12 research outputs found
Sample-optimal classical shadows for pure states
We consider the classical shadows task for pure states in the setting of both
joint and independent measurements. The task is to measure few copies of an
unknown pure state in order to learn a classical description which
suffices to later estimate expectation values of observables. Specifically, the
goal is to approximate for any Hermitian observable
to within additive error provided and
. Our main result applies to the joint measurement
setting, where we show
samples of are necessary and sufficient to succeed with high
probability. The upper bound is a quadratic improvement on the previous best
sample complexity known for this problem. For the lower bound, we see that the
bottleneck is not how fast we can learn the state but rather how much any
classical description of can be compressed for observable estimation. In
the independent measurement setting, we show that samples suffice. Notably, this implies that the
random Clifford measurements algorithm of Huang, Kueng, and Preskill, which is
sample-optimal for mixed states, is not optimal for pure states. Interestingly,
our result also uses the same random Clifford measurements but employs a
different estimator.Comment: 28 page
Non-Exponential Behaviour in Logical Randomized Benchmarking
We construct a gate and time-independent noise model that results in the
output of a logical randomized benchmarking protocol oscillating rather than
decaying exponentially. To illustrate our idea, we first construct an example
in standard randomized benchmarking where we assume the existence of ``hidden''
qubits, permitting a choice of representation of the Clifford group that
contains multiplicities. We use the multiplicities to, with each gate
application, update a hidden memory of the gate history that we use to
circumvent theorems which guarantee the output decays exponentially. In our
focal setting of logical randomized benchmarking, we show that the presence of
machinery associated with the implementation of quantum error correction can
facilitate non-exponential decay. Since, in logical randomized benchmarking,
the role of the hidden qubits is assigned to the syndrome qubits used in error
correction and these are strongly coupled to the logical qubits via a decoder.Comment: 8 pages + 3 pages of appendices, 7 figure
Fast estimation of outcome probabilities for quantum circuits
We present two classical algorithms for the simulation of universal quantum
circuits on qubits constructed from instances of Clifford gates and
arbitrary-angle -rotation gates such as gates. Our algorithms complement
each other by performing best in different parameter regimes. The
algorithm produces an additive precision estimate of the Born
rule probability of a chosen measurement outcome with the only source of
run-time inefficiency being a linear dependence on the stabilizer extent (which
scales like for gates). Our algorithm is state-of-the-art
for this task: as an example, in approximately hours (on a standard
desktop computer), we estimated the Born rule probability to within an additive
error of , for a qubit, non-Clifford gate quantum circuit with
more than Clifford gates. Our second algorithm, ,
calculates the probability of a chosen measurement outcome to machine precision
with run-time where is an efficiently computable,
circuit-specific quantity. With high probability, is very close to for random circuits with many Clifford gates, where is the
number of measured qubits. can be effective in surprisingly
challenging parameter regimes, e.g., we can randomly sample Clifford+
circuits with , , and gates, and then compute
the Born rule probability with a run-time consistently less than minutes
using a single core of a standard desktop computer. We provide a C+Python
implementation of our algorithms.Comment: 25+14 pages, 6 figures. Version 2 contains a minor correction to the
scaling of Theorem 3, a small improvement to the scaling of Theorem 2 and
various other improvements. Comments welcom
Shallow shadows: Expectation estimation using low-depth random Clifford circuits
We provide practical and powerful schemes for learning many properties of an
unknown n-qubit quantum state using a sparing number of copies of the state.
Specifically, we present a depth-modulated randomized measurement scheme that
interpolates between two known classical shadows schemes based on random Pauli
measurements and random Clifford measurements. These can be seen within our
scheme as the special cases of zero and infinite depth, respectively. We focus
on the regime where depth scales logarithmically in n and provide evidence that
this retains the desirable properties of both extremal schemes whilst, in
contrast to the random Clifford scheme, also being experimentally feasible. We
present methods for two key tasks; estimating expectation values of certain
observables from generated classical shadows and, computing upper bounds on the
depth-modulated shadow norm, thus providing rigorous guarantees on the accuracy
of the output estimates. We consider observables that can be written as a
linear combination of poly(n) Paulis and observables that can be written as a
low bond dimension matrix product operator. For the former class of observables
both tasks are solved efficiently in n. For the latter class, we do not
guarantee efficiency but present a method that works in practice; by
variationally computing a heralded approximate inverses of a tensor network
that can then be used for efficiently executing both these tasks.Comment: 22 pages, 12 figures. Version 2: new MPS variational inversion
algorithm and new numeric
Quantifying quantum speedups: improved classical simulation from tighter magic monotones
Consumption of magic states promotes the stabilizer model of computation to
universal quantum computation. Here, we propose three different classical
algorithms for simulating such universal quantum circuits, and characterize
them by establishing precise connections with a family of magic monotones. Our
first simulator introduces a new class of quasiprobability distributions and
connects its runtime to a generalized notion of negativity. We prove that this
algorithm has significantly improved exponential scaling compared to all prior
quasiprobability simulators for qubits. Our second simulator is a new variant
of the stabilizer-rank simulation algorithm, extended to work with mixed states
and with significantly improved runtime bounds. Our third simulator trades
precision for speed by discarding negative quasiprobabilities. We connect each
algorithm's performance to a corresponding magic monotone and, by
comprehensively characterizing the monotones, we obtain a precise understanding
of the simulation runtime and error bounds. Our analysis reveals a deep
connection between all three seemingly unrelated simulation techniques and
their associated monotones. For tensor products of single-qubit states, we
prove that our monotones are all equal to each other, multiplicative and
efficiently computable, allowing us to make clear-cut comparisons of the
simulators' performance scaling. Furthermore, our monotones establish several
asymptotic and non-asymptotic bounds on state interconversion and distillation
rates. Beyond the theory of magic states, our classical simulators can be
adapted to other resource theories under certain axioms, which we demonstrate
through an explicit application to the theory of quantum coherence.Comment: 24+13 pages, 8 figures; final author copy. Since v1: restructured
with additional discussion, proof sketches and examples. Since v3: minor
revisions to improve clarity, additional acknowledgment
On the classical simulability of quantum circuits
Whether a class of quantum circuits can be efficiently simulated with a classical computer, or is provably hard to simulate, depends quite critically on the precise notion of āclassical simulationā. We focus on two important notions of simulator, that we refer to as poly-boxes and EPSION-simulators and, discuss how other notions of simulation relate to these. A poly-box is a classical algorithm that outputs additive 1/poly precision estimates of Born probabilities and marginals. We present a general framework used to construct poly-boxes. This framework generalizes a number of recent works on simulation. As an application, we use the general framework to construct a classical additive 1/poly precision Born rule probability estimation algorithm for Clifford plus T circuits. Our algorithm scales exponentially in the number of T gates but polynomially in all other parameters and is intended to be state of the art for this estimation task. We expect this result to be particularly useful in the characterization and verification of near term quantum devices. We argue that the notion of classical simulation we call EPSION-simulation, captures the essence of possessing āequivalent computational powerā to the quantum system it simulates: It is statistically impossible to distinguish an agent with access to an EPSION-simulator from one possessing the simulated quantum system. We relate EPSION-simulation to various alternative notions of simulation predominantly focusing on its relation to poly-boxes. Accepting some plausible computational theoretic assumptions, we show that EPSION-simulation is strictly stronger than a poly-box by showing that IQP circuits and unconditioned magic-state injected Clifford circuits are both hard to EPSION-simulate and yet admit a poly-box. In contrast, we also show that these two notions are equivalent under an additional assumption on the sparsity of the output distribution (poly-sparsity)
From estimation of quantum probabilities to simulation of quantum circuits
Investigating the classical simulability of quantum circuits provides a promising avenue towards understanding the computational power of quantum systems. Whether a class of quantum circuits can be efficiently simulated with a probabilistic classical computer, or is provably hard to simulate, depends quite critically on the precise notion of classical simulation and in particular on the required accuracy. We argue that a notion of classical simulation, which we call epsilon-simulation (or epsilon-simulation for short), captures the essence of possessing equivalent computational power as the quantum system it simulates: It is statistically impossible to distinguish an agent with access to an epsilon-simulator from one possessing the simulated quantum system. We relate epsilon-simulation to various alternative notions of simulation predominantly focusing on a simulator we call a poly-box. A poly-box outputs 1/poly precision additive estimates of Born probabilities and marginals. This notion of simulation has gained prominence through a number of recent simulability results. Accepting some plausible computational theoretic assumptions, we show that epsilon-simulation is strictly stronger than a poly-box by showing that IQP circuits and unconditioned magic-state injected Clifford circuits are both hard to epsilon-simulate and yet admit a poly-box. In contrast, we also show that these two notions are equivalent under an additional assumption on the sparsity of the output distribution (poly-sparsity)