Noisy Magic States

A factory cannot deliver the ideal $|T\rangle$. Whatever circuit prepares it is itself faulty, so the output is a mixed state $\rho$ close to but not equal to $|T\rangle\langle T|$. To reason about distillation we need a clean, quantitative noise model and a single number that captures "how good" a noisy copy is.

The depolarizing model

The simplest isotropic noise model mixes the target with the maximally mixed state:

Here $p$ is the depolarizing parameter: with probability $p$ the qubit is replaced by white noise. On the Bloch sphere this shrinks the Bloch vector toward the center without rotating it. If $\hat n = \tfrac{1}{\sqrt2}(1,1,0)$ is the ideal $|T\rangle$ direction, the noisy state has Bloch vector $(1-p)\,\hat n$ — same direction, length reduced from $1$ to $1-p$.

Fidelity and input error

The figure of merit is the fidelity to the ideal magic state, which for a pure target reduces to the expectation value

It is more convenient to track the input error (or infidelity)

A factory producing depolarized copies at parameter $p = 0.08$ thus delivers magic states with error $\varepsilon = 0.04$. The whole point of distillation is to drive $\varepsilon$ down by orders of magnitude.

Stochastic vs. coherent errors

A subtlety worth flagging: the depolarizing model treats the error as stochastic (a probabilistic mixture). Real preparation faults can be coherent — a small over-rotation $|T\rangle \to e^{-i\theta Z/2}|T\rangle$ — which to first order also gives fidelity $F \approx 1 - O(\theta^2)$ but behaves differently under accumulation. Distillation protocols are designed to suppress the leading-order stochastic error; coherent errors are typically converted to stochastic ones by the Clifford twirl implicit in the protocol. For this module we take the depolarized $\varepsilon = 1 - F$ as the input quality, the same convention used in the Bravyi–Kitaev analysis.[^bravyi-kitaev]

Where this is heading

We now have the language to state the distillation problem precisely. A protocol takes $n$ copies of a state with input error $\varepsilon_{\text{in}}$ and, using only stabilizer operations, outputs $k$ copies with output error $\varepsilon_{\text{out}}$. A good protocol has $\varepsilon_{\text{out}} \ll \varepsilon_{\text{in}}$ whenever $\varepsilon_{\text{in}}$ is below a threshold. The canonical example — the $15$-to-$1$ protocol — is next.

Try it

For a depolarized magic state with $p = 0.08$, compute the input error $\varepsilon = 1 - F$ where $F = \langle T|\rho|T\rangle$. Derive $F$ from the density matrix; do not hard-code it.

[^bravyi-kitaev]: Bravyi, Kitaev, Universal quantum computation with ideal Clifford gates and noisy ancillas, arXiv:quant-ph/0403025.

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