[Paper Review] Deterministic construction of arbitrary $W$ states with quadratically increasing number of two-qubit gates

Fırat Diker|arXiv (Cornell University)|Jun 28, 2016

Quantum Information and Cryptography25 references7 citations

TL;DR

This paper proposes a deterministic, scalable method to construct arbitrary $W$ states using only $cNOT$ and $F$ two-qubit gates, with the number of two-qubit gates scaling quadratically with qubit count. The approach enables deterministic generation of $n$-qubit $W$ states via iterative circuit enhancement, offering a practical optical implementation with feasible gate counts and minimal use of multi-qubit operations.

ABSTRACT

We propose a quantum circuit composed of $cNOT$ gates and four single-qubit gates to generate a $W$ state of three qubits. This circuit was then enhanced by integrating two-qubit gates to create a $W$ state of four and five qubits. After a couple of enhancements, we show that an arbitrary $W$ state can be generated depending only on the degree of enhancement. The generalized formula for the number of two-qubit gates required is given, showing that an $n$-qubit $W$-state generation can be achieved with quadratically increasing number of two-qubit gates. Also, the practical feasibility is discussed regarding photon sources and various applications of $cNOT$ gates.

Motivation & Objective

• To develop a deterministic, scalable quantum circuit for generating arbitrary $W$ states using only two-qubit gates.
• To minimize resource overhead by avoiding three- or higher-body entangling gates.
• To provide a practical optical implementation using $cNOT$ and $F$ gates with feasible gate counts.
• To analyze the feasibility of the scheme under experimental constraints such as photon loss and gate fidelity.
• To derive a generalized formula for the number of two-qubit gates required as a function of $n$, the number of qubits.

Proposed method

• The method starts with a three-qubit $W$-state generation circuit using two $F$ gates and two $cNOT$ gates, which is then enhanced iteratively to produce larger $W$ states.
• Each enhancement step increases the input qubit count by one and applies a sequence of $cNOT$ and $F$ gates to extend the entanglement deterministically.
• The $F$ gate is implemented using four half-wave plates (HWPs) and a $cNOT$ gate, acting as a controlled operation on the target qubit when the control is vertically polarized.
• The circuit structure is generalized to $n$ qubits by recursively applying the same gate pattern, ensuring deterministic output.
• The number of $cNOT$ gates is derived as $\frac{n(n+1)}{2} - 2$, showing quadratic scaling with $n$, while $cZ$ gates are used sparingly and require only two HWPs and a $cNOT$.
• The scheme assumes ideal gate operation and precise HWP angle settings, with error analysis considering angular deviation and photon loss.

Experimental results

Research questions

• RQ1 Can a deterministic, scalable $W$ state of arbitrary size be constructed using only two-qubit gates?
• RQ2 What is the optimal gate count scaling for generating $n$-qubit $W$ states using a recursive circuit enhancement approach?
• RQ3 How does the success probability of the $cNOT$ gate in linear optics affect the overall feasibility of large-scale $W$ state generation?
• RQ4 What are the main experimental challenges—such as angular precision and photon loss—that could disrupt the deterministic construction of large $W$ states?
• RQ5 Can the proposed optical circuit be implemented with current technology, particularly using parametric down-conversion for single-photon sources?

Key findings

• The number of $cNOT$ gates required to generate an $n$-qubit $W$ state scales quadratically as $\frac{n(n+1)}{2} - 2$, with the total number of two-qubit gates growing as $\mathcal{O}(n^2)$.
• The scheme uses only two-qubit gates ($cNOT$ and $cZ$), avoiding the need for three- or higher-body entangling gates, which simplifies experimental implementation.
• The success probability of generating an $n$-qubit $W$ state is $\left(\frac{1}{9}\right)^{\frac{n(n+1)-4}{2}}$, which drops rapidly with increasing $n$, especially for $n \geq 3$, due to the low success rate of linear optical $cNOT$ gates.
• The method is experimentally feasible using parametric down-conversion for single-photon sources, with error rates scaling as $O(\gamma^n \delta)$, where $\delta \sim 10^{-4}$, making errors negligible compared to desired events for small $n$.
• Small angular deviations in HWP settings—e.g., from $22.05^\circ$ to $21.5^\circ$—can lead to failure in state preparation, highlighting the need for high-precision control in large-scale implementations.
• The use of weak cross-Kerr nonlinearities is identified as a promising route to improve $cNOT$ gate fidelity and reduce resource overhead, enhancing scalability.