Quantum State Encoding with a Color Code¶

This tutorial showcases the basic functionality of Tsim.

tsim is a circuit sampler for Clifford+T circuits, based on stabilizer rank decomposition and ZX-calculus techniques. It closely follows the API of stim and directly uses stim's circuit format.

In contrast to Stim, Tsim supports T and T_DAG instructions.

The following circuit demonstrates this by preparing and measuring the state $$H\,T\,|+\rangle = e^{i\pi/8}\Big[\cos\!\left(\tfrac{\pi}{8}\right)\,|0\rangle \;-\; i\,\sin\!\left(\tfrac{\pi}{8}\right)\,|1\rangle\Big].$$

To sample from this circuit, we first compile it into a sampler:

We can now sample bitstrings from the measurement instructions:

Let's run a large number of shots to estimate the probability of measuring 1.

As expected, the probability is close to $\sin(\pi/8)^2 \approx 0.1464$.

Detectors and Observables¶

Next, we consider a more complex example: an encoding circuit for the [[7,1,3]] Steane code. This circuit prepares the logical state

$$\frac{1}{2}\Big[(1 + e^{i\pi/4})|\bar{0}\rangle + (1 - e^{i\pi/4})|\bar{1}\rangle\Big]$$

tsim supports multiple visualization methods. The default is a ZX diagram, where measurement vertices are annotated with rec[i], and detector and observable vertices are annotated with det[i] and obs[i], respectively.

tsim also wraps stim's visualization functions:

To sample detection events and logical observables, we can compile a detector sampler, similar to stim.

Since the circuit is just a logical encoding of the 1-qubit circuit from the beginning of the tutorial, the logical observable should behave exactly like the physical qubit in the first example, i.e., the observable should be 1 with probability $\sin(\pi/8)^2 \approx 0.1464$.

Additionally, since the circuit is noiseless, we should not observe any non-zero detection events:

Adding Noise¶

A core capability of tsim is its support for Pauli noise channels.

Let's look at a simple example. We'll insert a depolarizing channel DEPOLARIZE1(0.01) before the final stabilizer measurements.

In the ZX diagram, the noise is represented by parametrized vertices with binary parameters e0, e1, etc. Since a depolarizing channel either applies X, Y, Z gates, each channel requires two bits, i.e. an X and a Z vertex.

Now we compile the sampler for the noisy circuit.

Sampling from the noisy circuit, we expect to see some non-zero detector events.

We again calculate the probability of measuring a logical $|\bar{1}\rangle$. Due to the noise, it deviates from the ideal value of $\sin(\pi/8)^2 \approx 0.1464$.

Error detection¶

One simple error correction strategy is post-selection: we discard any shots where a detector fired (indicating an error occurred). This effectively projects us back to the code space, but reduces the success rate (yield).

Error correction¶

To actively correct errors, we need a decoder. A decoder takes the detector syndrome and predicts whether the observable should be flipped.

In this example, we will use the tesseract decoder.

After correction, we see that the the probability of getting a logical $|\bar{1}\rangle$ is close to the ideal value.

Monte Carlo Simulation with sinter¶

tsim is compatible with sinter, a tool for performing large Monte Carlo simulations. We can use sinter to sample and decode over a range of physical error rates.

sinter provides a number of convenient plotting tools. Here, we use them to plot the observable flip rate as a function of the physical error rate. We observe that the decoded probability approaches the expected value of $\sin(\pi/8)^2$ much faster.