5,617 research outputs found
Sparse Quantum Codes from Quantum Circuits
Sparse quantum codes are analogous to LDPC codes in that their check operators require examining only a constant number of qubits. In contrast to LDPC codes, good sparse quantum codes are not known, and even to encode a single qubit, the best known distance is O(√n log(n)), due to Freedman, Meyer and Luo.
We construct a new family of sparse quantum subsystem codes with minimum distance n[superscript 1 - ε] for ε = O(1/√log n). A variant of these codes exists in D spatial dimensions and has d = n[superscript 1 - ε - 1/D], nearly saturating a bound due to Bravyi and Terhal.
Our construction is based on a new general method for turning quantum circuits into sparse quantum subsystem codes. Using this prescription, we can map an arbitrary stabilizer code into a new subsystem code with the same distance and number of encoded qubits but where all the generators have constant weight, at the cost of adding some ancilla qubits. With an additional overhead of ancilla qubits, the new code can also be made spatially local.National Science Foundation (U.S.) (Grant CCF-1111382)United States. Army Research Office (Contract W911NF-12-1-0486
Robust sparse IQP sampling in constant depth
Between NISQ (noisy intermediate scale quantum) approaches without any proof
of robust quantum advantage and fully fault-tolerant quantum computation, we
propose a scheme to achieve a provable superpolynomial quantum advantage (under
some widely accepted complexity conjectures) that is robust to noise with
minimal error correction requirements. We choose a class of sampling problems
with commuting gates known as sparse IQP (Instantaneous Quantum
Polynomial-time) circuits and we ensure its fault-tolerant implementation by
introducing the tetrahelix code. This new code is obtained by merging several
tetrahedral codes (3D color codes) and has the following properties: each
sparse IQP gate admits a transversal implementation, and the depth of the
logical circuit can be traded for its width. Combining those, we obtain a
depth-1 implementation of any sparse IQP circuit up to the preparation of
encoded states. This comes at the cost of a space overhead which is only
polylogarithmic in the width of the original circuit. We furthermore show that
the state preparation can also be performed in constant depth with a single
step of feed-forward from classical computation. Our construction thus exhibits
a robust superpolynomial quantum advantage for a sampling problem implemented
on a constant depth circuit with a single round of measurement and
feed-forward
Trading classical and quantum computational resources
We propose examples of a hybrid quantum-classical simulation where a
classical computer assisted by a small quantum processor can efficiently
simulate a larger quantum system. First we consider sparse quantum circuits
such that each qubit participates in O(1) two-qubit gates. It is shown that any
sparse circuit on n+k qubits can be simulated by sparse circuits on n qubits
and a classical processing that takes time . Secondly, we
study Pauli-based computation (PBC) where allowed operations are
non-destructive eigenvalue measurements of n-qubit Pauli operators. The
computation begins by initializing each qubit in the so-called magic state.
This model is known to be equivalent to the universal quantum computer. We show
that any PBC on n+k qubits can be simulated by PBCs on n qubits and a classical
processing that takes time . Finally, we propose a purely
classical algorithm that can simulate a PBC on n qubits in a time where . This improves upon the brute-force simulation
method which takes time . Our algorithm exploits the fact that
n-fold tensor products of magic states admit a low-rank decomposition into
n-qubit stabilizer states.Comment: 14 pages, 4 figure
Topological Order and Memory Time in Marginally Self-Correcting Quantum Memory
We examine two proposals for marginally self-correcting quantum memory, the
cubic code by Haah and the welded code by Michnicki. In particular, we prove
explicitly that they are absent of topological order above zero temperature, as
their Gibbs ensembles can be prepared via a short-depth quantum circuit from
classical ensembles. Our proof technique naturally gives rise to the notion of
free energy associated with excitations. Further, we develop a framework for an
ergodic decomposition of Davies generators in CSS codes which enables formal
reduction to simpler classical memory problems. We then show that memory time
in the welded code is doubly exponential in inverse temperature via the Peierls
argument. These results introduce further connections between thermal
topological order and self-correction from the viewpoint of free energy and
quantum circuit depth.Comment: 19 pages, 18 figure
Good approximate quantum LDPC codes from spacetime circuit Hamiltonians
We study approximate quantum low-density parity-check (QLDPC) codes, which
are approximate quantum error-correcting codes specified as the ground space of
a frustration-free local Hamiltonian, whose terms do not necessarily commute.
Such codes generalize stabilizer QLDPC codes, which are exact quantum
error-correcting codes with sparse, low-weight stabilizer generators (i.e. each
stabilizer generator acts on a few qubits, and each qubit participates in a few
stabilizer generators). Our investigation is motivated by an important question
in Hamiltonian complexity and quantum coding theory: do stabilizer QLDPC codes
with constant rate, linear distance, and constant-weight stabilizers exist?
We show that obtaining such optimal scaling of parameters (modulo
polylogarithmic corrections) is possible if we go beyond stabilizer codes: we
prove the existence of a family of approximate QLDPC
codes that encode logical qubits into physical
qubits with distance and approximation infidelity
. The code space is
stabilized by a set of 10-local noncommuting projectors, with each physical
qubit only participating in projectors. We
prove the existence of an efficient encoding map, and we show that arbitrary
Pauli errors can be locally detected by circuits of polylogarithmic depth.
Finally, we show that the spectral gap of the code Hamiltonian is
by analyzing a spacetime circuit-to-Hamiltonian
construction for a bitonic sorting network architecture that is spatially local
in dimensions.Comment: 51 pages, 13 figure
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