Measurement-Based Quantum Computation

Measurement-based quantum computation (MBQC) represents a finite-qubit computation as an open graph resource consumed by local measurements. Its central object is not an individual measurement branch but the corrected channel obtained after outcome-dependent byproducts are applied or tracked. This page develops XY-plane qubit patterns, branch maps, flow and gflow certificates, ideal wire, rotation, and entangling kernels, and branch-complete verification. It stops before generic graph-state theory, generic dynamic-circuit semantics, hardware scheduling, delegated-computation security, fault-tolerance thresholds, and computational-advantage claims.

Required background. Graph States supplies finite-qubit graph states, CZ preparation, and stabilizer semantics. Mid-Circuit Measurement and Feedforward supplies adaptive histories, classical records, and the distinction between physical corrections and frame tracking.

Measurement Patterns as Quantum Computations

An MBQC pattern combines an open graph, a preparation prescription, local measurement commands, a classical dependency relation, and output corrections. Entangling operations create the resource before the computation is consumed. Each measurement produces a classical bit, changes the quantum branch, and may alter later measurement commands. The output therefore depends on both the quantum branch map and the classical record.

A useful claim has three separate layers:

• the graph and commands define a family of unnormalized branch maps;
• outcome-conditioned corrections or a tracked Pauli frame merge those branches into a logical channel; and
• a determinism certificate establishes that every accepted branch realizes the same logical isometry, up to probability and global phase.

Neither an acyclic command list nor a normalized state from one branch proves the third layer. A complete pattern record must expose every dependency, probability, correction, and resource currency needed to audit the claim.

The Ten-Field MBQC Pattern Record

Use the following record for a proposed pattern, a numerical audit, or an implementation result. A field is never left blank: give its value or explain why it is not applicable.

1. MBQC task and licensed claim — Name the input-to-output map or sampling task, distinguish an ideal identity from an empirical result, and state exactly what the record licenses.
2. Open graph, registers, inputs, outputs, and order — Give $V$, $E$, $I$, $O$, the measured set, register dimensions, and the tensor or readout order.
3. Resource state, input injection, preparation, and promises — Separate supplied input systems from prepared $|+\rangle$ vertices, entanglers, accepted-resource promises, and unsupported physical assumptions.
4. Measurement planes, angles, outcomes, and basis conventions — State the measurement plane, angle sign, plus/minus basis, bit assignment, and every branch-dependent commanded angle.
5. Classical dependencies, flow or gflow, and causal order — Give dependency parities, a valid flow or gflow certificate when claimed, logical order, physical measurement layers, and any signal shifts.
6. Branch maps, probabilities, byproducts, and frame updates — Give unnormalized branch maps, branch probabilities, normalization, branch phases where relevant, and the accumulated Pauli frame.
7. Output channel, corrections, readout, and postprocessing — State the corrected channel or distribution, physical-versus-tracked corrections, final measurement command, and reported-bit processing.
8. Resource currencies, adaptive depth, and implementation assumptions — Count supplied inputs, prepared vertices, entanglers, measurements, classical bits, measurement and adaptive layers, frame operations, accepted-run cost, and unavailable hardware currencies separately.
9. Verification data, metric, tolerance, uncertainty, and reproducibility — State exact identities, enumerated data, numerical metrics, tolerances, roundoff, sampling uncertainty or its justification as N/A, and reproducible evaluator details.
10. Conclusion, stopping point, and canonical handoff — State what the record establishes, what it does not establish, and which canonical owner receives the next question.

This record prevents a logical pattern-size statement from silently becoming a hardware-cost statement. It also keeps a supplied unknown input distinct from ancillas that the resource-state preparation actually creates.

Open Graphs, Resource States, and Input Injection

Let $G=(V,E)$ be a finite simple undirected graph. Let $I$ be an ordered input set, $O$ an ordered output set, and

$$M=V\setminus O$$

the measured set. A record must say whether $I$ and $O$ overlap. The examples below use disjoint input and output sets. Tensor factors are always ordered by the displayed vertex tuple before amplitudes or branch maps are written.

The graph-state entangler is

$$E_G=\prod_{\{u,v\}\in E}CZ_{uv}.$$

All graph-edge CZ gates commute, so the product is independent of the order chosen for this ideal operator. For a supplied input $|\psi\rangle_I$, the input-injected resource is

$$|G(\psi)\rangle =E_G\left( |\psi\rangle_I\otimes |+\rangle_{V\setminus I} \right).$$

The equation declares one supplied system for each input vertex and one prepared $|+\rangle$ state for each non-input vertex. It does not provide a native gate decomposition, an entangler schedule, a preparation fidelity, a loss model, or an accepted-resource probability. Those must be supplied separately if claimed.

This page imports the graph-state preparation object but gives it a specifically computational interpretation: inputs are injected, non-output vertices are consumed, and the remaining quantum systems carry a record-dependent output frame. General graph-state transformations and entanglement structure remain with Graph States.

Equatorial Measurements and One-Bit Teleportation

Throughout, the XY-plane basis and bit convention are

$$|\pm_\alpha\rangle =\frac{|0\rangle\pm e^{i\alpha}|1\rangle}{\sqrt 2}, \qquad s= \begin{cases} 0,&+\\ 1,&- \end{cases}.$$

Define

$$R_z(\theta)=e^{-i\theta Z/2}, \qquad J(\alpha)=H R_z(-\alpha).$$

For input vertex 1 and prepared output vertex 2, direct projection gives the exact one-bit kernel

$$(\langle s_\alpha|_1\otimes I_2) CZ_{12} (|\psi\rangle_1\otimes|+\rangle_2) = \frac{e^{-i\alpha/2}}{\sqrt 2} X_2^sJ(\alpha)|\psi\rangle_2.$$

The factor $e^{-i\alpha/2}/\sqrt2$ is part of the exact unnormalized branch map. Only after that equality is recorded may the normalized output be described, up to global phase, as $X^sJ(\alpha)|\psi\rangle$. Both outcomes have probability $1/2$, independent of the input, and the logical $J(\alpha)$ operation is recovered by applying or tracking $X^s$.

This identity is often called one-bit teleportation, but it is a local graph-wire kernel. The Bell-pair protocol, its two-bit message, and communication benchmarks belong to Quantum Teleportation. Basis rotations, unread outcomes, and shot estimators belong to Measurement in Circuits.

Byproducts, Pauli Frames, and Adaptive Angles

Suppose a logical qubit reaches measurement vertex $i$ with incoming frame $X_i^xZ_i^z$. With the basis convention above,

$$M_i^\theta X_i^xZ_i^z =M_i^{(-1)^x\theta+z\pi}.$$

To implement algorithmic angle $\alpha$, command

$$\theta=(-1)^x\alpha+z\pi \pmod {2\pi}.$$

The $z\pi$ term swaps the plus and minus labels, so it may always be implemented as a classical flip of the reported outcome bit. The $X$ term changes the angle sign. Except at special Pauli angles or under an explicitly proved symmetry, that sign change cannot be replaced by an outcome flip.

For a causal outcome string $\mathbf s$, define the branch map from the input Hilbert space to the output Hilbert space by

$$K_{\mathbf s} =\left( \bigotimes_{i\in M} \langle s_i{}_{\alpha_i(\mathbf s_{<i})}|_i \right) E_G \left( I_I\otimes|+\rangle_{V\setminus I} \right).$$

For input density operator $\rho$,

$$p(\mathbf s) =\operatorname{Tr}\!\left( K_{\mathbf s}\rho K_{\mathbf s}^{\dagger} \right).$$

If $C_{\mathbf s}$ is the declared physical output correction or the mathematical representative of a tracked frame, the corrected channel is

$$\mathcal E(\rho) =\sum_{\mathbf s} C_{\mathbf s}K_{\mathbf s}\rho K_{\mathbf s}^{\dagger}C_{\mathbf s}^{\dagger}.$$

A corrected deterministic isometry $U$ requires every accepted branch to obey

$$C_{\mathbf s}K_{\mathbf s} =e^{i\phi_{\mathbf s}} \sqrt{p_{\mathbf s}}\,U,$$

with $p_{\mathbf s}$ independent of the input. This condition does not imply that different branches have equal probabilities. The value $p_{\mathbf s}=2^{-|M|}$ is justified only by a pattern-specific argument or explicit enumeration.

A tracked frame is classical information, not a gate that was physically performed. Its operational effect appears when a later non-Pauli command, output correction, or readout basis consumes it.

Flow, gflow, Causal Order, and Determinism

Write $O^c=V\setminus O$, $I^c=V\setminus I$, and let $N(v)$ denote the neighborhood of $v$. For an XY-plane open graph, a flow is a map

$$f:O^c\longrightarrow I^c$$

and a strict partial order $\prec$ such that, for every $i\in O^c$,

$$i\sim f(i), \qquad i\prec f(i),$$

and

$$i\prec k \quad \text{for every }k\in N(f(i))\setminus\{i\}.$$

Outcome $s_i$ then generates the correction

$$X_{f(i)}^{s_i} \prod_{k\in N(f(i))\setminus\{i\}} Z_k^{s_i}.$$

The adjacency condition and the extra-neighbor order condition are essential. A topological ordering by itself is not a flow certificate.

Generalized flow permits set-valued correction targets. Define

$$g:O^c\longrightarrow\mathcal P(I^c),$$

and

$$\operatorname{Odd}(S) =\left\{ v\in V: |N(v)\cap S|=1\pmod 2 \right\}.$$

For XY-plane measurements, require all three conditions

$$j\in g(i) \Longrightarrow i\prec j,$$ $$j\in\operatorname{Odd}(g(i))\setminus\{i\} \Longrightarrow i\prec j,$$

and

$$i\in\operatorname{Odd}(g(i)).$$

Flow is a compact sufficient certificate; gflow reaches a broader deterministic class through set-valued corrections. Any theorem invocation must state its measurement plane and its extensivity, uniformity, strong-determinism, and stepwise assumptions. Neither certificate is necessary beyond its stated model, and neither converts an ideal partial order into a physical latency guarantee.

Logical Wires, Rotations, and Entangling Patterns

Concatenating one-bit kernels produces a logical wire. On the three-vertex path $1$–$2$–$3$, first measure vertex 1 at $\alpha$, then command

$$\beta'=(-1)^{s_1}\beta$$

at vertex 2. Every branch has probability $1/4$, and the normalized raw output is, up to a branch global phase,

$$X_3^{s_2}Z_3^{s_1} J(\beta)J(\alpha)|\psi\rangle.$$

The sign dependency is the smallest non-Pauli adaptivity audit: fixing the second angle generally changes the logical operation rather than merely relabeling a bit. Longer paths compose further $J$ steps while propagating outcome parities into later signs and $\pi$ shifts.

A two-wire entangling kernel uses inputs $1,2$, outputs $3,4$, and edges

$$E=\bigl\{ \{1,3\},\{2,4\},\{3,4\} \bigr\}.$$

Measuring both inputs in the X basis gives four equiprobable branches. The logical target is

$$CZ_{34}(H\otimes H),$$

and the output frame is

$$\left(X_3^{s_1}Z_3^{s_2}\right) \otimes \left(Z_4^{s_1}X_4^{s_2}\right).$$

Thus the reusable ideal kernels are a $J(\alpha)$ step, a multistep wire or rotation, and a two-wire CZ pattern. They specify logical maps and frames, not a graph-generation protocol or hardware schedule.

Universality, Compilation, and Resource Accounting

Because arbitrary equatorial angles provide $J(\alpha)=HR_z(-\alpha)$ and graph edges provide CZ, these kernels support ideal qubit universality under the declared preparation and measurement capabilities. This model-level statement is not an efficient graph compiler, a fault-tolerant universal set, a physical gate set, or evidence of quantum advantage. Exact and approximate gate-set criteria remain with Universal Gate Sets, while ordinary wire-and-gate composition remains with the Circuit Model.

Every resource ledger separates:

• supplied input width from the $|V\setminus I|$ prepared non-input vertices;
• graph vertices from accepted prepared-resource states;
• entangler count $|E|$ from entangler depth and physical success probability;
• measured vertices $|V\setminus O|$ from physical measurement layers;
• classical outcomes from parity or XOR operations, live memory, and feedforward dependencies;
• generic adaptive depth from Pauli-angle signal shifting and final frame postprocessing;
• physical corrections from tracked Pauli frames;
• logical pattern size from circuit-to-pattern compilation expansion; and
• per-attempt resources from accepted-run and retry cost.

Native gate counts, wall-clock latency, loss, calibration, thresholds, throughput, and implementation fidelity are N/A unless an explicit specialist model supplies them. Program representation, lowering, scheduling, and executable packaging belong to the Quantum Software Stack and Circuit Intermediate Representations.

Verification, Noise Boundaries, and Canonical Handoffs

An auditable MBQC calculation starts with exact branch operators. Check completeness by summing all branch probabilities, and compare corrected branches only after fixing tensor order and aligning their irrelevant global phases. Exact symbolic identities are normative. A numerical enumeration must name its runtime, arithmetic, tensor order, phase-alignment convention, metric, and tolerance. Sampling uncertainty is N/A when all branches are enumerated; a finite-shot confidence statement belongs only to actual sampled data.

Ideal correctness does not license a noise, threshold, or device claim. Generic Pauli and Clifford bookkeeping belongs to the Stabilizer Formalism, and generic non-Pauli equatorial measurements leave stabilizer closure. Encoded gadgets and overhead belong to Fault-Tolerant Gates, while threshold claims belong to the Threshold Theorem.

Physical graph generation, fusion, loss, detectors, delay lines, and experimental evidence belong to Photonic Qubits. Continuous-Variable Quantum Computation owns CV-cluster computation semantics, nullifiers, finite-squeezing logical noise, the Gaussian/non-Gaussian universality boundary, and the abstract model-resource record; Continuous-Variable Platforms retains physical mode-resource generation, loss, detectors, delay lines, calibration, and evidence. Blindness, leakage, traps, and trust assumptions belong to Blind and Delegated Quantum Computation. None follows from an ideal deterministic pattern.

Worked Audit: One-Bit Teleportation at a Nonzero Angle

This audit enumerates an ideal two-vertex pattern. Its numerical check is a roundoff-level reproduction of the exact algebra, not sampled or device data.

1. MBQC task and licensed claim — Audit exact one-bit teleportation of a supplied $|+\rangle$ input at $\alpha=\pi/3$. The record licenses only the ideal corrected $J(\pi/3)$ channel and the consequence of discarding its branch record.

2. Open graph, registers, inputs, outputs, and order — Use $V=(1,2)$, $E=\{\{1,2\}\}$, $I=\{1\}$, $O=\{2\}$, $M=\{1\}$, two qubits, and tensor order $(1,2)$. Input and output are disjoint.

3. Resource state, input injection, preparation, and promises — Supply $|+\rangle_1$, prepare $|+\rangle_2$, and apply ideal $CZ_{12}$. Preparation success, a native-gate realization, and a noise model are N/A because this is an ideal pattern audit.

4. Measurement planes, angles, outcomes, and basis conventions — Measure vertex 1 in the declared XY basis at $\alpha=\pi/3$; $|+_{\pi/3}\rangle$ gives $s=0$ and $|-_{\pi/3}\rangle$ gives $s=1$.

5. Classical dependencies, flow or gflow, and causal order — No angle depends on an earlier result. Flow is $f(1)=2$ with $1\prec2$; there is one logical and physical measurement layer. Signal shifting is N/A because there is only one measurement.

6. Branch maps, probabilities, byproducts, and frame updates — The exact branch operators are

   $$K_s =2^{-1/2}e^{-i\pi/6} X^sJ(\pi/3),$$

   so $p_0=p_1=1/2$ and the raw frame is $X^s$. The displayed scalar retains both the branch normalization and global phase.

7. Output channel, corrections, readout, and postprocessing — Applying or tracking $X^s$ gives

   $$J(\pi/3)|+\rangle =\frac{\sqrt3}{2}|0\rangle +\frac{i}{2}|1\rangle$$

   in both corrected branches. If the record is forgotten and no correction is made, the output is $I_2/2$. Its purity is $1/2$, its squared fidelity with the target is $1/2$, and its trace distance from the target is $1/2$. A final readout is N/A for this state-output audit.

8. Resource currencies, adaptive depth, and implementation assumptions — Count two qubits total, one supplied input, one prepared $|+\rangle$, one CZ, one measurement, one bit, measurement depth one, adaptive depth zero, and at most one $X$ correction or frame update. Accepted-run retry cost, hardware timing, native-gate cost, and throughput are N/A because no preparation-success or implementation model is declared.

9. Verification data, metric, tolerance, uncertainty, and reproducibility — The exact target amplitudes in tensor order $(1,2)$ after output extraction are $(0.8660254037844386,0.5i)$. A Node.js 26.4.0 binary64 enumeration, aligning each normalized branch by the unit phase that minimizes its Euclidean residual, used tolerance $2\times10^{-14}$. It found maximum probability error $3.33\times10^{-16}$, probability-sum defect $6.66\times10^{-16}$, and maximum phase-aligned state residual $2.31\times10^{-16}$. Exact algebra is normative; sampling uncertainty is N/A because both branches are enumerated.

10. Conclusion, stopping point, and canonical handoff — The audit verifies the ideal one-bit kernel, its frame, and the information lost by forgetting the outcome. It does not verify a device, communication protocol, compiler, or fault-tolerant gadget; those questions pass respectively to the relevant hardware page, Quantum Teleportation, the Quantum Software Stack, or Fault-Tolerant Gates.

Worked Audit: An Adaptive Three-Qubit Rotation

The second audit isolates the smallest generic angle dependency. It compares the correct adaptive command with a deliberately fixed second angle.

1. MBQC task and licensed claim — Audit an adaptive three-vertex rotation with input $|+\rangle$, $\alpha=\pi/3$, and $\beta=\pi/4$. The record licenses the ideal corrected $J(\beta)J(\alpha)$ channel and quantifies the error caused by omitting the sign dependency.

2. Open graph, registers, inputs, outputs, and order — Use path $V=(1,2,3)$, $E=\{\{1,2\},\{2,3\}\}$, $I=\{1\}$, $O=\{3\}$, $M=\{1,2\}$, three qubits, and tensor order $(1,2,3)$. Input and output are disjoint.

3. Resource state, input injection, preparation, and promises — Supply $|+\rangle_1$, prepare $|+\rangle_2|+\rangle_3$, and apply ideal $CZ_{12}CZ_{23}$. Physical preparation success, a native schedule, and a noise promise are N/A.

4. Measurement planes, angles, outcomes, and basis conventions — Measure vertex 1 at $\alpha=\pi/3$ and vertex 2 at commanded angle $\beta'=(-1)^{s_1}\pi/4$, using the declared XY basis with plus/minus encoded as zero/one.

5. Classical dependencies, flow or gflow, and causal order — Use flow $f(1)=2$, $f(2)=3$, order $1\prec2\prec3$, and the dependency $s_1\to\beta'$. The generic logical and physical measurement depth is two; no Pauli-angle signal shift removes this non-Pauli dependency.

6. Branch maps, probabilities, byproducts, and frame updates — For every $(s_1,s_2)$, $p(s_1,s_2)=1/4$. Up to a branch global phase, the normalized raw output is

   $$X^{s_2}Z^{s_1} J(\pi/4)J(\pi/3)|+\rangle.$$

   The accumulated frame is $X^{s_2}Z^{s_1}$.

7. Output channel, corrections, readout, and postprocessing — After applying or tracking the frame, the target amplitudes are

   $$a_0 = \frac{ \sqrt3e^{i\pi/8} +ie^{-i\pi/8} }{2\sqrt2} = 0.701057384650 +0.560985526797i,$$ $$a_1 = \frac{ \sqrt3e^{i\pi/8} -ie^{-i\pi/8} }{2\sqrt2} = 0.430459334577 -0.092295955641i.$$

   Corrected squared fidelity is one in every branch. If the second angle is incorrectly fixed at $+\pi/4$, both $s_1=0$ branches retain squared fidelity one, both $s_1=1$ branches have squared fidelity $5/8$, and the branch average is $13/16$. Up to global phase, the failing-branch overlap is $3/4+i/4$. Final readout is N/A because the audit compares state channels.

8. Resource currencies, adaptive depth, and implementation assumptions — Count three qubits total, one supplied input, two prepared $|+\rangle$ states, two CZ gates, two measurements, two bits, generic measurement and adaptive depth two, one sign dependency, and a final $X/Z$ frame. Accepted-run retry cost, native timing, and hardware latency are N/A.

9. Verification data, metric, tolerance, uncertainty, and reproducibility — A Node.js 26.4.0 binary64 enumeration in tensor order $(1,2,3)$, using unit-phase-aligned Euclidean state residuals and squared fidelities, used tolerance $2\times10^{-14}$. It gives maximum probability error $2.50\times10^{-16}$, probability-sum defect $8.88\times10^{-16}$, maximum phase-aligned corrected-state residual $2.31\times10^{-16}$, and maximum absolute corrected-infidelity residual $4.44\times10^{-16}$. Exact algebra is normative; sampling uncertainty is N/A because all four branches are enumerated.

10. Conclusion, stopping point, and canonical handoff — The audit verifies the parity-controlled sign update and quantifies its omission. It does not establish a general compiler, hardware latency, fault tolerance, or an advantage claim; those questions belong to their specialist owners.

Common Failure Modes and Ownership Boundaries

• Dropping the exact branch factor. The one-bit equality includes $e^{-i\alpha/2}/\sqrt2$. Removing it is permissible only after normalization and only when the remaining state is explicitly stated up to global phase.
• Changing only half of a convention. Replacing $e^{i\alpha}$ by $e^{-i\alpha}$ in the equatorial basis also changes the sign in the associated $J$ operation. A convention translation must update both.
• Reporting only normalized branches. A normalized state cannot reveal its occurrence probability. Give the unnormalized branch map and probability before comparing corrected states.
• Turning every frame into a gate. An output-dependent Pauli frame is classical side information unless a physical correction is explicitly chosen and counted.
• Replacing every sign dependency by a bit flip. A $z\pi$ command shift swaps outcome labels. A generic $X$-induced angle sign reversal changes the basis and is not the same operation.
• Calling any order a certificate. Flow requires adjacency and neighborhood checks; gflow requires its odd-neighborhood conditions. A topological ordering alone proves neither.
• Inferring uniformity from determinism. Branch-independent corrected action does not by itself force equal branch probabilities. Uniformity needs a separate proof or enumeration.
• Calling ideal simultaneity a schedule. Graph connectivity and commuting measurements do not specify native entanglers, controller timing, routing, detector response, or a hardware layer.
• Miscounting the supplied input. A supplied input occupies a graph vertex but is not a prepared $|+\rangle$ ancilla. State both currencies.
• Collapsing unlike resource currencies. Logical pattern size, accepted-resource cost, native gates, adaptive depth, and wall-clock time answer different questions and must remain separate.
• Overclaiming from an ideal identity. A branch-complete MBQC calculation is not evidence of hardware readiness, security, fault tolerance, or computational advantage.

Exercises

Translate an Equatorial-Basis Convention

An author uses

$$|\pm_\alpha\rangle_{\mathrm{alt}} = \frac{|0\rangle\pm e^{-i\alpha}|1\rangle}{\sqrt2}.$$

Translate the one-bit kernel into this convention and identify the logical $J$ operation.

Solution

The alternative state at angle $\alpha$ is the page’s state at angle $-\alpha$:

$$|\pm_\alpha\rangle_{\mathrm{alt}} =|\pm_{-\alpha}\rangle.$$

Substitute $-\alpha$ into the exact one-bit identity:

$$(\langle s_\alpha|_{\mathrm{alt}}\otimes I) CZ (|\psi\rangle\otimes|+\rangle) = \frac{e^{i\alpha/2}}{\sqrt2} X^sJ(-\alpha)|\psi\rangle.$$

Thus the normalized corrected action is $J(-\alpha)=HR_z(\alpha)$, up to global phase. The sign change records a relabeling of the basis parameter, not a different physical projector or channel.

Parallelize a Three-Node Pauli Wire

Set both algorithmic angles on a three-vertex path to zero. Determine the four branch probabilities, logical action, output frame, and physical measurement depth.

Solution

Here $J(0)=H$, so the corrected logical action is

$$J(0)J(0)=H^2=I.$$

The three-path identity gives probability $1/4$ for each $(s_1,s_2)$ and raw output

$$X_3^{s_2}Z_3^{s_1}|\psi\rangle.$$

The second commanded angle is $(-1)^{s_1}0=0$, independent of $s_1$. Both physical measurements are therefore X measurements and may occupy one measurement layer. The logical flow order $1\prec2\prec3$ remains useful for interpreting and processing the output frame; physical parallelism does not erase that dependency structure.

Consume an Incoming Pauli Frame

A logical qubit arrives with frame $X^xZ^z$. Derive the command that implements algorithmic equatorial angle $\alpha$, and determine which dependency may be converted into a reported-bit flip.

Solution

Commuting the incoming frame into the measurement command gives

$$M^\theta X^xZ^z =M^{(-1)^x\theta+z\pi}.$$

Equivalently, the physical command that realizes the desired logical angle is

$$\theta=(-1)^x\alpha+z\pi \pmod{2\pi}.$$

Adding $\pi$ exchanges $|+_\theta\rangle$ and $|-_\theta\rangle$, so one may omit the $z\pi$ shift and report $s=r\mathbin{\oplus}z$, where $r$ is the raw detector bit. By contrast, changing $\alpha$ to $-\alpha$ generally rotates to a different basis. The $x$ dependency cannot be replaced by a bit flip except at special Pauli angles or under a declared symmetry.

Derive a Two-Wire Entangling Pattern

Let $V=\{1,2,3,4\}$, $I=\{1,2\}$, $O=\{3,4\}$, and

$$E=\bigl\{ \{1,3\},\{2,4\},\{3,4\} \bigr\}.$$

Measure inputs 1 and 2 in the X basis. Find the branch probabilities, corrected target, and raw output frame.

Solution

Each input-to-output edge implements a one-bit step, while $CZ_{34}$ commutes with the two input projections. The two independent X-basis outcomes therefore give four branches with

$$p(s_1,s_2)=\frac14.$$

The logical one-bit steps supply $H\otimes H$, and the output edge supplies the entangler, so the corrected target is

$$CZ_{34}(H\otimes H).$$

Flow choices $f(1)=3$ and $f(2)=4$ give the direct $X_3^{s_1}$ and $X_4^{s_2}$ corrections. Because output 4 is the other neighbor of $f(1)=3$, outcome $s_1$ also contributes $Z_4^{s_1}$. Similarly, $s_2$ contributes $Z_3^{s_2}$. The raw frame is therefore

$$\left(X_3^{s_1}Z_3^{s_2}\right) \otimes \left(Z_4^{s_1}X_4^{s_2}\right).$$

Verify Flow on a Five-Vertex Path

For the path $1$–$2$–$3$–$4$–$5$ with input 1, output 5, and algorithmic angles $\alpha_1,\ldots,\alpha_4$, verify $f(i)=i+1$ and derive the commanded angles, output frame, and generic adaptive depth.

Solution

For each measured $i$, vertex $f(i)=i+1$ is adjacent to $i$ and later in the order

$$1\prec2\prec3\prec4\prec5.$$

The only other neighbor of $f(i)$, when it exists, is $i+2$, which is also later than $i$. All flow conditions hold. Propagating the $X$ and $Z$ components from earlier outcomes gives

$$\theta_1=\alpha_1, \qquad \theta_2=(-1)^{s_1}\alpha_2,$$ $$\theta_3=(-1)^{s_2}\alpha_3+s_1\pi, \qquad \theta_4=(-1)^{s_3}\alpha_4+s_2\pi.$$

The remaining output frame is

$$X_5^{s_4}Z_5^{s_3}.$$

For generic non-Pauli angles, each measurement command can depend on the preceding record, so the adaptive measurement depth is four. Special Pauli angles can permit signal shifting, but they do not change the generic count.

Read Out a Framed Logical Qubit

A logical output carries frame $X^xZ^z$ and is to be measured at algorithmic equatorial angle $\alpha$. Give two equivalent correct readout procedures and reject the tempting generic shortcut.

Solution

One may send the fully repaired command

$$\theta=(-1)^x\alpha+z\pi \pmod{2\pi}$$

to the measurement device and use its returned bit directly. Equivalently, command

$$\theta=(-1)^x\alpha$$

and convert the raw result $r$ to the logical result $s=r\mathbin{\oplus}z$. The alternatives are equivalent because the omitted $z\pi$ term only swaps the two basis labels.

There is no generic procedure that commands $\alpha$ and compensates $x$ with a bit flip: the $X$ frame changes $\alpha$ to $-\alpha$, which is generally a different equatorial basis.

Count Accepted Resource States

Every preparation attempt consumes 25 vertices, 32 entanglers, and 23 measurements. Attempts are independent, each resource is accepted with probability $0.8$, and 1,000 accepted outputs are required. Find the expected number and variance of attempts, its standard deviation, and the expected resource totals.

Solution

For the number $N$ of Bernoulli attempts required to obtain $r=1000$ acceptances with $p=0.8$, the negative-binomial moments are

$$\mathbb E[N] =\frac{r}{p} =1250,$$ $$\operatorname{Var}(N) =\frac{r(1-p)}{p^2} =312.5, \qquad \sigma_N =17.67766953.$$

Every attempt spends all listed resources, including rejected attempts. Multiplying the per-attempt costs by $\mathbb E[N]$ gives

$$\mathbb E[\text{vertices}] =31{,}250,$$ $$\mathbb E[\text{entanglers}] =40{,}000, \qquad \mathbb E[\text{measurements}] =28{,}750.$$

These are expected costs, not a fixed-run guarantee. A time per attempt, parallelism model, and controller model were not supplied, so throughput and wall-clock latency are N/A.

Complete a Ten-Field Four-Vertex Pattern Record

Complete the full audit record for the path $1$–$2$–$3$–$4$ carrying a supplied $|+\rangle$ input from vertex 1 to vertex 4 with zero algorithmic angles. Use Pauli signal shifting where it is valid.

Solution

1. MBQC task and licensed claim — Audit an ideal four-vertex Pauli wire carrying a supplied $|+\rangle$ input to corrected output $|0\rangle$. The record licenses only the exact logical $H$ map and its branch and frame record.

2. Open graph, registers, inputs, outputs, and order — Use path $V=(1,2,3,4)$, $E=\{\{1,2\},\{2,3\},\{3,4\}\}$, $I=\{1\}$, $O=\{4\}$, $M=\{1,2,3\}$, four qubits, and tensor order $(1,2,3,4)$. Input and output are disjoint.

3. Resource state, input injection, preparation, and promises — Supply $|+\rangle_1$, prepare $|+\rangle_2|+\rangle_3|+\rangle_4$, and apply the three ideal path-edge CZ gates. Preparation success, noise, and native scheduling are N/A.

4. Measurement planes, angles, outcomes, and basis conventions — Use algorithmic angles zero and commanded angles $\theta_1=0$, $\theta_2=0$, and $\theta_3=s_1\pi$ in the declared XY basis, with plus/minus encoded as zero/one.

5. Classical dependencies, flow or gflow, and causal order — Use $f(1)=2$, $f(2)=3$, $f(3)=4$, and logical order $1\prec2\prec3\prec4$. Because $\theta_3$ differs from zero only by $\pi$, all three vertices may be physically X-measured in one layer and the commanded bit is $s_3=r_3\mathbin{\oplus}s_1$. This signal shift is specific to the Pauli angle and does not extend to generic non-Pauli angles.

6. Branch maps, probabilities, byproducts, and frame updates — Every one of the eight branches has probability $1/8$. Up to branch global phase, the normalized raw output for arbitrary input $|\psi\rangle$ is

   $$X_4^{s_3}Z_4^{s_2} H|\psi\rangle.$$

   The final frame is $X_4^{s_3}Z_4^{s_2}$; the earlier $s_1$ dependence has been absorbed into the commanded bit.

7. Output channel, corrections, readout, and postprocessing — Apply or track $X_4^{s_3}Z_4^{s_2}$. Since $H|+\rangle=|0\rangle$, every corrected branch returns $|0\rangle$. The only required signal postprocessing is $s_3=r_3\mathbin{\oplus}s_1$; a further output readout is N/A for this state-output task.

8. Resource currencies, adaptive depth, and implementation assumptions — Count four qubits total, one supplied input, three prepared $|+\rangle$ states, three CZ gates, three measurements, three bits, one physical Pauli-measurement layer, no adaptive measurement latency, one XOR, and a final $X/Z$ frame. Entangler depth, native cost, accepted-run retry cost, hardware latency, and throughput are N/A without an implementation model.

9. Verification data, metric, tolerance, uncertainty, and reproducibility — Symbolically, $J(0)^3=H^3=H$. Exhaustive exact enumeration gives eight probabilities equal to $1/8$, probability sum one, and corrected squared fidelity one in every branch. A binary64 runtime, numerical tolerance, and roundoff metric are N/A because no floating-point result is used; sampling uncertainty is N/A because the symbolic record is exhaustive.

10. Conclusion, stopping point, and canonical handoff — The record verifies an ideal branch-complete Pauli wire and a legal Pauli signal shift. It does not verify a graph-state source, physical parallelism, fault tolerance, compilation efficiency, or speedup; those claims pass to their canonical owners.

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