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A Three-Factor Solovay–Kitaev Error Budget

Statement

Suppose a fixed three-factor unitary block is $U = G_{3} G_{2} G_{1}$ , where each $G_{j} = e^{i ϕ_{j}} S_{j}$ with $ϕ_{j} \in R$ and $S_{j} \in S U (2)$ . If unitary gate words $W_{j}$ satisfy $∥ S_{j} - W_{j} ∥ \leq ε / 3$ , then the word product $C = W_{3} W_{2} W_{1}$ satisfies $\min_{∣ λ ∣ = 1} ∥ U - λ C ∥ \leq ε$ .

Example

Take three one-qubit factors in a fixed decomposition of a block. Each Solovay–Kitaev call is assigned one third of the total error tolerance. The scalar phases are recorded as one phase for the whole unconditional block.

Verification

Given: $G_{j} = e^{i ϕ_{j}} S_{j}$ and unitary words $W_{j}$ with $∥ S_{j} - W_{j} ∥ \leq ε / 3$ for $j = 1, 2, 3$ .

[F1] The operator norm is submultiplicative (The operator norm is submultiplicative and satisfies ||T^*T|| = ||T||^2).

[F2] The operator norm is the maximum over unit vectors (The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||), and the inner-product vector norm satisfies the triangle inequality (The inner-product norm is definite, homogeneous, and satisfies the triangle inequality).

1.1givenalgebra

Insert one factor at a time: $S_{3} S_{2} S_{1} - W_{3} W_{2} W_{1} = (S_{3} - W_{3}) S_{2} S_{1} + W_{3} (S_{2} - W_{2}) S_{1} + W_{3} W_{2} (S_{1} - W_{1})$ .

2.1F1F2step 1.1

Each $S_{j}$ and $W_{j}$ is unitary by the Given. For any unitary $Q$ , $∥ Q v ∥ = ∥ v ∥$ for every unit vector $v$ , so [F2] gives $∥ Q ∥ = 1$ . The submultiplicativity in [F1] and the triangle inequality in [F2] therefore bound the norm of the right side by $∥ S_{3} - W_{3} ∥ + ∥ S_{2} - W_{2} ∥ + ∥ S_{1} - W_{1} ∥ \leq ε$ .

3.1givenstep 2.1algebra∎

By Given, the scalar factors in $G_{3} G_{2} G_{1}$ multiply to $λ = e^{i (ϕ_{1} + ϕ_{2} + ϕ_{3})}$ . Therefore $∥ U - λ C ∥ = ∥ S_{3} S_{2} S_{1} - W_{3} W_{2} W_{1} ∥ \leq ε$ by step 2.1. This is an unconditional-block guarantee; it does not license discarding $λ$ inside a later control.

Depends on

Used by

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Sources

Christopher M. Dawson and Michael A. Nielsen, The Solovay–Kitaev Algorithm, arXiv:quant-ph/0505030v2 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415v6 (standard reference, not scraped)