7 research outputs found
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 [[N,k,d,ε]] approximate QLDPC codes that encode k = Ω(N) logical qubits into N physical qubits with distance d = Ω(N) and approximation infidelity ε = 1/(N). The code space is stabilized by a set of 10-local noncommuting projectors, with each physical qubit only participating in N projectors. We prove the existence of an efficient encoding map and show that the spectral gap of the code Hamiltonian scales as Ω(N^(−3.09)). We also show that arbitrary Pauli errors can be locally detected by circuits of polylogarithmic depth.
Our family of approximate QLDPC codes is based on applying a recent connection between circuit Hamiltonians and approximate quantum codes (Nirkhe, et al., ICALP 2018) to a result showing that random Clifford circuits of polylogarithmic depth yield asymptotically good quantum codes (Brown and Fawzi, ISIT 2013). Then, in order to obtain a code with sparse checks and strong detection of local errors, we use a spacetime circuit-to-Hamiltonian construction in order to take advantage of the parallelism of the Brown-Fawzi circuits. Because of this, we call our codes spacetime codes.
The analysis of the spectral gap of the code Hamiltonian is the main technical contribution of this work. We show that for any depth D quantum circuit on n qubits there is an associated spacetime circuit-to-Hamiltonian construction with spectral gap Ω(n^(−3.09)D⁻² log⁻⁶ (n)). To lower bound this gap we use a Markov chain decomposition method to divide the state space of partially completed circuit configurations into overlapping subsets corresponding to uniform circuit segments of depth logn, which are based on bitonic sorting circuits. We use the combinatorial properties of these circuit configurations to show rapid mixing between the subsets, and within the subsets we develop a novel isomorphism between the local update Markov chain on bitonic circuit configurations and the edge-flip Markov chain on equal-area dyadic tilings, whose mixing time was recently shown to be polynomial (Cannon, Levin, and Stauffer, RANDOM 2017). Previous lower bounds on the spectral gap of spacetime circuit Hamiltonians have all been based on a connection to exactly solvable quantum spin chains and applied only to 1+1 dimensional nearest-neighbor quantum circuits with at least linear depth
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
Quantum LDPC codes from intersecting subsets
This paper introduces a construction of quantum CSS codes from a tuple of
component CSS codes and two collections of subsets. The resulting codes have
parallelizable encoding and syndrome measurement circuits and built-in
redundancy in the syndrome measurements. In a certain subfamily of the general
construction, the resulting codes are related to a natural generalization of
classical Reed-Muller codes, and this leads to a formula for the quantum code
distance. The paper gives a number of examples of codes with block size , and with syndrome measurements involving 2, 4 or 8 qubits. These
include codes for which the distance exceeds the syndrome measurement weight,
as well as codes which provide asymmetric protection against bit flip and phase
flip errors
Decoding algorithms for surface codes
Quantum technologies have the potential to solve computationally hard
problems that are intractable via classical means. Unfortunately, the unstable
nature of quantum information makes it prone to errors. For this reason,
quantum error correction is an invaluable tool to make quantum information
reliable and enable the ultimate goal of fault-tolerant quantum computing.
Surface codes currently stand as the most promising candidates to build error
corrected qubits given their two-dimensional architecture, a requirement of
only local operations, and high tolerance to quantum noise. Decoding algorithms
are an integral component of any error correction scheme, as they are tasked
with producing accurate estimates of the errors that affect quantum
information, so that it can subsequently be corrected. A critical aspect of
decoding algorithms is their speed, since the quantum state will suffer
additional errors with the passage of time. This poses a connundrum-like
tradeoff, where decoding performance is improved at the expense of complexity
and viceversa. In this review, a thorough discussion of state-of-the-art
surface code decoding algorithms is provided. The core operation of these
methods is described along with existing variants that show promise for
improved results. In addition, both the decoding performance, in terms of error
correction capability, and decoding complexity, are compared. A review of the
existing software tools regarding surface code decoding is also provided.Comment: 54 pages, 31 figure
Polylog-LDPC Capacity Achieving Codes for the Noisy Quantum Erasure Channel
We provide polylog sparse quantum codes for correcting the erasure channel arbitrarily close to the capacity. Specifically, we provide [[n, k, d]] quantum stabilizer codes that correct for the erasure channel arbitrarily close to the capacity if the erasure probability is at least 0.33, and with a generating set hS1, S2, . . . Sn−ki such that |Si | ≤ log2+ζ (n) for all i and for any ζ > 0 with high probability. In this work we show that the result of Delfosse et al. [5] is tight: one can construct capacity approaching codes with weight almost O(1)
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum