Quantum Circuit Design for Decoded Quantum Interferometry

TL;DR

This work provides a concrete quantum circuit realization of Decoded Quantum Interferometry (DQI) for Max-XORSAT by integrating Unary Amplitude Encoding, deterministic Dicke-state preparation via Split-and-Cyclic Shift (SCS), phase and constraint encoding, and a reversible Gauss-Jordan elimination decoder. The circuit supports coherent decoding over superpositions, with a Hadamard transform on the syndrome register and postselection to yield approximate solutions. Gate counts and circuit depth scale roughly as $O(m^2)$ while requiring about $2m$ qubits, and simulations validate the approach up to $m=15$ (30 qubits). The results show a practical path toward scalable DQI implementations and point to future work on improved decoding, Dicke-state prep, and hardware validation to harness potential quantum advantages in structured optimization problems.

Abstract

Decoded Quantum Interferometry (DQI) is a recently proposed quantum algorithm for approximating solutions to combinatorial optimization problems by reducing instances of linear satisfiability to bounded-distance decoding over superpositions of quantum states. A central challenge in realizing DQI is the design of a decoder that operates coherently on quantum superpositions. In this work, we present a concrete quantum circuit implementation of DQI, with a focus on the decoding subroutine. Our design leverages a reversible Gauss-Jordan elimination circuit for the decoding stage. We analyze the circuit's depth and gate complexity and validate its performance through simulations on systems with up to 30 qubits. These results establish a concrete foundation for scalable implementations of DQI and open the door to future algorithmic refinements and hardware-level realizations.

Figures (5)

• Figure 1: DQI algorithm state evolution diagram.
• Figure 2: Inductive construction of $U_{m,\ell}$ from a sequence of $\mathrm{SCS}_{m,\ell}$
• Figure 3: Total gate count for each circuit component versus instance size, with $p=2$, $r=1$, and $\ell=2$.
• Figure 4: Circuit depth versus instance size, with $p=2$, $r=1$, and $\ell=2$.
• Figure 5: Result plot for the 6-bit instance. The $x$-axis lists all possible 6-bit states. The blue line (left $y$-axis) shows the classical objective value for each state, while the red bars (right $y$-axis) display the measured probability (post-selected on $\ket{0}$). The coincidence of the objective function peaks with large measurement probabilities shows clearly that DQI finds optimal (or near-optimal) peaks.

Theorems & Definitions (2)

• Definition 1: Dicke-State Unitary $U_{n,h}$
• Definition 2: Split and Cyclic Shift (SCS)