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arXiv:2512.15901

Closed-Form Optimal Quantum Circuits for Single-Query Identification of Boolean Functions

Leonardo Bohac

quant-phSubmitted Dec 17, 2025• Announced Dec 19, 2025• 0 citations

Abstract

We study minimum-error identification of an unknown single-bit Boolean function given black-box (oracle) access with one allowed query. Rather than stopping at an abstract optimal measurement, we give a fully constructive solution: an explicit state preparation and an explicit measurement unitary whose computational-basis readout achieves the Helstrom-optimal success probability 3/4 for distinguishing the four possible functions. The resulting circuit is low depth, uses a fixed gate set, and (in this smallest setting) requires no entanglement in the input state. Beyond the specific example, the main message is operational. It highlights a regime in which optimal oracle discrimination is not only well-defined but implementably explicit: the optimal POVM collapses to a compact gate-level primitive that can be compiled, verified, and composed inside larger routines. Motivated by this, we discuss a "what if" question that is open in spirit: for fixed (n,m,k), could optimal k-query identification (possibly for large hypothesis classes) admit deterministic, closed-form descriptions of the inter-query unitaries and the final measurement unitary acting on the natural n+m-qubit input--output registers (and, if needed, small work registers)? Even when such descriptions are not compact and do not evade known circuit-complexity barriers for generic Boolean functions, making the optimum constructive at the circuit level would be valuable for theory-to-hardware translation and for clarifying which forms of "oracle access" are physically meaningful.

Comments: 7 pages, 1 figure, 2 tables. Includes explicit circuit decomposition

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We study minimum-error identification of an unknown single-bit Boolean function given black-box (oracle) access with one allowed query. Rather than stopping at an abstract optimal measurement, we give a fully constructive solution: an explicit state preparation and an explicit measurement unitary whose computational-basis readout achieves the Helstrom-optimal success probability 3/4 for distinguishing the four possible functions. The resulting circuit is low depth, uses a fixed gate set, and (in this smallest setting) requires no entanglement in the input state. Beyond the specific example, the main message is operational. It highlights a regime in which optimal oracle discrimination is not only well-defined but implementably explicit: the optimal POVM collapses to a compact gate-level primitive that can be compiled, verified, and composed inside larger routines. Motivated by this, we discuss a "what if" question that is open in spirit: for fixed (n,m,k), could optimal k-query identification (possibly for large hypothesis classes) admit deterministic, closed-form descriptions of the inter-query unitaries and the final measurement unitary acting on the natural n+m-qubit input--output registers (and, if needed, small work registers)? Even when such descriptions are not compact and do not evade known circuit-complexity barriers for generic Boolean functions, making the optimum constructive at the circuit level would be valuable for theory-to-hardware translation and for clarifying which forms of "oracle access" are physically meaningful.

Hamming-weight computation and symmetric Boolean functions are basic primitives for quantum arithmetic and state preparation, where circuit depth and qubit overhead matter. We construct an exact circuit computing the Hamming weight of an

n

-qubit register with depth

O(\log n)

and size

O(n\log n)

, using only the

n

input qubits and

m=\lceil\log_2(n+1)\rceil

binary output qubits. The circuit uses arbitrary one-qubit gates and CNOTs with all-to-all connectivity. Both depth and total width are optimal. Our constructions combine tree structures and pipelining, temporarily using input qubits as workspace. We also give exact circuits for every symmetric Boolean function with the same depth and size bounds, using only

m

ancillary qubits returned to zero. Finally, we apply the counter to exact preparation of balanced Dicke states for even

n

. With only

O(\log n)

ancillary qubits, we obtain expected depth

O(\log^2 n)

using measurements and classical feedforward, and depth

O(n^{1/4}\log n)

using unitary gates alone.

One-time programs (OTPs) aim to let a user evaluate a program on a single input while revealing nothing else. Classical OTPs require hardware. Quantum measurement suggests a physical basis for one-time use, yet deterministic functionalities remain impossible due to gentle-measurement attacks (Broadbent, Gutoski and Stebila, 2013). Recent constructions cover randomized functionalities with high-entropy outputs (Gunn, Movassagh 2025; Gupte et al., 2025), but the strongest achievable security remains unclear. Inspired by classical obfuscation, we ask for a "best-possible" one-time compiler that, for any functionality, leaks the least information compared with any other one-time implementation. We prove that such a generic compiler cannot exist even for classical randomized functionalities assuming

\mathsf{SZK} \not\subseteq \mathsf{BQP}

. We then identify a natural subclass: testable OTP compilers, which output quantum states augmented with reflection programs for themselves. We formulate a simplified, generalized Single-Effective-Query (SEQ) simulation security notion for quantum channels, using self-adjoint implementations whose behavior under arbitrary quantum interactions depends only on the channel. SEQ security implies best-possible testable one-time security. We construct SEQ-secure OTPs for all quantum functionalities in the classical oracle model, giving the first positive results for arbitrary quantum channels beyond classical randomized functionalities. SEQ security could thus serve as a testable one-time analogue of virtual black-box (VBB) security. Finally, we propose stateful quantum indistinguishability obfuscation (stateful quantum iO): quantum state obfuscation for stateful quantum programs. It implies best-possible testable OTPs and is achievable in the classical oracle model, offering an approach towards best-possible testable OTPs in the plain model.

I study a promise problem for an unknown unitary operator acting on an

n

-qubit system. The operator is promised to take one of two forms: either it implements a fixed permutation of computational basis states, or it implements the same permutation together with a conditional sign change determined by a designated input qubit. I show that these two cases can be distinguished with certainty using a single query to the unknown operator and a measurement of only one qubit. The procedure requires no ancilla qubits and uses only

n+1

Hadamard gates in addition to the oracle call. The promise is intrinsically quantum, since the two cases differ only in their relative-phase structure and therefore have no direct classical counterpart in the usual black-box model.

We study the computation complexity of Boolean functions in the quantum black box model.In this model our task is to compute a function f : {0, 1} → {0, 1} on an input x ∈ {0, 1} n that can be accessed by querying the black box.Quantum algorithms are inherently probabilistic; we are interested in the lowest possible probability that the algorithm outputs incorrect answer (the error probability) for a fixed number of queries.We show that the lowest possible error probability for ANDn and EQUALITY n+1 is 1 2 -n n 2 +1 .

An answer partition specifies which oracle instances share an output, while query complexity also depends on how those instances are accessed. We formulate exact one-query answer-partition resolution as block discrimination of a unitary oracle family. Applied to Deutsch's problem with a controlled response unitary V, the criterion shows that one query suffices if and only if -1 is an eigenvalue of V; cyclic addition in odd response dimension therefore raises the exact quantum cost to two queries, matching the classical cost while preserving the same computational-basis point values. Holding the standard Boolean point oracle fixed, we further show that balanced answer partitions with equal-sized cells can have different exact classical and quantum query complexities. A complete exact classification is given for all 35 balanced three-bit partitions, while the unresolved four-bit cases are reported explicitly as numerical candidates. These results separate the combinatorial specification of an answer from the oracle-dependent distinguishability that determines query complexity.

Abstract In this paper, we present a generic construction for synthesizing an optimal T-depth quantum circuit for any arbitrary n -input, m -output Boolean function f : {0, 1 } n → {0, 1} m with algebraic degree k ≤ n , achieving an exact Toffoli (consequently T) depth of ⌈log 2 k ⌉. This broadly generalizes the recent result establishing the optimal Toffoli (and T) depth for multi-controlled Toffoli decompositions (Dutta et al., Phys. Rev. A, 2025). The optimality of T-depth in this initiative is considered in the context of implementing an n -MCT, assuming the decomposition via Clifford plus Toffoli gates. The key technique involves inspecting the Algebraic Normal Form (ANF) of the Boolean function. Obtaining a benchmark for the minimum T-depth of such circuits is crucial for the efficient implementation of quantum algorithms by enabling greater parallelism, reducing time complexity, and minimizing circuit latency, making them suitable for near-term quantum devices with limited coherence times. The broader implications of our results include a provable lower bounds on T-depth for S-box and block cipher implementations, such as AES. Finally, we also explain the impact of our result in identifying the T-depth for the generic cryptanalysis of block ciphers using Grover’s algorithm.