrix and vector entries. Finally, we
establish that the number of circuit repetitions needed for estimating the
quantum Fisher information matrix is asymptotically negligible for an
increasing number of iterations and qubits.Comment: 17 pages, 3 figure
Exponential Error Suppression for Near-Term Quantum Devices
As quantum computers mature, quantum error correcting codes (QECs) will be
adopted in order to suppress errors to any desired level at a cost in
qubit-count that is merely poly-logarithmic in . However in the NISQ
era, the complexity and scale required to adopt even the smallest QEC is
prohibitive. Instead, error mitigation techniques have been employed; typically
these do not require an increase in qubit-count but cannot provide exponential
error suppression. Here we show that, for the crucial case of estimating
expectation values of observables (key to almost all NISQ algorithms) one can
indeed achieve an effective exponential suppression. We introduce the Error
Suppression by Derangement (ESD) approach: by increasing the qubit count by a
factor of , the error is suppressed exponentially as where
is a suppression factor that depends on the entropy of the errors. The ESD
approach takes independently-prepared circuit outputs and applies a
controlled derangement operator to create a state whose symmetries prevent
erroneous states from contributing to expected values. The approach is
therefore `NISQ-friendly' as it is modular in the main computation and requires
only a shallow circuit that bridges the copies immediately prior to
measurement. Imperfections in our derangement circuit do degrade performance
and therefore we propose an approach to mitigate this effect to arbitrary
precision due to the remarkable properties of derangements. a) they decompose
into a linear number of elementary gates -- limiting the impact of noise b)
they are highly resilient to noise and the effect of imperfections on them is
(almost) trivial. In numerical simulations validating our approach we confirm
error suppression below for circuits consisting of several hundred
noisy gates (two-qubit gate error ) using no more than circuit
copies.Comment: 34 pages, 12 figure
Continuous phase-space representations for finite-dimensional quantum states and their tomography
Continuous phase spaces have become a powerful tool for describing,
analyzing, and tomographically reconstructing quantum states in quantum optics
and beyond. A plethora of these phase-space techniques are known, however a
thorough understanding of their relations was still lacking for
finite-dimensional quantum states. We present a unified approach to continuous
phase-space representations which highlights their relations and tomography.
The infinite-dimensional case from quantum optics is then recovered in the
large-spin limit.Comment: 15 pages, 9 figures, v4: extended tomography analysis, added
references and figure
Training variational quantum circuits with CoVaR: covariance root finding with classical shadows
Exploiting near-term quantum computers and achieving practical value is a
considerable and exciting challenge. Most prominent candidates as variational
algorithms typically aim to find the ground state of a Hamiltonian by
minimising a single classical (energy) surface which is sampled from by a
quantum computer. Here we introduce a method we call CoVaR, an alternative
means to exploit the power of variational circuits: We find eigenstates by
finding joint roots of a polynomially growing number of properties of the
quantum state as covariance functions between the Hamiltonian and an operator
pool of our choice. The most remarkable feature of our CoVaR approach is that
it allows us to fully exploit the extremely powerful classical shadow
techniques, i.e., we simultaneously estimate a very large number
of covariances. We randomly select covariances and estimate analytical
derivatives at each iteration applying a stochastic Levenberg-Marquardt step
via a large but tractable linear system of equations that we solve with a
classical computer. We prove that the cost in quantum resources per iteration
is comparable to a standard gradient estimation, however, we observe in
numerical simulations a very significant improvement by many orders of
magnitude in convergence speed. CoVaR is directly analogous to stochastic
gradient-based optimisations of paramount importance to classical machine
learning while we also offload significant but tractable work onto the
classical processor.Comment: 25 pages, 9 figure
Quantum natural gradient generalised to non-unitary circuits
Variational quantum circuits are promising tools whose efficacy depends on
their optimisation method. For noise-free unitary circuits, the quantum
generalisation of natural gradient descent was recently introduced. The method
can be shown to be equivalent to imaginary time evolution, and is highly
effective due to a metric tensor reconciling the classical parameter space to
the device's Hilbert space. Here we generalise quantum natural gradient to
consider arbitrary quantum states (both mixed and pure) via completely positive
maps; thus our circuits can incorporate both imperfect unitary gates and
fundamentally non-unitary operations such as measurements. Whereas the unitary
variant relates to classical Fisher information, here we find that quantum
Fisher information defines the core metric in the space of density operators.
Numerical simulations indicate that our approach can outperform other
variational techniques when circuit noise is present. We finally assess the
practical feasibility of our implementation and argue that its scalability is
only limited by the number and quality of imperfect gates and not by the number
of qubits.Comment: 20 pages, 6 figure
Quantum Analytic Descent
Variational algorithms have particular relevance for near-term quantum
computers but require non-trivial parameter optimisations. Here we propose
Analytic Descent: Given that the energy landscape must have a certain simple
form in the local region around any reference point, it can be efficiently
approximated in its entirety by a classical model -- we support these
observations with rigorous, complexity-theoretic arguments. One can classically
analyse this approximate function in order to directly `jump' to the
(estimated) minimum, before determining a more refined function if necessary.
We verify our technique using numerical simulations: each analytic jump can be
equivalent to many thousands of steps of canonical gradient descent.Comment: 14 pages, 4 figure
Probabilistic Interpolation of Quantum Rotation Angles
Quantum computing requires a universal set of gate operations; regarding
gates as rotations, any rotation angle must be possible. However a real device
may only be capable of bits of resolution, i.e. it might support only
possible variants of a given physical gate. Naive discretization of an
algorithm's gates to the nearest available options causes coherent errors,
while decomposing an impermissible gate into several allowed operations
increases circuit depth. Conversely, demanding higher can greatly
complexify hardware. Here we explore an alternative: Probabilistic Angle
Interpolation (PAI). This effectively implements any desired, continuously
parametrised rotation by randomly choosing one of three discretised gate
settings and postprocessing individual circuit outputs. The approach is
particularly relevant for near-term applications where one would in any case
average over many runs of circuit executions to estimate expected values. While
PAI increases that sampling cost, we prove that a) the approach is optimal in
the sense that PAI achieves the least possible overhead and c) the overhead is
remarkably modest even with thousands of parametrised gates and only bits
of resolution available. This is a profound relaxation of engineering
requirements for first generation quantum computers where even bits of
resolution may suffice and, as we demonstrate, the approach is many orders of
magnitude more efficient than prior techniques. Moreover we conclude that, even
for more mature late-NISQ hardware, no more than bits will be necessary.Comment: 15 pages, 5 figures -- includes proof of optimality of protocol,
generalisation to non-uniform settings et
Fast computation of spherical phase-space functions of quantum many-body states
Quantum devices are preparing increasingly more complex entangled quantum
states. How can one effectively study these states in light of their increasing
dimensions? Phase spaces such as Wigner functions provide a suitable framework.
We focus on phase spaces for finite-dimensional quantum states of single qudits
or permutationally symmetric states of multiple qubits. We present methods to
efficiently compute the corresponding phase-space functions which are at least
an order of magnitude faster than traditional methods. Quantum many-body states
in much larger dimensions can now be effectively studied by experimentalists
and theorists using these phase-space techniques.Comment: 12 pages, 3 figure
Distributed Simulation of Statevectors and Density Matrices
Classical simulation of quantum computers is an irreplaceable step in the
design of quantum algorithms. Exponential simulation costs demand the use of
high-performance computing techniques, and in particular distribution, whereby
the quantum state description is partitioned between a network of cooperating
computers - necessary for the exact simulation of more than approximately 30
qubits. Distributed computing is notoriously difficult, requiring bespoke
algorithms dissimilar to their serial counterparts with different resource
considerations, and which appear to restrict the utilities of a quantum
simulator. This manuscript presents a plethora of novel algorithms for
distributed full-state simulation of gates, operators, noise channels and other
calculations in digital quantum computers. We show how a simple, common but
seemingly restrictive distribution model actually permits a rich set of
advanced facilities including Pauli gadgets, many-controlled many-target
general unitaries, density matrices, general decoherence channels, and partial
traces. These algorithms include asymptotically, polynomially improved
simulations of exotic gates, and thorough motivations for high-performance
computing techniques which will be useful for even non-distributed simulators.
Our results are derived in language familiar to a quantum information theory
audience, and our algorithms formalised for the scientific simulation
community. We have implemented all algorithms herein presented into an
isolated, minimalist C++ project, hosted open-source on Github with a
permissive MIT license, and extensive testing. This manuscript aims both to
significantly improve the high-performance quantum simulation tools available,
and offer a thorough introduction to, and derivation of, full-state simulation
techniques.Comment: 56 pages, 18 figures, 28 algorithms, 1 tabl