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arXiv:1008.1029 · Physical Review A

Subsystem codes with spatially local generators

Sergey B. Bravyi

Published Jan 26, 2011• 54 citations
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Abstract

We study subsystem codes whose gauge group has local generators in two-dimensional (2D) geometry. It is shown that there exists a family of such codes defined on lattices of size $L\ifmmode\times\else\texttimes\fi{}L$ with the number of logical qubits $k$ and the minimum distance $d$ both proportional to $L$. The gauge group of these codes involves only two-qubit generators of type $\mathit{XX}$ and $\mathit{ZZ}$ coupling nearest-neighbor qubits (and some auxiliary one-qubit generators). Our proof is not constructive as it relies on a certain version of the Gilbert-Varshamov bound for classical codes. Along the way, we introduce and study properties of generalized Bacon-Shor codes that might be of independent interest. Secondly, we prove that any 2D subsystem $[n,k,d]$ code with spatially local generators obeys upper bounds $\mathit{kd}=O(n)$ and ${d}^{2}=O(n)$. The analogous upper bound proved recently for 2D stabilizer codes is ${\mathit{kd}}^{2}=O(n)$. Our results thus demonstrate that subsystem codes can be more powerful than stabilizer codes under the spatial locality constraint.

DOI
10.1103/physreva.83.012320

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We study subsystem codes whose gauge group has local generators in two-dimensional (2D) geometry. It is shown that there exists a family of such codes defined on lattices of size L\ifmmode×\else\texttimes\fiLL\ifmmode\times\else\texttimes\fi{}LL\ifmmode×\else\texttimes\fiL with the number of logical qubits kkk and the minimum distance ddd both proportional to LLL. The gauge group of these codes involves only two-qubit generators of type XX\mathit{XX}XX and ZZ\mathit{ZZ}ZZ coupling nearest-neighbor qubits (and some auxiliary one-qubit generators). Our proof is not constructive as it relies on a certain version of the Gilbert-Varshamov bound for classical codes. Along the way, we introduce and study properties of generalized Bacon-Shor codes that might be of independent interest. Secondly, we prove that any 2D subsystem [n,k,d][n,k,d][n,k,d] code with spatially local generators obeys upper bounds kd=O(n)\mathit{kd}=O(n)kd=O(n) and d2=O(n){d}^{2}=O(n)d2=O(n). The analogous upper bound proved recently for 2D stabilizer codes is kd2=O(n){\mathit{kd}}^{2}=O(n)kd2=O(n). Our results thus demonstrate that subsystem codes can be more powerful than stabilizer codes under the spatial locality constraint.
We derive new tradeoffs for reliable quantum information storage in a 2D local architecture based on subsystem quantum codes. Our results apply to stabilizer subsystem codes, that is, stabilizer codes in which part of the logical qubits does not encode any information. A stabilizer subsystem code can be specified by its gauge group - a subgroup of the Pauli group that includes the stabilizers and the logical operators on the unused logical qubits. We assume that the physical qubits are arranged on a two-dimensional grid and the gauge group has spatially local generators such that each generator acts only on a few qubits located close to each other. Our main result is an upper bound kd = O(n), where k is the number of encoded qubits, d is the minimal distance, and n is the number of physical qubits. In the special case when both gauge group and the stabilizer group have spatially local generators, we derive a stronger bound kd2= O(n) which is tight up to a constant factor.
We study properties of stabilizer codes that permit a local description on a regular D -dimensional lattice. Specifically, we assume that the stabilizer group of a code (the gauge group for subsystem codes) can be generated by local Pauli operators such that the support of any generator is bounded by a hypercube of size O (1). Our first result concerns the optimal scaling of the distance d with the linear size of the lattice L . We prove an upper bound d = O ( L D -1 ) which is tight for D =1, 2. This bound applies to both subspace and subsystem stabilizer codes. Secondly, we analyze the suitability of stabilizer codes for building a self-correcting quantum memory. Any stabilizer code with geometrically local generators can be naturally transformed to a local Hamiltonian penalizing states that violate the stabilizer condition. A degenerate ground state of this Hamiltonian corresponds to the logical subspace of the code. We prove that for D =1, 2, different logical states can be mapped into each other by a sequence of single-qubit Pauli errors such that the energy of all intermediate states is upper bounded by a constant independent of the lattice size L . The same result holds if there are unused logical qubits that are treated as 'gauge qubits'. It demonstrates that a self-correcting quantum memory cannot be built using stabilizer codes in dimensions D =1, 2. This result is in sharp contrast with the existence of a classical self-correcting memory in the form of a two-dimensional (2D) ferromagnet. Our results leave open the possibility for a self-correcting quantum memory based on 2D subsystem codes or on 3D subspace or subsystem codes.
We describe a general method for turning quantum circuits into sparse quantum subsystem codes. The idea is to turn each circuit element into a set of low-weight gauge generators that enforce the input-output relations of that circuit element. 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. Applying our construction to certain concatenated stabilizer codes yields families of subsystem codes with constant weight generators and with minimum distance d = n1-∈, where E = O(1/√log n). For spatially local codes in D dimensions, we nearly saturate a bound due to Bravyi and Terhal and achieve d = n1-∈-1/D. Previously the best code distance achievable with constant-weight generators in any dimension, due to Freedman, Meyer, and Luo, was O(√n log n) for a stabilizer code.
We introduce a family of two-dimensional (2D) topological subsystem quantum error-correcting codes. The gauge group is generated by two-local Pauli operators, so that two-local measurements are enough to recover the error syndrome. We study the computational power of code deformation in these codes and show that boundaries cannot be introduced in the usual way. In addition, we give a general mapping connecting suitable classical statistical mechanical models to optimal error correction in subsystem stabilizer codes that suffer from depolarizing noise.
The compass model on a square lattice provides a natural template for building subsystem stabilizer codes.The surface code and the Bacon-Shor code represent two extremes of possible codes depending on how many gauge qubits are fixed.We explore threshold behavior in this broad class of local codes by trading locality for asymmetry and gauge degrees of freedom for stabilizer syndrome information.We analyze these codes with asymmetric and spatially inhomogeneous Pauli noise in the code capacity and phenomenological models.In these idealized settings, we observe considerably higher thresholds against asymmetric noise.At the circuit level, these codes inherit the bare-ancilla fault tolerance of the Bacon-Shor code.
We investigate various aspects of operator quantum error-correcting codes or, as we prefer to call them, subsystem codes. We give various methods to derive subsystem codes from classical codes. We give a proof for the existence of subsystem codes using a counting argument similar to the quantum Gilbert-Varshamov bound. We derive linear programming bounds and other upper bounds. We answer the question whether or not there exist [[n,n-2d+2,r>0,d]]q subsystem codes. Finally, we compare stabilizer and subsystem codes with respect to the required number of syndrome qudits.