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Circuit and pattern translations are branch preserving and universal

Statement

Every finite J/CZ circuit has an adaptive open-graph pattern whose corrected branches implement the same unitary channel and whose transcript branches preserve the exact probabilities. Conversely, every finite open-graph pattern has a raw measurement-circuit presentation with exactly its raw branch operators $K_{b}$ , probabilities, and positive-probability conditional outputs. Appending its specified and separately charged correction suffixes gives a corrected measurement-circuit presentation with exactly its corrected branch operators $C (b) K_{b}$ , unchanged probabilities, and corrected conditional outputs $C (b) ρ_{b} C (b)^{*}$ . The corresponding raw instruments agree transcript by transcript, as do the corresponding corrected instruments; the corrected channels agree after forgetting the transcript. For every $n \geq 0$ , each unitary channel on the ordered $n$ -qubit register $Q_{n}$ has a J/CZ circuit representative and hence an adaptive pattern implementing the same channel; for $n \geq 1$ the unitary matrix has size $2^{n}$ , and for $n = 0$ the channel is the identity on $C$ . Global scalar phases are retained in branch maps and do not alter the induced channel.

Facts & Assumptions

Given: A finite unitary circuit or a finite open-graph pattern, with the ordered wire and branch conventions of the assigned definitions.

[F1]

Every finite J/CZ circuit is represented by the circuit-to-pattern construction, with branch maps, probabilities and corrected channel as stated (A finite J/CZ circuit is simulated branch by branch by its adaptive pattern).

[F2]

Every finite open-graph pattern has a raw measurement circuit with the same exact ordered $K_{b}$ , outcome law and raw conditional states. Appending the specified separately charged suffix gives a corrected circuit with the same $C (b) K_{b}$ and corrected conditional states as the corrected pattern; the raw and corrected transcript-labelled instruments agree with their respective counterparts (Every finite open-graph pattern is a measurement circuit).

[F3]

$J (α) = H P_{α}$ , $J (0) = H$ , and $P_{α} = H J (α)$ (J gate and ordered equatorial measurement vectors, J-gate basis and exact phase identities).

[F4]

The angle- $π$ controlled phase is $C Z = diag (1, 1, 1, - 1)$ (Controlled Phase Rotation).

[F5]

Define local notation $R_{z}^{E} (ϕ) : = R_{z} (- ϕ)$ and $R_{y}^{E} (θ) : = R_{y} (- θ)$ using the QC07 convention. Thus $R_{z}^{E} (ϕ) = diag (e^{i ϕ / 2}, e^{- i ϕ / 2})$ and $R_{y}^{E} (θ) = (\begin{array}{cc} \cos (θ / 2) & \sin (θ / 2) \\ - \sin (θ / 2) & \cos (θ / 2) \end{array})$ . The actual Euler theorem One-Qubit Euler Decomposition gives $U = e^{i δ} R_{z} (a) R_{y} (t) R_{z} (b)$ . Setting $α = - a$ , $θ = - t$ , $β = - b$ yields $U = e^{i δ} R_{z}^{E} (α) R_{y}^{E} (θ) R_{z}^{E} (β)$ , retaining the same scalar. The superscript $E$ is notation local to this proof, with the opposite angle signs from the supplier's rotations.

[F6]

The one-qubit matrix definition fixes $H = 2^{- 1 / 2} (\begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array})$ , and the Pauli definition fixes $X, Y, Z$ (Exact one-qubit gates and rotations, The Pauli matrices).

[F7]

Every finite-dimensional unitary channel has an exact finite circuit with no ancillas using arbitrary one-qubit gates and CNOT gates; the $n = 0$ channel is the identity (Exact finite unitary synthesis from cosine-sine recursion). This supplier proves existence from the unoptimized cosine-sine recursion and makes no gate-count, finite-bit, or synthesis-time claim.

Proof

1.1F1construct

For a finite J/CZ circuit, [F1] supplies the open-graph pattern and gives the equality of every transcript branch map, the branch probabilities and the corrected unitary channel. This proves the circuit-to-pattern direction with the arbitrary input unchanged.

1.2F2algebra

For a finite open-graph pattern, [F2] first lists its ancilla preparations, edge gates, ordered projective measurements and transcript-controlled basis choices as a raw measurement circuit. Its exact branch operator is $K_{b}$ , giving the same $p_{b}$ and the same raw conditional state $ρ_{b}$ whenever $p_{b} > 0$ . The separately charged terminal correction then gives $C (b) K_{b}$ and the conditional state $C (b) ρ_{b} C (b)^{*}$ , exactly as for the corrected pattern. Unitarity of $C (b)$ leaves $p_{b}$ unchanged. Therefore the respective raw and corrected transcript instruments agree, and summing corrected branches gives the same corrected channel. No normalized state is assigned on a zero-probability branch.

1.3F3F4F5F6F7algebra

By [F7], choose a finite exact synthesis circuit. For each one-qubit factor, [F5] gives its Euler product. Direct multiplication of the Pauli matrices gives $S : = P_{π / 2} = diag (1, i)$ , $S X S^{†} = Y$ , and $H Z H = X$ . Therefore $R_{y}^{E} (θ) = S H R_{z}^{E} (θ) H S^{†}$ : indeed, conjugating $R_{z}^{E} (θ) = e^{i θ Z / 2}$ by $H$ gives $e^{i θ X / 2}$ , and conjugating that by $S$ gives $e^{i θ Y / 2}$ , whose matrix is the one in [F5]. Also $R_{z}^{E} (ϕ) = e^{i ϕ / 2} P_{- ϕ}$ . By [F3], each $P_{- ϕ}$ is $H J (- ϕ)$ , $H = J (0)$ , $S = P_{π / 2}$ is $H J (π / 2)$ , and $S^{†} = P_{- π / 2}$ is $H J (- π / 2)$ . Thus each Euler factor is a finite product of J gates times its explicitly retained scalar. On basis inputs $∣ 0, y ⟩$ , $(I \otimes H) C Z (I \otimes H)$ fixes the target; on $∣ 1, y ⟩$ it applies $H Z H = X$ to the target. Hence this circuit maps $∣ x, y ⟩$ to $∣ x, y \oplus x ⟩$ and equals CNOT. Replacing each one-qubit gate and CNOT in the finite decomposition therefore gives a finite J/CZ circuit for the same unitary channel.

2.1step 1.3F1algebracases∎

Apply [F1] to that J/CZ circuit. The resulting finite pattern has the same corrected unitary channel and exact branch probabilities; its scalar phase remains part of each branch map. This proves universality at the unitary-channel level, including the one-dimensional identity case with an empty circuit and empty transcript.

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Sources

Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415, §20.6 (standard reference, not scraped)
Robert Raussendorf and Hans J. Briegel, Quantum computing via measurements only, pp. 2–4 (standard reference, not scraped)
Robert Raussendorf, Daniel E. Browne, and Hans J. Briegel, Measurement-based quantum computation on cluster states, §§II F–G (standard reference, not scraped)