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BQP is the class of promise problems decided by uniform polynomial-size quantum circuits with bounded error. Completeness $2/3$ and soundness $1/3$ are conventional; BQP error reduction shows that other fixed separated thresholds give the same class.

Expand a quantum circuit matrix element by inserting computational basis resolutions between gates, then enumerate the intermediate strings depth first. The active stack stores one basis label and accumulator per quantum circuit layer, using polynomial space for polynomially many qubits and gates. Recompute entries rather than storing an exponential matrix or state vector. The output probability requires the Born rule sum of squared amplitude moduli.

The least worst-case number of input-oracle calls made by a quantum circuit that computes a Boolean function with zero error on every input. Known unitary gates and classical processing do not count toward this quantity; their runtime is a separate cost.

A Feynman-Kitaev Hamiltonian penalizes incorrect input initialization, disagreement between successive quantum circuit steps and clock labels, and optionally a rejecting output. Its propagation quadratic form is a sum of $\frac{1}{2}\Vert {\eta }_t-U_t{\eta }_{t-1}{\Vert }^2$. Without output penalty, its zero-energy space consists of correctly initialized computational history states. A nonlocal quantum clock makes the propagation formula simple but does not supply fixed qubit locality by itself.

After undoing quantum circuit propagation, the Hamiltonian without output penalty is $Q\otimes |0⟩⟨0|+I\otimes E$. The common kernel is the valid input space times the uniform clock vector. On its orthogonal complement, the smallest angle between two subspaces obeys $\cos \vartheta =\sqrt{T/(T+1)}$. The Kitaev geometrical lemma and path gap $2/(T+1{)}^2$ give $\Delta \ge (T+1{)}^{-3}$. Degenerate valid quantum witnesses must be removed together when computing this angle.

Hamiltonian simulation constructs a quantum circuit approximating the time-evolution operator $e^{-itH}$ generated by a Hamiltonian operator $H$, with cost controlled by the evolution time, requested precision, and structure of $H$.

A nonlocal quantum clock records quantum circuit time in an abstract $T+1$ dimensional register. A binary encoding uses logarithmically many qubits, but transitions $|t⟩⟨t-1|$ need not have bounded qubit locality. Extra diagonal penalties may exclude unused binary labels. A unary quantum clock provides a different encoding for fixed-locality constructions.

A deterministic finite-dimensional universal programmable quantum gate cannot implement every unitary exactly: programs for physically distinct unitaries would have to be mutually orthogonal, but a finite-dimensional program register contains only finitely many mutually orthogonal states.

For an even-length Boolean string, one phase query followed by a known unitary extension can produce a zero-pair outcome with amplitude proportional to the zero-minus-one imbalance, or an index pair with amplitude proportional to the difference of their phase signs. A zero-pair observation certifies nonbalance. Every nonzero pair has opposite bits, so deleting it preserves the imbalance and allows recursion on two fewer indices. Ending with an empty list certifies balance. This quantum circuit is exact on all inputs and uses at most half the original length in queries, even without the constant-or-balanced promise.

Replace every CNOT gate by $H_j{E}_{ij}{H}_j$. The resulting quantum circuit contains only Hadamard gates and Controlled-Z gates. Realize each Hadamard gate by a fresh one-bit teleportation link measured at angle zero, so every internal measurement is in the fixed $X$ basis. Insert a graph edge between the current wire vertices for each Controlled-Z gate. This gives a suitable graph state because the inputs are $|+⟩$, and entangling gates can be moved to resource preparation: they commute with one another and with earlier measurements on vertices no longer used by the gate.

Track a Pauli frame ${⨂}_i{X}_i^{p_i}{Z}_i^{q_i}$. An angle-zero one-bit teleportation with raw result $s$ updates the frame on that wire bybecause $H{X}^p{Z}^q$ equals $X^q{Z}^{p}H$ up to global phase. A Controlled-Z gate updates $q_i\leftarrow q_i\oplus p_j$ and $q_j\leftarrow q_j\oplus p_i$, leaving the $p$ bits unchanged. These are classical binary updates; no measurement angle depends on them.

All the internal quantum measurements consequently have predetermined bases. Their projectors act on distinct resource vertices and commute, so all internal measurements can be performed simultaneously: the logical measurement depth is at most one. If the output is to be read in the computational basis, those fixed-basis measurements can occur in the same layer; a raw output $t_i$ is relabelled $t_i\oplus p_i$. If quantum outputs are retained, the same calculation gives the desired output in a known Pauli frame. This nonadaptive Hadamard–CNOT measurement pattern concerns measurement depth, not the depth of resource preparation or classical frame computation.

The quantum circuit has one control wire per binary digit and one target quantum register. Every control starts in $|0⟩$, receives a Hadamard gate, controls the corresponding power $U^{2^j}$, and then enters the collective inverse quantum Fourier transform. Only the control register is measured; the target stays in $|\psi ⟩$.

The omitted intermediate wires continue the pattern $j=0,1,\dots ,m-1$. The inverse quantum Fourier transform black box is defined on the integer basis $|k⟩$ with $k={\sum }_j{2}^j{k}_j$, so the displayed ordering introduces no hidden bit reversal. The measured bits satisfyEach controlled power box represents $2^j$ sequential controlled-$U$ uses when only that primitive is available.

The amplitude amplification theorem concerns a known preparation quantum circuit $A$ and a good-subspace projector $\Pi$. Write $A|0⟩=\sin \theta |g⟩+\cos \theta |b⟩$, with ${\sin }^2\theta =p=\Vert \Pi A|0⟩{\Vert }^2$. Let $S_0=I-2|0⟩⟨0|$ and $S_{\chi }=I-2\Pi$. ThenThe reflection operators preserve the two-dimensional good-bad plane and rotate it by $2\theta$, so $O(1/\sqrt{p})$ iterations amplify a small known success probability to a constant close to one. Each iteration uses one good-subspace phase test and one use each of $A,A^{†}$, together with a known reflection. Amplitude amplification provides a quadratic improvement in the number of repetitions of a successful preparation. The query cost of the preparation and its inverse must be included when they themselves use the input oracle. Exact amplitude amplification uses additional known-overlap preparation or phase matching to avoid integer-iteration overshoot.

An explicit measurement-based quantum computation pattern uses six vertices $u_0,u_1,u_2,v_0,v_1,v_2$. Prepare a graph state with every vertex in $|+⟩$ and apply a Controlled-Z gate for each edgeThe first two links on each wire permit graph-state preparation of a computational-basis input followed by the logical J gate. Use the following single-qubit measurements:

All entangling edges can be made at preparation time because Controlled-Z gates commute. A future edge that does not touch a currently measured vertex can equivalently be deferred, which allows the one-bit teleportation identities to be applied in their logical order.

The two initial $X$ measurements implement $H|+⟩=|0⟩$ with Pauli frames $X^{s_0},X^{t_0}$ on the logical inputs. The first adaptive J gate then has output frame $X^{s_1}{Z}^{s_0}$ on $u_2$. Propagating through $E_{u_2{v}_1}$ gives framesup to branchwise global phase. The second adaptive J gate converts the latter intoA $Z$ correction does not alter a computational-basis measurement, while an $X$ correction flips its bit. Consequently the deterministic classical postprocessing isThis reproduces the output-bit distribution of the original quantum circuit, including its known byproduct corrections.

There is an additional simplification for these particular zero inputs. Since $J(\alpha )|0⟩=|+⟩$, $E(|+⟩|0⟩)=|+⟩|0⟩$, and $J(\beta )|0⟩=|+⟩$, the exact final state is $|+⟩|+⟩$, independently of the angles. The requested bit is therefore fair. A single isolated graph-state vertex measured in $Z$ already simulates that bit distribution; the six-vertex pattern also explicitly realizes the circuit and its corrections.

Use the operator norm induced by the usual vector norm, and assume the input quantum state is normalized. Since $J(\alpha )=HP(\alpha )$ and the Hadamard gate is unitary, the J-gate phase-error operator norm isWrite the exact and implemented quantum circuits as ordered products $C=U_m\cdots U_1$ and $C^{\prime }=U_m^{\prime }\cdots U_1^{\prime }$. The quantum circuit gate-error telescoping bound follows fromEvery surrounding factor is unitary, including gates tensored with identities on other qubits, so the triangle inequality and the submultiplicativity of the operator norm giveThe exact Controlled-Z gates contribute zero to that sum. ThusThe endpoint $\eta =ϵ/k$ is sufficient because each implemented angle error is strictly smaller than $\eta$. If $k=0$, the circuits are identical and any positive $\eta$ works. The bound controls the stated vector distance with actual gate phases retained, so no adjustment of the global phase of one output is needed.