Subsystem Codes Bacon Shor

Short answer: The Bacon-Shor code is a subsystem quantum error-correcting code on a two-dimensional grid of qubits. It encodes one logical qubit in m by m physical qubits and needs only weight-2 measurements on neighbouring pairs, a property papers describe as a hardware simplification. A 2007 paper proved a threshold lower bound for it, and a 2018 paper reported a simulated comparison with the surface code at distance 3. These are theory and simulation results.

What a subsystem code is

The Qubit Zoo entry on the code, a community reference, describes the Bacon-Shor code as a family of subsystem codes defined on a 2D grid of physical qubits. It says the physical Hilbert space splits into a logical subsystem, a gauge subsystem and a stabilizer-fixed subsystem. Only the logical subsystem encodes information, and the gauge subsystem is "don't care" degrees of freedom.

The layout

The entry says the code encodes one logical qubit in m by m physical qubits with distance m, using only two-body gauge measurements: weight-2 XX and ZZ checks on neighbouring pairs. The X stabilizers are products of XX along each pair of adjacent rows, and the Z stabilizers are products of ZZ along each pair of adjacent columns. The stabilizers have weight 2m, but syndrome extraction measures only the weight-2 gauge operators and computes the stabilizer syndrome classically from their products. The entry calls this a major hardware simplification. A 2018 paper in Physical Review A gives the same structure for an L by L lattice, with 2(L-1) stabilizers of weight 2L normally inferred from 2L(L-1) weight-2 gauge operators.

Fault-tolerant error correction

A 2007 Physical Review Letters paper, "Subsystem Fault Tolerance with the Bacon-Shor Code", says the gauge subsystems lead to remarkably simple and efficient methods for fault-tolerant error correction. Most notably, it says, error correction does not require entangled ancillary states and can be implemented with nearest-neighbour two-qubit measurements. Using these methods, the authors prove a lower bound on the quantum accuracy threshold of 1.94 x 10^-4 for adversarial stochastic noise, which they say improves previous lower bounds by nearly an order of magnitude. That is a proven lower bound under a stated noise model, not a measured hardware threshold.

A comparison with the surface code

The 2018 paper shows that the stabilizers can be measured directly and fault tolerantly with bare ancillary qubits, by constructing circuits that follow the pattern of the gauge operators. It then compares distance-3 versions of the rotated surface code and the Bacon-Shor code under the standard depolarizing model and in the context of a trapped-ion quantum computer. For a simple circuit of prepare, error correct and measure, the authors find that the Bacon-Shor code outperforms the surface code by needing fewer qubits, taking less time and having a lower error rate. The result is specific to small codes and the circuit studied.

Biased noise

The Qubit Zoo entry adds that rectangular variants can correct more X than Z errors, or the reverse, which it says matches biased-noise hardware.

What this does not tell you

These sources do not show the code running on hardware at scale, and the distance-3 comparison does not settle which code is better at larger sizes.

FAQ

What is the Bacon-Shor code?

A subsystem quantum error-correcting code on a 2D grid, encoding one logical qubit in m by m physical qubits.

Why is it considered simple?

It needs only weight-2 measurements on neighbouring qubits, per the Qubit Zoo entry and the 2007 paper.

Does it need entangled ancillas?

The 2007 paper says its methods do not require entangled ancillary states.

Does it beat the surface code?

A 2018 paper reports that at distance 3, for one simple circuit, it did in simulation. That does not cover larger codes.

What threshold was proven?

A lower bound of 1.94 x 10^-4 for adversarial stochastic noise, per the 2007 paper.