Controlled Operations

A controlled operation is one joint transformation on a control register and a target register. It is not shorthand for measuring the control and then deciding which gate to run. When the control is coherent, the branch labels remain quantum amplitudes, branch phases can interfere, and the operation can entangle the control with the target.

This page develops the semantic and verification contract for such operations. It treats projector-controlled blocks, open and multiple controls, SELECT operators, branch phases, eigenstate action, and the resources required to add a control. Named CNOT, CZ, and Toffoli matrices remain with the gate-family page; constructive multiplexor networks remain with compilation; reusable phase-transduction conversions remain with their dedicated concept owner; and phase-estimation or algorithm-specific oracle consequences remain with their algorithmic owners.

Required background. Multi-Qubit Gates supplies tensor order, control–target conventions, common controlled gates, and their entangling behavior. For review, Circuit Model distinguishes coherent and classical wires, while Single-Qubit Gates fixes target matrices and exact, global-phase, and channel-level equality.

Controlled Operations as Coherent Conditionals

Let $C$ be a control system and $T$ a target system. A coherent conditional applies different target operations in orthogonal control sectors while preserving superpositions among those sectors. In the simplest qubit case,

C_1(U) = |0\rangle\langle0|_C\otimes I_T +|1\rangle\langle1|_C\otimes U_T.

With control first and target second, this is the block matrix

C_1(U) = \begin{pmatrix} I&0\\ 0&U \end{pmatrix}.

If the input control is $a|0\rangle+b|1\rangle$ , then

C_1(U) \bigl(a|0\rangle+b|1\rangle\bigr)|\psi\rangle = a|0\rangle|\psi\rangle +b|1\rangle U|\psi\rangle.

No measurement has occurred. If $U|\psi\rangle$ is not proportional to $|\psi\rangle$ and both amplitudes are nonzero, the result is generally entangled. If a measurement record instead selects a later operation, the overall process is an instrument or channel with different coherence properties.

Before using a controlled-gate symbol, declare:

the control and target registers and their tensor order;
the basis in which the control condition is defined;
the exact target operator, including its phase representative;
whether access is a known circuit, a controlled primitive, a black box, or a physical subspace construction;
whether cost means gates, queries, depth, ancillas, or physical evolution time.

The Gates, Circuits, and Computation Models guide places this contract among the surrounding gate, measurement, and computation-model pages.

The Ten-Field Controlled-Operation Record

Every implementation or theorem claim should fill this record. Write N/A with a reason when a field genuinely does not apply; a blank field leaves the access or phase convention underdetermined.

Controlled task and licensed claim — State the conditional transformation and the strongest conclusion sought.
Registers, dimensions, order, and basis — Name every register, fix tensor order and bit significance, and declare the control basis.
Control condition, projectors, and promises — Give the active values or predicate, a complete projector resolution, and any input promise.
Target operation and access model — Specify every branch operator and whether it is known, synthesized, queried, native, or supplied through extra physical structure.
Input state or subspace and preparation — State the tested or licensed domain, including reference entanglement and ancilla preparation.
Branch map, phase, inverse, and power conventions — Fix branch phases, operator order, inverse action, and the meaning and cost of powers.
Ancillas, synthesis assumptions, and resource currencies — Separate width, gates, depth, queries, time, and fault-tolerant or hardware resources.
Observable, metric, and verification evidence — Name truth-table, interference, matrix, state, channel, or residual checks and their tolerances.
Alternatives, failure modes, and uncertainty — Record classical-feedforward alternatives, phase ambiguity, leakage, approximation, and unresolved access assumptions.
Conclusion, stopping point, and canonical handoff — License only what the evidence establishes and route synthesis, algorithms, or hardware onward.

A basis truth table is never enough when relative phase matters. At minimum, a coherent-control audit combines branchwise basis tests with one phase-sensitive superposition test and an explicit inverse or unitarity check.

Projector and Block-Diagonal Semantics

Let $\{P_j\}$ be a complete orthogonal resolution of the control space:

P_jP_k=\delta_{jk}P_j, \qquad \sum_jP_j=I_C.

Let every $U_j$ be unitary on the same target space. The general block-controlled operator is

W=\sum_jP_j\otimes U_j.

Orthogonality removes cross terms:

\begin{aligned} W^\dagger W &= \sum_{j,k}P_jP_k\otimes U_j^\dagger U_k\\ &= \sum_jP_j\otimes I_T =I_C\otimes I_T. \end{aligned}

Its inverse is branchwise,

W^{-1}=W^\dagger = \sum_jP_j\otimes U_j^\dagger.

If the projectors overlap, fail to span the declared control space, or select blocks of different target dimension, this formula does not yet define the advertised complete unitary. A promised action on only part of the space needs either a full extension or an explicit out-of-promise contract.

Register order is part of the definition. In control–target order, $C_1(U)=\operatorname{diag}(I,U)$ . In target–control order the matrix is related by the SWAP permutation, not by silently reinterpreting indices. Operators on Composite Systems owns the general tensor-product and projector algebra.

Open, Negative, and Multiple Controls

An open control activates on computational-basis value zero:

C_0(U) = P_0\otimes U+P_1\otimes I = (X\otimes I)C_1(U)(X\otimes I),

where $P_b=|b\rangle\langle b|$ . The word “negative” refers to this declared basis convention; it is not a basis-independent physical property.

For a Boolean predicate $g(c)$ on a multi-bit control string,

P_g = \sum_{c:g(c)=1}|c\rangle\langle c|,

and

C_g(U) = (I-P_g)\otimes I+P_g\otimes U.

A conventional Toffoli is the special case $g(c_1,c_0)=c_1c_0$ with $U=X$ , but Multi-Qubit Gates remains the canonical home for its matrix, entangling behavior, and native-versus-compiled interpretation. Reversible Computation owns Boolean embeddings, constants, garbage, and uncomputation.

Mixed-polarity controls are obtained by conjugating the appropriate control wires with $X$ . Their resource cost depends on the declared alphabet: a diagram may count one mixed-polarity primitive, while a decomposition into positive controls counts the surrounding flips. State which model is being used.

Multiplexed and SELECT Operations

A selector register can choose among many coherent branches. For $k$ selector qubits and target unitaries $U_j$ ,

\operatorname{SELECT}(U) = \sum_{j=0}^{2^k-1} |j\rangle\langle j|\otimes U_j.

This is a unitary exactly when the control projectors form a complete orthogonal resolution and every branch is unitary on the common target space. Its inverse is

\operatorname{SELECT}(U)^{-1} = \sum_j|j\rangle\langle j|\otimes U_j^\dagger.

Branch phases are operational. Replacing $U_j$ by $e^{i\gamma_j}U_j$ multiplies the SELECT operator by a diagonal phase operation on the selector register. Equal isolated channels therefore do not imply equal multiplexed gates.

One abstract SELECT call is not automatically one elementary gate or one oracle query. It may conceal branch logic, data access, ancillas, routing, and approximation. Gate Decomposition owns Gray-code and multiplexor networks, topology-aware synthesis, and compiled cost; Algorithmic Primitives owns SELECT and controlled access as algorithmic interfaces.

Coherent Control versus Classical Feedforward

A controlled unitary preserves amplitudes between control sectors. For a control state $a|0\rangle+b|1\rangle$ and target state $|\psi\rangle$ ,

C_1(U) \bigl(a|0\rangle+b|1\rangle\bigr)|\psi\rangle = a|0\rangle|\psi\rangle +b|1\rangle U|\psi\rangle.

The two branches can therefore interfere later. A circuit control dot denotes this coherent block action; it does not secretly measure the control.

Measurement followed by classical feedforward is a different process. If the computational-basis outcome is forgotten, its channel is

\mathcal E(\rho) = \sum_{b=0}^{1} (P_b\otimes U_b)\rho(P_b\otimes U_b)^\dagger, \qquad U_0=I, \quad U_1=U.

This sum deletes the off-diagonal control blocks $P_0\rho P_1$ and $P_1\rho P_0$ . Retaining the measurement record gives a classical–quantum record rather than restoring the lost branch coherence. Coherent control and feedforward may agree on computational-basis truth tables while differing on every phase-sensitive test.

The Circuit Model owns wires, measurement, reset, classical records, and dynamic feedforward. Quantum Instruments owns measured outcomes together with their state updates. This page owns the audit that distinguishes those processes from a block-unitary conditional.

Measurement in Circuits owns the measured-record side of this boundary: basis and outcome encoding, selected or unread outputs, terminal or destructive use, shot statistics, and postprocessing. This page retains the coherence and block-unitary audit.

Mid-Circuit Measurement and Feedforward owns the causal measured-branch program—record availability, conditional operations, reset, repeated branching, and branch resources—while this page retains the coherent block-unitary comparator.

Global Phases Become Relative

Two target operators $U$ and $e^{i\gamma}U$ induce the same isolated unitary channel. Adding a control promotes their phase difference to a control-relative operation:

C_1(e^{i\gamma}U) = \bigl(P(\gamma)\otimes I\bigr)C_1(U), \qquad P(\gamma)=P_0+e^{i\gamma}P_1.

Only a common phase multiplying the entire joint operator is globally irrelevant. For example,

C_1(-X)=(Z_C\otimes I)C_1(X),

so the two controlled gates can give orthogonal control states even though $X$ and $-X$ define the same isolated channel. Likewise, independent SELECT branch phases $e^{i\gamma_j}$ implement the selector-register phase

\sum_j e^{i\gamma_j}|j\rangle\langle j|.

An access contract must therefore distinguish exact operator equality, equality up to one global phase, and equality of isolated channels. Single-Qubit Gates owns the target-gate phase conventions, while Rays and Global Phase owns the state-ray equivalence. Here the issue is specifically the relative phase between controlled branches.

Quantum Fourier Transform uses this phase-sensitive contract to assemble known controlled rotations into a binary phase pattern; it owns the transform factorization, circuit order, swap convention, and approximate-QFT cutoff rather than the general semantics of coherent control.

Eigenstate Kickback and Target Entanglement

For input $|+\rangle|\psi\rangle$ , define

z=\langle\psi|U|\psi\rangle.

After the controlled operation, the reduced control state is

\rho_C = \frac12 \begin{pmatrix} 1&z^*\\ z&1 \end{pmatrix}, \qquad \operatorname{Tr}\rho_C^2 = \frac{1+|z|^2}{2}.

More generally, for control amplitudes $a,b$ , the determinant is

\det\rho_C = |ab|^2\left(1-|\langle\psi|U|\psi\rangle|^2\right).

When both control amplitudes are nonzero, the output factors exactly when $U|\psi\rangle$ is proportional to $|\psi\rangle$ . Otherwise the distinct target branches entangle control and target. Failure to entangle one chosen input does not prove that the gate is local or non-entangling on all inputs.

If $U|u\rangle=e^{i\theta}|u\rangle$ , then

C_1(U)|+\rangle|u\rangle = \frac{|0\rangle+e^{i\theta}|1\rangle}{\sqrt2} \otimes|u\rangle.

The eigenphase appears on the control while the target factors. This page retains that generic two-branch identity and the factorization criterion. Phase Kickback owns the reusable common-eigenstate and character-state phase-transduction contract, Boolean and modular conversions, and phase-sensitive verification. Algorithmic Primitives and Quantum Phase Estimation retain algorithm composition, powered access, precision, success probability, and costs.

For integer $r$ ,

C_1(U)^r=C_1(U^r),

but this algebraic identity says nothing about whether $U^r$ costs one powered query, $r$ controlled-base queries explicitly supplied by the access model, depth $r$ , or physical evolution time $rt$ . Those are different access models and resource currencies.

Controlled Access, Verification, and Canonical Handoffs

If a phase-fixed known circuit is declared as $U=G_m\cdots G_1$ , controlling every factor gives

C_1(U)=C_1(G_m)\cdots C_1(G_1).

A decomposition valid only up to a target-global phase needs the corresponding control phase restored. Gate-by-gate control also carries the decomposition’s ancilla, depth, connectivity, and approximation assumptions; it is not a free resource statement.

Completely unknown black-box access is more restrictive. A channel oracle cannot distinguish $U$ from $e^{i\gamma}U$ , yet their controlled operations differ. Standard circuit access to an arbitrary unknown unitary therefore does not canonically supply $C_1(U)$ . This is an access-model obstruction, not a claim that the block-diagonal matrix fails to exist.

Quantum Oracles records whether access is supplied as a channel, a fixed unitary representative, a controlled unitary, or an interface with an adequate phase reference. This page owns why controlization is not automatic; the Oracle page owns the taxonomy and capability declaration used by later algorithms.

Practical schemes can evade those assumptions by supplying extra structure: for example, a known invariant subspace on which the operation acts trivially, a path degree of freedom, or an interferometric phase reference. Such a physical interface fixes information absent from a channel-only black box. It must be named as an additional resource, not advertised as universal standard-circuit controlization.

A semantic verification should include all of the following:

register order, basis, active values, and a complete projector resolution;
branch unitarity and a full inverse, including phases and ancilla cleanup;
basis-state tests plus at least one superposition or interference test;
a declared operator, state, or channel metric and its comparison domain;
separate gate, query, depth, ancilla, time, and hardware currencies;
an explicit statement of promises and out-of-promise behavior.

With identical control projectors and a fixed relative-phase convention,

\|C_1(U)-C_1(V)\|=\|U-V\|

in operator norm. Independently rephasing a target block changes the controlled operation and invalidates that comparison.

Universal Gate Sets owns exact and approximate reachability. Gate Decomposition owns synthesis, multiplexor networks, topology, ancillas, and compiled error. Quantum Gates supplies compact lookup formulas. Hardware chapters own native interactions, pulses, leakage, crosstalk, calibration, and experimental fidelity.

Worked Audit: A Phase-Sensitive Controlled Rotation

This synthetic audit checks an exact ideal semantic record. It makes no compilation or hardware claim.

Controlled task and licensed claim — Verify the exact action and branch phase of a controlled rotation, including a falsifiable control-interference prediction.
Registers, dimensions, order, and basis — Use qubit control $c$ first and qubit target $t$ second, in computational order $|00\rangle,|01\rangle,|10\rangle,|11\rangle$ .
Control condition, projectors, and promises — Activate on $c=1$ with $P_0+P_1=I_c$ . There is no input-subspace promise.
Target operation and access model — Use the known, phase-referenced ideal gate
$U=e^{i\pi/5}R_z(2\pi/3) =\operatorname{diag} \left(e^{-2\pi i/15},e^{8\pi i/15}\right).$
Input state or subspace and preparation — Prepare $|+\rangle_c|1\rangle_t$ exactly in the ideal model.
Branch map, phase, inverse, and power conventions — The inactive branch is $I_t$ , the active branch is the displayed phase representative, and
$C_1(U) =\operatorname{diag} \left(1,1,e^{-2\pi i/15},e^{8\pi i/15}\right), \qquad C_1(U)^{-1}=C_1(U^\dagger).$
Powers are not requested.
Ancillas, synthesis assumptions, and resource currencies — Count one known ideal controlled gate and no ancilla. Decomposition, native availability, fault-tolerant cost, and physical time are N/A because no implementation is claimed.
Observable, metric, and verification evidence — The output is
$\frac{|0\rangle+e^{8\pi i/15}|1\rangle}{\sqrt2} \otimes|1\rangle,$
so
$\langle X_c\rangle =\cos(8\pi/15)=-0.104528, \qquad \langle Y_c\rangle =\sin(8\pi/15)=0.994522.$
Exact matrix unitarity, inverse, state-norm, and analytic-versus-direct-state residuals are zero.
Alternatives, failure modes, and uncertainty — Incorrectly discarding $e^{i\pi/5}$ before adding the control predicts instead
$\langle X_c\rangle=\cos(\pi/3)=0.500000, \qquad \langle Y_c\rangle=\sin(\pi/3)=0.866025.$
The ideal arithmetic has no statistical uncertainty; phase-reference and implementation uncertainty are unresolved rather than silently set to zero.
Conclusion, stopping point, and canonical handoff — License the exact ideal controlled matrix and its phase-sensitive prediction. Stop before claiming a synthesis, native primitive, calibration, or hardware fidelity; route those questions to Gate Decomposition and the relevant hardware owner.

The basis-label map alone would miss the error in field 9. The control interference is what tests the phase promoted by coherent control.

Worked Audit: A Four-Branch SELECT Operation

The second synthetic record tests a complete selector resolution and the reduced state produced by coherent branch selection.

Controlled task and licensed claim — Verify the exact four-branch SELECT semantics, inverse, and reduced-target prediction for one declared input.
Registers, dimensions, order, and basis — Order two selector qubits as $(c_1,c_0)$ followed by target $t$ . Use big-endian selector labels $00,01,10,11$ and the computational basis throughout.
Control condition, projectors, and promises — Use the complete projectors $|j\rangle\langle j|$ for all four selector strings. There is no promise outside this full selector space.
Target operation and access model — Declare exact known ideal branches
$U_{00}=I, \qquad U_{01}=X, \qquad U_{10}=Z, \qquad U_{11}=H,$
and one abstract semantic call
$W=\sum_{j=0}^{3}|j\rangle\langle j|\otimes U_j.$
Input state or subspace and preparation — Prepare $|++\rangle_C|0\rangle_T$ in the exact ideal model.
Branch map, phase, inverse, and power conventions — All branch phases are exactly those of the displayed standard matrices. The output is
$|\Psi\rangle =\frac12\left( |00\rangle|0\rangle +|01\rangle|1\rangle +|10\rangle|0\rangle +|11\rangle|+\rangle \right).$
The inverse is $W^{-1}=\sum_j|j\rangle\langle j|\otimes U_j^\dagger$ . Here every branch is a Hermitian involution, so $W^{-1}=W$ and $W^2=I$ .
Ancillas, synthesis assumptions, and resource currencies — Report width three and one abstract SELECT call. Gate count, query count, depth, ancillas, routing, and native cost are N/A until a synthesis or access model is declared.
Observable, metric, and verification evidence — Tracing out the orthogonal selector gives
$\rho_T =\frac14\left( 2|0\rangle\langle0| +|1\rangle\langle1| +|+\rangle\langle+| \right) =\frac18 \begin{pmatrix} 5&1\\ 1&3 \end{pmatrix}.$
Therefore
$\lambda_\pm =\frac{1\pm1/\sqrt8}{2}, \qquad \operatorname{Tr}\rho_T^2=\frac9{16}.$
Projector completeness and branch unitarity prove $W^\dagger W=I$ . Matrix, inverse, trace, positivity, eigenvalue-sum, and purity residuals are exactly zero.
Alternatives, failure modes, and uncertainty — A classically sampled branch would produce a different control record and no coherent SELECT unitary. Unknown branch phases, incomplete selectors, synthesized approximation, and implementation noise are outside this exact record.
Conclusion, stopping point, and canonical handoff — License the ideal four-branch operator, inverse, output state, and reduced-target prediction. Stop before assigning generic synthesis, oracle, routing, fault-tolerant, or hardware cost.

The reduced target is mixed because it is correlated with orthogonal selector labels. That observation does not itself identify a compilation or implementation.

Common Failure Modes

Dropping a target-global phase before control. That phase becomes relative between the inactive and active branches. Fix one exact representative or include the compensating control phase.
Treating a control dot as measurement. A coherent block unitary preserves control-sector off-diagonal terms; measurement and feedforward do not.
Leaving register order implicit. Control–target and target–control matrices differ by a permutation. Declare tensor order, bit significance, and basis first.
Using incomplete or overlapping control projectors. A partial truth table does not define a full unitary. Complete the projector resolution or state a promise and extension.
Calling an open control basis independent. Open and closed values are labels in a declared basis; arbitrary-basis predicates require their own projectors.
Verifying only basis labels. Truth-table tests cannot detect branch phases. Include a superposition or interferometric test.
Assuming controlled access is free. A base-unitary query, a powered query, and a controlled query are distinct interfaces unless a reduction is supplied.
Counting one SELECT call as one gate. Abstract block access hides decoding, routing, ancillas, topology, and approximation.
Equating an algebraic power with a cost model. $C(U)^r=C(U^r)$ does not make $U^r$ a unit-cost primitive.
Ignoring promised-subspace behavior. A correct action on promised inputs does not define the rest of the Hilbert space.
Undoing labels but not phases or ancillas. The inverse must reverse the complete joint operation and return work registers as declared.
Promoting ideal semantics to a native-control claim. Pulse design, leakage, crosstalk, calibration, and measured fidelity require separate hardware evidence.

The stopping rule is simple: if the control predicate, phase reference, access interface, comparison metric, or resource currency is unresolved, license only the formal block operator—not an implementation, algorithmic speedup, or experimental gate.

Exercises

Exercise 1: Projector Proof and Inverse

Let $W=\sum_jP_j\otimes U_j$ . Prove that $W$ is unitary and derive its inverse when the $P_j$ form a complete orthogonal resolution and every $U_j$ is unitary. Identify what fails if projectors overlap or one branch is nonunitary.

Solution

Using $P_j^\dagger=P_j$ and $P_jP_k=\delta_{jk}P_j$ ,

W^\dagger W = \sum_{j,k}P_jP_k\otimes U_j^\dagger U_k = \sum_jP_j\otimes I = I.

The same calculation gives $WW^\dagger=I$ , so

W^{-1}=W^\dagger = \sum_jP_j\otimes U_j^\dagger.

If projectors overlap, cross terms $P_jP_k\otimes U_j^\dagger U_k$ survive and the branches are not an orthogonal block decomposition. If a branch is nonunitary, its diagonal block contributes $P_j\otimes U_j^\dagger U_j\ne P_j\otimes I$ . If the projectors are orthogonal but incomplete, the sum acts as zero on the omitted control subspace rather than as a complete unitary.

Exercise 2: Control Coherence and Entanglement

For input $|+\rangle|\psi\rangle$ , derive the reduced control state after $C_1(U)$ , its purity, and the exact factorization condition. Explain why a controlled gate need not entangle every input.

Solution

The joint output is

|\Psi\rangle = \frac{|0\rangle|\psi\rangle+|1\rangle U|\psi\rangle}{\sqrt2}.

Writing $z=\langle\psi|U|\psi\rangle$ and tracing out the target gives

\rho_C = \frac12 \begin{pmatrix} 1&z^*\\ z&1 \end{pmatrix}, \qquad \operatorname{Tr}\rho_C^2 = \frac{1+|z|^2}{2}.

The output is pure on the control, hence factorized, exactly when $|z|=1$ . Equality in Cauchy–Schwarz for normalized $|\psi\rangle$ and $U|\psi\rangle$ then requires $U|\psi\rangle=e^{i\theta}|\psi\rangle$ . Thus an eigenstate produces phase kickback without entanglement, while a generic target state can entangle. One non-entangling input does not classify the whole gate as local.

Exercise 3: A Mixed-Polarity 10 Control

Controls are ordered $(c_1,c_0)$ before target $t$ . Activate $X_t$ only for $c_1c_0=10$ . Give the full truth table, a decomposition using positive-control $CCX$ , its abstract resources, and the output on $|++0\rangle$ .

Solution

With $P=|10\rangle\langle10|$ ,

W=(I-P)\otimes I+P\otimes X.

It fixes $000,001,010,011,110,111$ and swaps

100\longleftrightarrow101.

Using right-to-left action,

W = (I\otimes X\otimes I) CCX (I\otimes X\otimes I).

The two $X$ gates open and close the $c_0$ control. In the abstract alphabet $\{X,CCX\}$ this has width three, size three, depth three, and no ancilla. A native mixed-polarity primitive may be counted as size and depth one only if that alphabet is explicitly declared.

On the stated input,

W|++0\rangle = \frac{|000\rangle+|010\rangle+|101\rangle+|110\rangle}{2}.

Tracing out the controls leaves target probabilities $3/4$ and $1/4$ , so

\operatorname{Tr}\rho_t^2 = \left(\frac34\right)^2 +\left(\frac14\right)^2 =\frac58,

and

S(\rho_t) = h_2(1/4) =0.811278\ \text{bits}.

Exercise 4: Distinguishing X from Minus X

Explain why $X$ and $-X$ define the same isolated unitary channel but different controlled channels. Give a control-superposition and target-eigenstate test that distinguishes them exactly.

Solution

The isolated channels agree because

(-X)\rho(-X)^\dagger=X\rho X.

After control is added,

C_1(-X)=(Z_C\otimes I)C_1(X).

Choose control $|+\rangle$ and target $|+\rangle$ , for which $X|+\rangle=|+\rangle$ . Then

C_1(X)|+\rangle|+\rangle =|+\rangle|+\rangle,

whereas

C_1(-X)|+\rangle|+\rangle =|-\rangle|+\rangle.

The outputs are orthogonal. An $X$ -basis measurement of the control distinguishes them with certainty, demonstrating that branch-relative phase is observable.

Exercise 5: A Transferred SELECT Calculation

Replace the $H$ branch in the four-branch audit by $S=\operatorname{diag}(1,i)$ and prepare $|++\rangle_C|+\rangle_T$ . Find the output, reduced target state, spectrum, purity, and inverse. Is the new SELECT operator an involution?

Solution

|\Psi\rangle = \frac12\left( |00\rangle|+\rangle +|01\rangle|+\rangle +|10\rangle|-\rangle +|11\rangle|s\rangle \right).

Tracing out the selector gives

\rho_T = \frac18 \begin{pmatrix} 4&1-i\\ 1+i&4 \end{pmatrix}.

Its eigenvalues and purity are

\lambda_\pm = \frac{1\pm1/\sqrt8}{2}, \qquad \operatorname{Tr}\rho_T^2=\frac9{16}.

Trace one and positive eigenvalues verify a valid reduced state. The exact inverse replaces every branch by its adjoint:

W^{-1} = |00\rangle\langle00|\otimes I +|01\rangle\langle01|\otimes X +|10\rangle\langle10|\otimes Z +|11\rangle\langle11|\otimes S^\dagger.

Because $S^2=Z\ne I$ , the new $W$ is not an involution. No elementary-gate or oracle cost follows from this semantic block description.

Exercise 6: Controlled Powers and Resource Currencies

Suppose $U|u\rangle=e^{2\pi i(3/16)}|u\rangle$ . For $r=4$ , find the kicked-back phase under controlled $U^r$ . Then compare a powered-query interface, repetition of a known base circuit, and controlled physical time evolution.

Solution

The powered eigenphase is

\left(e^{2\pi i(3/16)}\right)^4 =e^{2\pi i(3/4)} =-i.

Thus $C_1(U^4)|+\rangle|u\rangle$ places the relative phase $-i$ on the active control branch. The algebra alone does not fix cost:

a declared powered oracle may count one query to $U^4$ ;
a model that explicitly supplies controlled- $U$ as its base oracle counts four sequential controlled- $U$ queries; plain unknown- $U$ access does not supply this interface;
a known circuit may repeat its controlled decomposition four times, with four times its size and usually its depth unless parallel structure is proved;
if $U=e^{-iHt/\hbar}$ , direct controlled evolution for $U^4=e^{-iH(4t)/\hbar}$ uses physical evolution time $4t$ unless another simulation interface is supplied.

Query count, gate count, depth, and evolution time must therefore be reported separately.

Exercise 7: Diagnosing Unknown-Black-Box Control

Use $U$ and $e^{i\gamma}U$ to explain why channel-only black-box access does not specify one controlled operation. Give one additional physical resource that can make a practical controlization scheme possible.

Solution

The two black-box channels are identical:

U\rho U^\dagger = (e^{i\gamma}U)\rho(e^{i\gamma}U)^\dagger.

Their controlled operators differ by

C_1(e^{i\gamma}U) = \bigl(P(\gamma)\otimes I\bigr)C_1(U),

which is observable on a control superposition. A channel-only interface therefore contains too little information to choose a unique branch phase, so it cannot universally define controlled $U$ in the standard circuit model.

One admissible extra resource is a known target subspace on which the device acts as the identity. A path degree of freedom can route one branch through that reference subspace and the other through the unknown action, thereby supplying structure absent from the abstract channel oracle. An interferometric phase reference or another enlarged physical interface can play a similar role. Such a construction is conditional on the added structure; it is not a universal black-box theorem.

Exercise 8: A Complete Controlled-Evolution Record

Complete the ten-field record for $H=(\hbar\omega/2)Z$ , $U(t)=e^{-iHt/\hbar}$ , control–target order, input $|+\rangle_c|0\rangle_t$ , and $\omega t=\pi/2$ . Use a declared ideal controlled-evolution interface and explain the effect of $H\mapsto H+cI$ .

Solution

Controlled task and licensed claim — Verify the exact ideal conditional evolution and its phase-sensitive control prediction; do not infer a physical implementation.
Registers, dimensions, order, and basis — Use qubit control $c$ first and qubit target $t$ second, both in the computational basis.
Control condition, projectors, and promises — Activate on $c=1$ with $P_0+P_1=I_c$ . The tested target is the $Z$ eigenstate $|0\rangle$ ; conclusions for arbitrary target inputs are not licensed by this one state.
Target operation and access model — Declare the ideal phase-referenced controlled-evolution interface for
$H=\frac{\hbar\omega}{2}Z, \qquad U(t)=e^{-iHt/\hbar}=e^{-i(\omega t/2)Z}, \qquad \omega t=\frac\pi2.$
Input state or subspace and preparation — Prepare $|+\rangle_c|0\rangle_t$ exactly. State-preparation error is N/A in this ideal algebraic record.
Branch map, phase, inverse, and power conventions — The inactive branch is $I_t$ and the active branch sends $|0\rangle$ to $e^{-i\pi/4}|0\rangle$ . The inverse is controlled $U(-t)=U(t)^\dagger$ . No powered interface is assumed.
Ancillas, synthesis assumptions, and resource currencies — Count one declared controlled evolution, no ancilla, and physical evolution time $t$ . Gate and base-query counts are N/A because no circuit or oracle decomposition is supplied.
Observable, metric, and verification evidence — The exact output is
$\frac{|0\rangle+e^{-i\pi/4}|1\rangle}{\sqrt2} \otimes|0\rangle,$
with
$\langle X_c\rangle=\frac1{\sqrt2}, \qquad \langle Y_c\rangle=-\frac1{\sqrt2}.$
Operator unitarity, inverse, and state-norm residuals are exactly zero.
Alternatives, failure modes, and uncertainty — Replacing $H$ by $H+cI$ changes isolated evolution only by the global phase $e^{-ict/\hbar}$ , but changes the controlled active branch by that relative phase. The phase reference must therefore be part of the interface. Approximation, calibration, leakage, and timing uncertainty are unresolved rather than licensed as zero.
Conclusion, stopping point, and canonical handoff — License the exact joint-unitary semantics and interference prediction for the declared interface. Route simulation construction and compiled resources to their canonical owners, and stop before a hardware-control or algorithmic-cost claim.

References

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The unknown-operation no-go and the practical extra-subspace construction use different access assumptions. Read the Araújo and Zhou results together rather than presenting either as an unconditional statement about all physical implementations.