Static and dynamic coherence fraction in the Bernstein-Vazirani algorithm

TL;DR

The paper introduces state and operator coherence fractions as resource-like quantities for quantum information and demonstrates their pivotal role in a generalized Bernstein-Vazirani (GBV) algorithm. It proves that the GBV success probability equals the initial state's coherence fraction, $P_{ ext{succ}}^{G}( ho)=C_{ ext{F}}( ho)$, and it analyzes how coherence evolves under the arbitrary initializing unitary, the oracle, and the Hadamard transform. The results show that the Hadamard gate maximizes coherence fraction, while the oracle modifies coherence in a state-dependent way; an explicit local-unitary example illustrates these dynamics. These findings connect coherence theory to computational performance, offering a practical framework for assessing quantum resources beyond entanglement and suggesting avenues to explore coherence-based optimization in other quantum algorithms and protocols.

Abstract

Quantum entanglement and coherence are crucial resources in quantum information theory. In some scenarios, however, it is not necessary to directly estimate entanglement or coherence measures to quantify the capabilities of a state in quantum information processing. Instead, fully entangled fraction and coherence fraction are two alternatives for entanglement and coherence in specific quantum tasks. Here, we establish a link between the coherence fraction and the Bernstein-Vazirani algorithm, which has several potential applications including cryptography and database search. We show that the success probability of the generalized Bernstein-Vazirani algorithm depends only on the coherence fraction of the initial state rather than its entanglement or coherence. Moreover, we discuss the coherence fraction dynamics and establish a relation between the operator's coherence fraction and the algorithm's success probability. Our findings highlight how quantum coherence fraction influences the efficiency of quantum algorithms.

Figures (3)

• Figure 1: Diagrammatic sketch of quantum coherence fraction in the Bernstein-Vazirani algorithm. The connections between the success probability of a generalized Bernstein-Vazirani algorithm and the coherence fraction, including both state and operator coherence fractions.
• Figure 2: Circuit diagram for GBV algorithm. Here two registers with $n+1$ qubits are initialized in the state $|0^n\rangle|1\rangle_{q}$. At first, apply an arbitrary unitary quantum gate $\mathcal{U}$ on the input state $|0^{n}\rangle$ of the first register to obtain an arbitrary initial state $|\psi\rangle$. Apply a Hadamard gate $H$ on the ancilla qubit $|1\rangle_{q}$ in the second register at the same time. Next, perform the oracle $\mathcal{O}_{\ell}$ on the system state and then apply Hadamard gates $H$ to all qubits. Finally, Measure the first register.
• Figure 3: The coherence fraction dynamics of the GBV algorithm. The link between $n$, parameters $(\alpha, \beta, \theta)$ and the coherence fraction of the states after three operators applied in the algorithm. (a). For a fixed $n=2$ and parameters $\alpha=\beta=0$, the three lines in the sub-figure correspond to $\theta=\pi/8$, $\pi/4$, and $\pi/3$, respectively. (b). For $n=2$, with parameters $\beta=\theta=\pi/4$, the sub-figure shows three lines for $\alpha=\pi/4$, $\pi/2$, and $\pi$. (c). With parameters $\alpha=\beta=\theta=\pi/4$, the three lines in the sub-figure demonstrate $n=2, 4$, and $8$, respectively. (d). With $\alpha=\pi$ and $\beta=\theta=\pi/4$, the sub-figure shows three lines corresponding to $n=2, 4$, and $8$.

Theorems & Definitions (10)

• Definition 1: State coherence fraction
• Theorem 1
• Proof
• Definition 2: Operator coherence fraction
• Theorem 2
• Proof
• Theorem 3
• Proof
• Theorem 4
• Proof