AlphaQubit Learning high-accuracy error decoding for quantum processors

Surface Code

Logical qubit in $d\times d$ grid called data qubit, $X$ and $Z$ stabilizer checks on groups of adjacent data qubits, with $d^2-1$ stabilizer qubits ($d^2$ logical qubit - 1 physical qubit)

Diagram: $5\times 5$ logical qubits, 12 stabilizer Z qubits, 12 stabilizer X qubits

Logical observable:

• $X_L$ is the product of $X$ operators along a row of data qubits.
• $Z_L$ is the product of $Z$ operators along a column of data qubits.
• Logical observable commutes with the stabilizer but anticommutes with each other.

Detection event: When two consecutive measurements of the same stabilizer give different parity outcomes.

Code distance: minimum length of logical observable -- the number of errors required to change the logical qubit without flipping a stabilizer check.

Performance measured in logical errors per round.

$E(n)=\frac{1}{2}(1-(1-2ϵ{)}^n$ where $ϵ$ is LER.

• $E(0)=0$
• $E(1)=ϵ$
• $E(N)\to \frac{1}{2}$ for large N

Decoder

Decoder state representation stores information about the syndrome history up to the current round.

• Pretraining: trained on samples from a generic noise model
• Finetuning: on the experimental samples

Memory Experiment

Encode a single qubit state $|{\psi }_i⟩$ as ${\rho }_i=|{\psi }_i⟩⟨{\psi }_i|$ (density matrix), perform multiple error-correction rounds, and perform a logical measurement on the resulting state ${\rho }_f$

Generalized Born's Rule

Instead of $p_i=|⟨\psi |{\varphi }_i⟩{|}^2$ (context of pure state)

• Measuring the observable that has eigenvector $|i⟩$: Projector - $P_i=|i⟩⟨i|$
• Pure state $p_i$ can be written as $p_i=⟨\psi |i⟩⟨i|\psi ⟩=⟨\psi |P_i|\psi ⟩$
• Using the cyclic property of the trace, $⟨\psi |P_i|\psi ⟩=\mathbf{Tr}(|\psi ⟩⟨\psi |P_i)$
• Given that ${\rho }_i=|{\psi }_i⟩⟨{\psi }_i|$, we get the generalized Born's rule.

Measurement Noise & Soft Measurement

• Readout of a qubit is not a perfect 0 or 1.
• the authors model the noise using Point Spread Functions (PSFs) for the states $|0⟩$, $|1⟩$, and the leaked state $|2⟩$.
• Decoder receives the probability $P(m_t=1)$
• Detection event: $m_t\oplus m_{t-1}=1$

Model architecture

• Uses both transformer (spatial processing) and RNN (temporal processing)