29 research outputs found
Is error detection helpful on IBM 5Q chips ?
This paper reports on experiments realized on several IBM 5Q chips which show
evidence for the advantage of using error detection and fault-tolerant design
of quantum circuits. We show an average improvement of the task of sampling
from states that can be fault-tolerantly prepared in the code, when
using a fault-tolerant technique well suited to the layout of the chip. By
showing that fault-tolerant quantum computation is already within our reach,
the author hopes to encourage this approach.Comment: 17 pages, 13 figures, 6 table
Sparse spectral approximations for computing polynomial functionals
We give a new fast method for evaluating sprectral approximations of
nonlinear polynomial functionals. We prove that the new algorithm is convergent
if the functions considered are smooth enough, under a general assumption on
the spectral eigenfunctions that turns out to be satisfied in many cases,
including the Fourier and Hermite basis
Quantum Pin Codes
arXiv: 1906.11394We introduce quantum pin codes: a class of quantum CSS codes. Quantum pin codes are a vast generalization of quantum color codes and Reed-Muller codes. A lot of the structure and properties of color codes carries over to pin codes. Pin codes have gauge operators, an unfolding procedure and their stabilizers form multi-orthogonal spaces. This last feature makes them interesting for devising magic-state distillation protocols. We study examples of these codes and their properties
Quantum Pin Codes
We introduce quantum pin codes: a class of quantum CSS codes. Quantum pin
codes are a generalization of quantum color codes and Reed-Muller codes and
share a lot of their structure and properties. Pin codes have gauge operators,
an unfolding procedure and their stabilizers form so-called -orthogonal
spaces meaning that the joint overlap between any stabilizer elements is
always even. This last feature makes them interesting for devising magic-state
distillation protocols, for instance by using puncturing techniques. We study
examples of these codes and their properties
Planar Floquet Codes
A protocol called the "honeycomb code", or generically a "Floquet code", was introduced by Hastings and Haah in \cite{hastings_dynamically_2021}. The honeycomb code is a subsystem code based on the honeycomb lattice with zero logical qubits but such that there exists a schedule for measuring two-body gauge checks leaving enough room at all times for two protected logical qubits. In this work we show a way to introduce boundaries to the system which curiously presents a rotating dynamics but has constant distance and is therefore not fault-tolerant
Quantum Error Correction with the Toric-GKP Code
We examine the performance of the single-mode GKP code and its concatenation
with the toric code for a noise model of Gaussian shifts, or displacement
errors. We show how one can optimize the tracking of errors in repeated noisy
error correction for the GKP code. We do this by examining the
maximum-likelihood problem for this setting and its mapping onto a 1D Euclidean
path-integral modeling a particle in a random cosine potential. We demonstrate
the efficiency of a minimum-energy decoding strategy as a proxy for the path
integral evaluation. In the second part of this paper, we analyze and
numerically assess the concatenation of the GKP code with the toric code. When
toric code measurements and GKP error correction measurements are perfect, we
find that by using GKP error information the toric code threshold improves from
to . When only the GKP error correction measurements are perfect
we observe a threshold at . In the more realistic setting when all error
information is noisy, we show how to represent the maximum likelihood decoding
problem for the toric-GKP code as a 3D compact QED model in the presence of a
quenched random gauge field, an extension of the random-plaquette gauge model
for the toric code. We present a new decoder for this problem which shows the
existence of a noise threshold at shift-error standard deviation for toric code measurements, data errors and GKP ancilla errors.
If the errors only come from having imperfect GKP states, this corresponds to
states with just 4 photons or more. Our last result is a no-go result for
linear oscillator codes, encoding oscillators into oscillators. For the
Gaussian displacement error model, we prove that encoding corresponds to
squeezing the shift errors. This shows that linear oscillator codes are useless
for quantum information protection against Gaussian shift errors.Comment: 50 pages, 14 figure
Homological Quantum Rotor Codes: Logical Qubits from Torsion
We formally define homological quantum rotor codes which use multiple quantum
rotors to encode logical information. These codes generalize homological or CSS
quantum codes for qubits or qudits, as well as linear oscillator codes which
encode logical oscillators. Unlike for qubits or oscillators, homological
quantum rotor codes allow one to encode both logical rotors and logical qudits,
depending on the homology of the underlying chain complex. In particular, such
a code based on the chain complex obtained from tessellating the real
projective plane or a M\"{o}bius strip encodes a qubit. We discuss the distance
scaling for such codes which can be more subtle than in the qubit case due to
the concept of logical operator spreading by continuous stabilizer
phase-shifts. We give constructions of homological quantum rotor codes based on
2D and 3D manifolds as well as products of chain complexes. Superconducting
devices being composed of islands with integer Cooper pair charges could form a
natural hardware platform for realizing these codes: we show that the
--qubit as well as Kitaev's current-mirror qubit -- also known as the
M\"{o}bius strip qubit -- are indeed small examples of such codes and discuss
possible extensions.Comment: 47 pages, 10 figures, 2 table
Optimal Hadamard gate count for Clifford synthesis of Pauli rotations sequences
The Clifford gate set is commonly used to perform universal quantum
computation. In such setup the gate is typically much more expensive to
implement in a fault-tolerant way than Clifford gates. To improve the
feasibility of fault-tolerant quantum computing it is then crucial to minimize
the number of gates. Many algorithms, yielding effective results, have been
designed to address this problem. It has been demonstrated that performing a
pre-processing step consisting of reducing the number of Hadamard gates in the
circuit can help to exploit the full potential of these algorithms and thereby
lead to a substantial -count reduction. Moreover, minimizing the number of
Hadamard gates also restrains the number of additional qubits and operations
resulting from the gadgetization of Hadamard gates, a procedure used by some
compilers to further reduce the number of gates. In this work we tackle the
Hadamard gate reduction problem, and propose an algorithm for synthesizing a
sequence of Pauli rotations with a minimal number of Hadamard gates. Based on
this result, we present an algorithm which optimally minimizes the number of
Hadamard gates lying between the first and the last gate of the circuit