Hamiltonian learning for quantum error correction

The efficient validation of quantum devices is critical for emerging technological applications. In a wide class of use cases the precise engineering of a Hamiltonian is required both for the implementation of gate-based quantum information processing as well as for reliable quantum memories. Inferring the experimentally realized Hamiltonian through a scalable number of measurements constitutes the challenging task of Hamiltonian learning. In particular, assessing the quality of the implementation of topological codes is essential for quantum error correction. Here, we introduce a neural-net-based approach to this challenge. We capitalize on a family of exactly solvable models to train our algorithm and generalize to a broad class of experimentally relevant sources of errors. We discuss how our algorithm scales with system size and analyze its resilience toward various noise sources.

Figure 1

Schematic of our method: We add background fields to the stabilizer Hamiltonian. We use expectation values of star operators, $A_i$, to determine ${\sigma }_z$ fields (shown in blue) and expectation values of plaquette operators, $B_i$, to determine ${\sigma }_x$ fields (shown in red).

Figure 2

Phase diagram of the modified toric code model. The field strength $\beta$ (blue axis) and configuration $\left\{{\lambda }_i\right\}$ are encoded as amplitude and angle in the phase diagram. Important field configurations are marked at the phase diagram boundary with an example field configuration and a small diagram. The small diagrams show the percentages of fields ${\lambda }_i$ being equal to $+1$, 0, or $-1$. A phase transition out of the topologically ordered phase (TO) into the non–topologically ordered phase (non-TO) occurs at the black phase boundary. The phase of the corresponding 2D classical spin model is denoted by PM (paramagnetic), FM (ferromagnetic), or AFM (antiferromagnetic). If all fields are equal to zero, the system corresponds to the pure toric code model (1). In the phase diagram, these configurations are indicated by a red dashed line with label “TC line”.

Figure 3

Visualization of the removal of the fields for a small lattice with arbitrary fields. The left panel shows a possible field distribution in Hamiltonian (8). The right panel shows the field distribution after five iterations of our protocol are performed. This illustrates the arrival at the toric code Hamiltonian (1).

Figure 4

Error probability for bit and phase flips (red and blue, respectively; left-hand axis) and Hamiltonian error (gray; right-hand axis) as a function of the iteration of the protocol; see text for details. At each iteration step the field configuration strength is shown (color scheme same as in Fig. 3).

Figure 5

Single qubit error probability and Hamiltonian error as a function of a standard deviation of the measurement noise. In red we show the bit flip error, in blue the phase error, and in gray the Hamiltonian error. The Hamiltonian error is plotted on the right-hand axis. The results are shown after $N=5$ iterations of our protocol.

Figure 6

Single qubit error probability and Hamiltonian error as a function of the lattice size. The used notation is the same as in Fig. 5.

Figure 7

The mapping to a classical (pseudo)spin model. The introduced pseudospins on the lattice sites are marked in green, the spins (qubits) on the edges in gray. The mapping is given by the relation ${\sigma }_i^z\left(g\right)={\theta }_s{\theta }_{s^{\prime }}$ for a spin and two adjacent pseudospins.

Figure 8

Phase diagram of the two probabilistic classical spin models. Temperature $T$ on the $y$ axis is plotted against dilution probabilities $q$ and $p$. The ferromagnetic (F) and paramagnetic (PM) phase and the correspondent topological order (TO) are marked for each model. The Boltzmann constant $k_B$ is set to 1.

Figure 9

The illustrated network consists of 3 input neurons (blue), 3 hidden layers with 128, 150, and 128 neurons, and one output neuron (red) to predict the absolute value of the field strength. The same network can be used to estimate the absolute value of $b_i^x$ with given inputs $\langle B_p\rangle , \langle B_{p^{\prime }}\rangle$, and $\langle B_p{B}_{p^{\prime }}\rangle$.

Figure 10

Training loss (blue) and evaluation loss (red) calculated as mean squared loss against number of training steps for a lattice of length $k=16$.

Figure 11

The relation between single qubit error rate $e_r$ on the $x$ axis and vertex flip rate $p$ on the $y$ axis obtained by numerical sampling is shown for the lattice sizes $k=8,12,16,20,24,28,32$.

Figure 12

The numerically obtained relation between single qubit error rate $e_r$ and vertex flip rate $p$ for $k=32$ are plotted together with the fitted curve (see text for details).