19,070 research outputs found
Low depth algorithms for quantum amplitude estimation
We design and analyze two new low depth algorithms for amplitude estimation
(AE) achieving an optimal tradeoff between the quantum speedup and circuit
depth. For , our algorithms require oracle calls and require the oracle to be called
sequentially times to perform amplitude
estimation within additive error . These algorithms interpolate
between the classical algorithm and the standard quantum algorithm
() and achieve a tradeoff . These algorithms
bring quantum speedups for Monte Carlo methods closer to realization, as they
can provide speedups with shallower circuits.
The first algorithm (Power law AE) uses power law schedules in the framework
introduced by Suzuki et al \cite{S20}. The algorithm works for and has provable correctness guarantees when the log-likelihood function
satisfies regularity conditions required for the Bernstein Von-Mises theorem.
The second algorithm (QoPrime AE) uses the Chinese remainder theorem for
combining lower depth estimates to achieve higher accuracy. The algorithm works
for discrete where is the number of distinct coprime
moduli used by the algorithm and , and has a fully rigorous
correctness proof. We analyze both algorithms in the presence of depolarizing
noise and provide experimental comparisons with the state of the art amplitude
estimation algorithms
Error Resilient Quantum Amplitude Estimation from Parallel Quantum Phase Estimation
We show how phase and amplitude estimation algorithms can be parallelized.
This can reduce the gate depth of the quantum circuits to that of a single
Grover operator with a small overhead. Further, we show that for quantum
amplitude estimation, the parallelization can lead to vast improvements in
resilience against quantum errors. The resilience is not caused by the lower
gate depth, but by the structure of the algorithm. Even in cases with errors
that make it impossible to read out the exact or approximate solutions from
conventional amplitude estimation, our parallel approach provided the correct
solution with high probability. The results on error resilience hold for the
standard version and for low depth versions of quantum amplitude estimation.
Methods presented are subject of a patent application [Quantum computing
device: Patent application EP 21207022.1]
Quantum unary approach to option pricing
We present a quantum algorithm for European option pricing in finance, where
the key idea is to work in the unary representation of the asset value. The
algorithm needs novel circuitry and is divided in three parts: first, the
amplitude distribution corresponding to the asset value at maturity is
generated using a low depth circuit; second, the computation of the expected
return is computed with simple controlled gates; and third, standard Amplitude
Estimation is used to gain quantum advantage. On the positive side, unary
representation remarkably simplifies the structure and depth of the quantum
circuit. Amplitude distributions uses quantum superposition to bypass the role
of classical Monte Carlo simulation. The unary representation also provides a
post-selection consistency check that allows for a substantial mitigation in
the error of the computation. On the negative side, unary representation
requires linearly many qubits to represent a target probability distribution,
as compared to the logarithmic scaling of binary algorithms. We compare the
performance of both unary vs. binary option pricing algorithms using error
maps, and find that unary representation may bring a relevant advantage in
practice for near-term devices.Comment: 14 (main) + 10 (appendix) pages, 22 figures. Final peer-reviewed
version, published in PRA. All suggestions from the referees have been
considered. We thank the referees and the journal for all the wor
Quantum Generative Adversarial Networks for Learning and Loading Random Distributions
Quantum algorithms have the potential to outperform their classical
counterparts in a variety of tasks. The realization of the advantage often
requires the ability to load classical data efficiently into quantum states.
However, the best known methods require gates to
load an exact representation of a generic data structure into an -qubit
state. This scaling can easily predominate the complexity of a quantum
algorithm and, thereby, impair potential quantum advantage. Our work presents a
hybrid quantum-classical algorithm for efficient, approximate quantum state
loading. More precisely, we use quantum Generative Adversarial Networks (qGANs)
to facilitate efficient learning and loading of generic probability
distributions -- implicitly given by data samples -- into quantum states.
Through the interplay of a quantum channel, such as a variational quantum
circuit, and a classical neural network, the qGAN can learn a representation of
the probability distribution underlying the data samples and load it into a
quantum state. The loading requires
gates and can, thus, enable the
use of potentially advantageous quantum algorithms, such as Quantum Amplitude
Estimation. We implement the qGAN distribution learning and loading method with
Qiskit and test it using a quantum simulation as well as actual quantum
processors provided by the IBM Q Experience. Furthermore, we employ quantum
simulation to demonstrate the use of the trained quantum channel in a quantum
finance application.Comment: 14 pages, 13 figure
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