Simulating Noisy Circuits

Every real quantum processor is coupled to its environment. Gates are imperfect, qubits leak energy, and stray fields shift phases. A noise model is a mathematical prescription that augments an ideal circuit with probabilistic error operations so that simulated statistics match measured hardware data.

From ideal to noisy: the insertion picture

The simplest simulation strategy is error insertion: after each ideal gate draw from the noise distribution and, if an error occurs, append the corresponding error gate. For a bit-flip channel with error probability $p$ this means appending $X$ with probability $p$ and doing nothing with probability $1-p$ .

Concretely, one simulation trajectory (or "shot") proceeds as:

Initialise all qubits to $|0\rangle$ .
Apply each ideal gate in order.
After each gate, sample whether an error occurred and, if so, insert the error gate.
Measure and record the outcome bit-string.

Repeating for many shots produces a sampled distribution whose histogram approximates the true noisy output probabilities.

A worked example: bit-flip noise on a Bell-adjacent circuit

Consider a two-qubit circuit whose ideal action is

|\psi_\text{ideal}\rangle = H \otimes I \, |00\rangle = |{+}\rangle|0\rangle = \tfrac{1}{\sqrt{2}}\bigl(|00\rangle + |10\rangle\bigr).

Measuring this ideal state yields the outcomes $00$ and $10$ each with probability $\tfrac{1}{2}$ .

Now suppose qubit 1 suffers a certain bit-flip error ( $p = 1$ ) after the gates. Inserting the Pauli $X$ on qubit 1 gives the noisy state

|\psi_\text{noisy}\rangle = I \otimes X \, |\psi_\text{ideal}\rangle = \tfrac{1}{\sqrt{2}}\bigl(|01\rangle + |11\rangle\bigr).

The measurement distribution shifts from $\{00, 10\}$ to $\{01, 11\}$ , each still with probability $\tfrac{1}{2}$ . The error is detectable: a classically expected distribution has no weight on $01$ or $11$ , so any observation in those bins signals a bit-flip on qubit 1.

Reading the sampled histogram

After running the noisy circuit for many shots the histogram shows:

| Outcome | Ideal probability | Noisy probability ( $p=1$ ) | |---------|-------------------|---------------------------| | $00$ | $\tfrac{1}{2}$ | $0$ | | $01$ | $0$ | $\tfrac{1}{2}$ | | $10$ | $\tfrac{1}{2}$ | $0$ | | $11$ | $0$ | $\tfrac{1}{2}$ |

The error has shifted every outcome bit-string by flipping qubit 1. In a probabilistic noise model ( $0 < p < 1$ ) all four bars would be non-zero, and a syndrome measurement (not a destructive readout) could diagnose the error without collapsing the encoded information.

From Kraus operators to circuit insertion

The bit-flip channel studied in an earlier lesson has Kraus operators

K_0 = \sqrt{1-p}\,I, \qquad K_1 = \sqrt{p}\,X.

Each shot samples one of these operators: with probability $1-p$ the identity acts (no error), and with probability $p$ the Pauli $X$ acts (bit flip). The circuit-insertion picture is a direct Monte Carlo realisation of this Kraus decomposition.

For more complex noise models — such as depolarizing or amplitude-damping — the same principle applies: each Kraus operator is mapped to a circuit fragment, and the stochastic choice between fragments is sampled at run time.

Try it

Build the two-qubit noisy circuit described above. Apply $H$ to qubit 0, then inject a certain bit-flip error on qubit 1 by applying $X$ to qubit 1, then measure. The grader checks that the output distribution places equal weight on $|01\rangle$ and $|11\rangle$ .

Run your code to see the quantum state.

Open the Probabilities tab: the bars for $00$ and $10$ should be absent, and the bars for $01$ and $11$ should each reach $0.5$ . That is the signature of a certain bit-flip error on qubit 1.