Externally Controlled Givens Rotations

• Externally controlled Givens rotations are defined as two-level unitary operations that are activated by external data, ensuring selective mixing in quantum state preparation.
• They employ additional controls on spectator qubits or parameters to prevent unwanted determinant mixing, enhancing precision in multiconfigurational simulations and numerical algorithms.
• Comparative studies with Sparse State Preparation reveal that while controlled Givens rotations offer chemical interpretability, they may incur higher gate counts and circuit depth under certain conditions.

Externally controlled Givens rotations are Givens rotations whose effective action is conditioned by information outside the two-dimensional subspace being mixed. In the most specific sense developed for second-quantized quantum chemistry, they are fermion-number-conserving, universal gates for preparing arbitrary multiconfigurational states, because a rotation between a reference determinant and an excited determinant generally must be controlled on qubits outside the excitation itself to avoid unwanted mixing. Related arXiv literature uses closely related language for hardware-level, externally tunable Givens rotations in cavity-qudit architectures and for numerical algorithms in which Givens sequences are steered by external inputs or stored control data (Greene-Diniz et al., 7 Aug 2025, Job, 2023).

1. Terminological scope and conceptual core

In Greene-Diniz et al., the external control is literal qubit control on modes not acted on by the Givens block. The objective is selective preparation of a linear combination of Slater determinants in the occupation-number basis. A bare Givens rotation mixes every basis state consistent with its local pattern, so additional controls on spectator qubits are used to restrict the action to the intended pair of determinants.

Across the other cited works, the phrase appears in broader but structurally related senses. In the cavity-qudit compilation setting, the externally controlled quantity is the rotation angle $\theta$, which is mapped analytically to a displacement amplitude $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$ in a five-pulse SNAP–displacement sequence. In compensated numerical linear algebra, the rotation is “externally controlled” by the choice of $(a,b)$ or by a prescribed angle $\phi$. In recycling-based QR updates for band-dominated operators, the external control is the stored sequence of Givens rotations whose row indices are shifted and reused as a window slides (Borges, 2021, Lindner et al., 2016).

This suggests that “externally controlled Givens rotation” is not a single canonical construction across all fields. Rather, the common theme is conditional deployment of a Givens transformation by information not contained solely in the local $2\times2$ block.

2. Quantum-chemistry formulation in the occupation-number basis

A Givens rotation is presented as the simplest particle-conserving excitation unitary. With Jordan–Wigner mapping, each spin–orbital mode is represented by a qubit bit $q\in\{0,1\}$. For a single-body excitation between orbitals $i,j$, the two-qubit block is

$$\mathcal{G}^2(\theta)= \begin{pmatrix} 1 & 0 & 0 & 0\ 0 & \cos\theta & \sin\theta & 0\ 0 & -\sin\theta & \cos\theta & 0\ 0 & 0 & 0 & 1 \end{pmatrix}$$

in the basis $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$, so that $|01\rangle$ and $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$0 are mixed while $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$1 and $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$2 are unchanged. More generally, a $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$3-body excitation has Hamming distance $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$4 between two occupation-number basis states and is embedded as a $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$5 Givens block in the full Hilbert space (Greene-Diniz et al., 7 Aug 2025).

The need for external controls follows from the structure of a multiconfigurational target state,

$\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$6

If one simply strings together Givens rotations, a gate acting on qubits $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$7 rotates every basis state whose bits on $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$8 match $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$9. That is too indiscriminate when the intended task is to mix the reference configuration $(a,b)$0 into one selected excited configuration at a time. Without additional controls, an intermediate rotation can inadvertently mix $(a,b)$1 and $(a,b)$2 for some $(a,b)$3, spoiling the desired linear combination.

The corrective idea is to control the rotation on one or more qubits $(a,b)$4 so that the gate fires only when $(a,b)$5 matches the reference bit $(a,b)$6. For Hamming distance $(a,b)$7, the control excludes other determinants that would otherwise be rotatable by the same local pattern. For Hamming distance $(a,b)$8, the same logic applies to two-body Givens blocks. The total preparation unitary is written as

$(a,b)$9

with each $\phi$0 carrying the external controls required for correctness.

3. Minimal-control synthesis and resource costs

The paper’s Algorithm 1 takes as input an ordered list $\phi$1 with $\phi$2 as reference and excites $\phi$3 sequentially for $\phi$4. For each target determinant, it computes the Hamming distance $\phi$5 and the set of differing indices $\phi$6. When $\phi$7, the procedure inspects each previous determinant $\phi$8, checks whether it would also be rotatable by the same two-qubit Givens block, and then chooses the lowest qubit index $\phi$9 with $2\times2$0 as an external control, conditioned on the state $2\times2$1. The $2\times2$2 case follows the same pattern on four qubits. For $2\times2$3, Algorithm 2 constructs a $2\times2$4 gadget based on controlled SWAPs, a central $2\times2$5, and an uncomputation stage; it uses controls on the minority qubit value so as to minimize the number of controls (Greene-Diniz et al., 7 Aug 2025).

The stated asymptotic point is that naively controlling on all $2\times2$6 spectators is prohibitive, whereas the minimal-control strategy reduces the overhead to only $2\times2$7. In the worst case of the higher-body gadget, one adds $2\times2$8 controls rather than $2\times2$9.

The resource model is given in a hardware-native compilation language. For a single-body Givens rotation on bare hardware, one needs 2 ZZMax gates plus 4 Phased-X; each external control adds another 2 ZZMax plus 2 $q\in\{0,1\}$0 gates via standard decomposition. For a double-body Givens rotation, one uses 14 ZZMax gates and 20 Phased-X. The $q\in\{0,1\}$1 gadget uses $q\in\{0,1\}$2 controlled SWAPs, each costing $q\in\{0,1\}$3 ZZMax, followed by one central $q\in\{0,1\}$4 and then $q\in\{0,1\}$5 SWAPs back. No ancillas are required. Circuit depth scales linearly in the number of Givens rotations, with each rotation contributing depth $q\in\{0,1\}$6 plus the depth of its external controls.

These constructions establish universality for multiconfigurational state preparation while keeping fermion number conserved throughout. A plausible implication is that the method is especially attractive when excitation structure itself is part of the scientific object being represented, rather than merely a route to a short circuit.

4. Comparison with sparsity-exploiting state preparation

Greene-Diniz et al. compare externally controlled Givens rotations to the Sparse State Preparation (SSP) method of Gleinig & Hoefler. SSP iteratively “merges” pairs of bit-strings in $q\in\{0,1\}$7, using a single-qubit rotation plus CNOT/NOT staircases to map $q\in\{0,1\}$8. Its gate count is stated as $q\in\{0,1\}$9, the same asymptotic scaling as minimal-control Givens rotations, but with no external multi-controls. Its circuit depth and two-qubit-gate count scale linearly in $i,j$0, independent of $i,j$1 or $i,j$2, and the paper reports that SSP circuits are order-of-magnitude smaller in all numerical experiments (Greene-Diniz et al., 7 Aug 2025).

Case	Externally controlled GR	SSP
4-qubit example	61 Phased-X, 4 $i,j$3, 44 ZZMax	10 Phased-X, 4 $i,j$4, 5 ZZMax
8 qubits, $i,j$5 at $i,j$6	174 Phased-X, 7 $i,j$7, 128 ZZMax	22 Phased-X, 6 $i,j$8, 17 ZZMax

The same section reports that circuit depths shrink by factors 3–10 under SSP. In a 4-qubit ground-state VQE study on C$i,j$9H$$\mathcal{G}^2(\theta)= \begin{pmatrix} 1 & 0 & 0 & 0\ 0 & \cos\theta & \sin\theta & 0\ 0 & -\sin\theta & \cos\theta & 0\ 0 & 0 & 0 & 1 \end{pmatrix}$$0, SSP gives $$\mathcal{G}^2(\theta)= \begin{pmatrix} 1 & 0 & 0 & 0\ 0 & \cos\theta & \sin\theta & 0\ 0 & -\sin\theta & \cos\theta & 0\ 0 & 0 & 0 & 1 \end{pmatrix}$$1 fewer measurement shots for the same 1 mHa accuracy because the circuit is shallower. For Hamiltonian moments using QCM4 and CMX2 on an 8-qubit active space, the $$\mathcal{G}^2(\theta)= \begin{pmatrix} 1 & 0 & 0 & 0\ 0 & \cos\theta & \sin\theta & 0\ 0 & -\sin\theta & \cos\theta & 0\ 0 & 0 & 0 & 1 \end{pmatrix}$$2 SSP state allows even the low-order CMX2 to match exact energies for all torsions, at a fraction of measurement cost. For Quantum Phase Estimation via QCELS on 12 qubits $$\mathcal{G}^2(\theta)= \begin{pmatrix} 1 & 0 & 0 & 0\ 0 & \cos\theta & \sin\theta & 0\ 0 & -\sin\theta & \cos\theta & 0\ 0 & 0 & 0 & 1 \end{pmatrix}$$3, an 8-term occupation-number state $$\mathcal{G}^2(\theta)= \begin{pmatrix} 1 & 0 & 0 & 0\ 0 & \cos\theta & \sin\theta & 0\ 0 & -\sin\theta & \cos\theta & 0\ 0 & 0 & 0 & 1 \end{pmatrix}$$4 has overlap 0.999 with the true ground state, and the reported SSP overhead is only 40 two-qubit gates versus $$\mathcal{G}^2(\theta)= \begin{pmatrix} 1 & 0 & 0 & 0\ 0 & \cos\theta & \sin\theta & 0\ 0 & -\sin\theta & \cos\theta & 0\ 0 & 0 & 0 & 1 \end{pmatrix}$$5 for a similarly controlled-Givens state. In Q-SCEOM excited-state calculations on 8 qubits, a Hamming-distance-4 matrix-element example uses only 3 ZZMax gates under SSP, whereas the Givens construction uses 14; over all 676 elements, SSP yields $$\mathcal{G}^2(\theta)= \begin{pmatrix} 1 & 0 & 0 & 0\ 0 & \cos\theta & \sin\theta & 0\ 0 & -\sin\theta & \cos\theta & 0\ 0 & 0 & 0 & 1 \end{pmatrix}$$6 fewer two-qubit gates.

The comparison clarifies a recurrent misconception: universality of controlled Givens rotations does not imply resource optimality on sparse chemical wavefunctions. The paper’s own practical advice is to use SSP whenever $$\mathcal{G}^2(\theta)= \begin{pmatrix} 1 & 0 & 0 & 0\ 0 & \cos\theta & \sin\theta & 0\ 0 & -\sin\theta & \cos\theta & 0\ 0 & 0 & 0 & 1 \end{pmatrix}$$7, while retaining the Givens formulation when chemical interpretability matters or when initialization at $$\mathcal{G}^2(\theta)= \begin{pmatrix} 1 & 0 & 0 & 0\ 0 & \cos\theta & \sin\theta & 0\ 0 & -\sin\theta & \cos\theta & 0\ 0 & 0 & 0 & 1 \end{pmatrix}$$8 should reproduce Hartree–Fock exactly. By contrast, the SSP $$\mathcal{G}^2(\theta)= \begin{pmatrix} 1 & 0 & 0 & 0\ 0 & \cos\theta & \sin\theta & 0\ 0 & -\sin\theta & \cos\theta & 0\ 0 & 0 & 0 & 1 \end{pmatrix}$$9 state can be a complicated superposition.

5. Hardware realization in cavity-qudit platforms

A separate line of work gives an analytic route for implementing a Givens rotation between adjacent Fock levels $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$0 using only displacement gates and SNAP phase gates. The core five-pulse sequence is

$\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$1

and comparison of its first-order expansion with the target two-level rotation yields the direct map

$\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$2

Previous publications had left the choice of $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$3 to numerical optimization at compile time; the map $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$4 removes that step and permits direct compilation of any $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$5-dimensional unitary into SNAP and displacement gates in $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$6 complex floating-point operations with a low constant prefactor (Job, 2023).

The same work states that the infidelity of the generated gate sequence $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$7 per Givens rotation scales approximately as $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$8. If each rotation is subdivided into $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$9 pieces of angle $|01\rangle$0, the full $|01\rangle$1 unitary infidelity in numerical studies decreases approximately as $|01\rangle$2. In a standard QR-style decomposition of $|01\rangle$3, there are $|01\rangle$4 adjacent Givens rotations, each implemented by 3 displacements and 2 SNAPs, so the raw pulse-schedule length is $|01\rangle$5 pulses, or $|01\rangle$6 when subdivision is used. The reported compile time for $|01\rangle$7 is $|01\rangle$8 on a modern laptop.

This literature uses “external control” in a hardware sense: the controller supplies $|01\rangle$9, evaluates $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$00 with negligible latency, and feeds the resulting amplitude into the SNAP–displacement chain. Practical remarks concern pulse shapes, bandwidth, ancilla coupling, calibration of $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$01, and decoherence trade-offs between lower unitary error and longer pulse sequences.

6. Numerical linear algebra, QR updates, and broader interpretations

In floating-point numerical linear algebra, a Givens rotation is often generated from a pair $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$02 through

$\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$03

or the same formula with the opposite sign convention for $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$04. Borges develops a compensated routine that computes the reciprocal hypotenuse using one division, one square root, and a sequence of FMAs, with correction terms formed from

$\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$05

so that the corrected reciprocal hypotenuse satisfies an error bound of order $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$06 rather than the $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$07 behavior typical of a naive implementation. The paper states an operation count of 1 division, 1 square-root, 9 FMAs, 1 plain add/sub, and 3 plain multiplies, for a total of approximately 15 flops; on a modern Intel® Core™ i7 (Skylake), the implementation runs in roughly 48–60 cycles total (Borges, 2021).

The same work explicitly describes external control in two senses. One may choose the pair $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$08 to be annihilated, in which case the routine returns $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$09 so that the second component is zeroed. Alternatively, if one wants a particular angle $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$10, one can set

$\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$11

and then recover the same $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$12 to high precision. Here the “external” quantity is not a spectator qubit but the user-specified input pair or angle.

A different external-control pattern appears in the recycling of Givens rotations for QR factorizations of sliding windows of band-dominated operators. For an initial factorization of a finite $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$13 section, Lindner and Schmidt store nested lists of Givens sequences. When the window shifts from $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$14 to $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$15, most rotations are retained, re-indexed, and applied again; only $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$16 rotations are removed, relocated, or newly computed. The stated consequence is $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$17 work per shift after an initial $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$18 factorization, rather than recomputing each QR factorization at $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$19 cost. Because every update is a multiplication by a unitary Givens matrix or its inverse, the method is described as backward-stable, with rounding-error accumulation of $\alpha=\Phi(\theta)=\theta/(4\sqrt{k+1})$20 over many shifts (Lindner et al., 2016).

Taken together, these results resolve another common misconception: “external control” does not denote a unique control mechanism. In multiconfigurational quantum chemistry it means spectator-qubit conditioning; in cavity-qudit compilation it means an externally tunable angle-to-pulse map; in compensated arithmetic it means user-specified annihilation data; in sliding-window QR it means explicit control by a stored sequence of rotations. The unifying structure is conditional selectivity: a local Givens action is made globally correct, hardware-realizable, numerically robust, or computationally reusable by information external to the nominal two-level rotation.