On Computational Power of Quantum Read-Once Branching Programs

In this paper we review our current results concerning the computational power of quantum read-once branching programs. First of all, based on the circuit presentation of quantum branching programs and our variant of quantum fingerprinting technique, we show that any Boolean function with linear polynomial presentation can be computed by a quantum read-once branching program using a relatively small (usually logarithmic in the size of input) number of qubits. Then we show that the described class of Boolean functions is closed under the polynomial projections.

💡 Research Summary

This paper investigates the computational power of quantum read‑once branching programs (QOBDDs), a quantum analogue of ordered binary decision diagrams (OBDDs). The authors first formalize quantum branching programs (QBPs) as sequences of classically controlled unitary operations acting on a Hilbert space of dimension d. The width of a QBP is defined as d, the length as the number of instructions, and the number of qubits required for a physical implementation is log d. They recall a known lower bound: any quantum OBDD that computes a Boolean function f with bounded error ε must have width at least log width(P)·2·log(1+1/ε), where P is the minimal deterministic OBDD for f. Consequently, for many natural functions whose deterministic OBDD width is exponential, the quantum width cannot be sub‑logarithmic.

The central technical contribution is a quantum fingerprinting technique based on “characteristic polynomials”. For a Boolean function f, a polynomial g_f over the ring ℤ_m is called characteristic if g_f(σ)=0 (mod m) exactly when f(σ)=1. When such a polynomial is linear, i.e., g_f(x)=c₀+∑_{i=1}^n c_i x_i, the authors show how to compute f with one‑sided error ε using a QOBDD of width O(log m) and only O(log log m) qubits.

The construction proceeds as follows. For a given input σ, the value g_f(σ) (mod m) is encoded into a quantum “fingerprint” state

|h_σ⟩ = (1/√t) ∑_{i=1}^t |i⟩ ⊗ ( cos(2πk_i·g_f(σ)/m) |0⟩ + sin(2πk_i·g_f(σ)/m) |1⟩ ),

where t = 2⌈log((2/ε)·ln m)⌉ and the integers k_i ∈ {1,…,m−1} form a “good” set K satisfying

(1/t) ∑_{i=1}^t cos²(2πk_i·b/m) < ε for every non‑zero b (mod m).

Such a set exists by a standard number‑theoretic argument. After preparing the equal superposition over the index register, the circuit applies a Hadamard transform on the index register (H⊗log t) and measures the whole system. If g_f(σ)=0, all cosine terms equal 1, so the measurement outcome |0⟩⊗|0…0⟩ occurs with probability 1, and the input is accepted. If g_f(σ)≠0, the average cosine squared is bounded by ε, giving acceptance probability ≤ ε. The whole process uses a single measurement at the end (measure‑once model).

To implement the fingerprint efficiently, the authors describe a circuit where each input bit x_j controls a rotation R_y(4πk_i·c_j/m) on the target qubit of each subspace i. The rotations are applied in parallel using controlled‑unitary gates, accumulating the sum ∑ c_j x_j in the rotation angle. After processing all input bits, a final rotation by the constant term c₀ is performed, followed by the global Hadamard on the index register. The resulting state matches the fingerprint described above.

Because the circuit depth is linear in n (the number of input variables) and the width is twice the number of qubits, the width is O(log m) and the total number of qubits is log t + 1 = O(log log m). Hence any Boolean function possessing a linear characteristic polynomial over ℤ_m can be computed by a quantum OBDD with logarithmic width and doubly‑logarithmic qubit count.

The paper lists several natural functions that admit such linear characteristic polynomials: MOD_m(x)=∑ x_i (mod m), MOD′_m(x)=∑ 2^{i−1} x_i (mod m), and EQ_n(x,y)=∑ (x_i⊕y_i) (mod 2). All these functions are thus computable with O(log log m) qubits and O(log m) width.

Finally, the authors prove that the class of functions efficiently computable by this fingerprinting method is closed under polynomial projections. If f has a linear characteristic polynomial and h is obtained from f by a polynomial substitution (i.e., h(x)=f(p₁(x),…,p_k(x)) where each p_i is a polynomial), then h also has a linear characteristic polynomial (obtained by composing the original polynomial with the substitutions) and can be computed with the same quantum fingerprinting technique. This mirrors the closure properties known for classical OBDDs and shows that the quantum model retains similar algebraic robustness.

In summary, the paper demonstrates that quantum read‑once branching programs can exploit quantum fingerprinting to evaluate a non‑trivial class of Boolean functions with extremely low space requirements: logarithmic width and doubly‑logarithmic qubit usage. It also establishes a matching lower bound based on deterministic OBDD width, thereby delineating the limits of the model. The work opens avenues for extending the fingerprinting approach to functions lacking linear characteristic polynomials, which remains an important direction for future research.