3-SAT solver for two-way quantum computers (2408.05812v1)

Abstract: While quantum computers assume existence of state preparation process $|0\rangle$, CPT symmetry of physics says that performing such process in CPT symmetry perspective, e.g. reversing used EM impulses ($V(t)\to V(-t)$), we should get its symmetric analog $\langle 0|$, referred here as state postparation - which should provide results as postselection, but with higher success rate. Two-way quantum computers (2WQC) assume having both $|0\rangle$ and $\langle 0|$ pre and postparation. In theory they allow to solve NP problems, however, basic approach would be more difficult than Shor algorithm, which is now far from being practical. This article discusses approach to make practical 2WQC 3-SAT solver, requiring exponential reduction of error rate, what should be achievable through linear increase of the numbers of gates. 2WQC also provides additional error correction capabilities, like more stable Grover algorithm, or mid-circuit enforcement of syndrome to zero, like proposed equalizer enforcing qubit equality.

• The paper presents a novel 3-SAT solver that uses CPT symmetry and state postparation to achieve potential exponential speedup in NP problem solving.
• It details a three-stage computation model involving superposition, reversible clause evaluation, and selective state postparation for enhanced solution accuracy.
• The research introduces strategic error correction methods, including mid-circuit equalizers, to balance fault-tolerance with computational depth in quantum processing.

Analysis of "3-SAT Solver for Two-Way Quantum Computers"

The paper presented focuses on the proposal and analysis of a 3-SAT solver specifically designed for two-way quantum computers (2WQC). The concept of 2WQC extends the functionality of standard one-way quantum computers (1WQC) by incorporating state postparation operations, which are introduced as the symmetric counterpart to state preparation when viewed through the lens of CPT symmetry. This paper explores the theoretical feasibility of employing such a model to solve specific NP problems, with an emphasis on the 3-SAT problem as a focal point.

Central Thesis and Approach

The core novelty of the work lies in the utilization of CPT symmetry—particularly the notion of state postparation ($\langle 0|$)—to enhance quantum computation capabilities. The approach predicates that by allowing both state preparation ($|0\rangle$) and its CPT symmetric counterpart, it's possible to achieve operations analogous to postselection—potentially realizing higher success rates in NP problem-solving than mere conventional methods. The objective is to achieve a substantial reduction in error rates while concurrently minimizing computational depth increase, thus theoretically offering an exponential speedup in computation for specific NP problems like 3-SAT.

Quantum 3-SAT Solver

The proposed 3-SAT solver is structured to work within the 2WQC framework. It capitalizes on postparation, which is conceptually akin to postselection but promises enhanced success probability. The computational model is detailed in three segments:

1. Superposition and Calculation: Similar to Shor's algorithm, a superposition of all potential states is initially created.
2. Clause Evaluation: Use of reversible quantum gates (C-ORs) that form the functional basis of clause satisfaction verification.
3. State Postparation: Implementation of $\langle 1|$ postparation that restricts the state space to solutions satisfying the 3-SAT constraints.

The realization of the gate operations, particularly the four-qubit C-OR gates as discussed, demands further refinement into fundamental quantum gates—an essential stride towards feasible implementations.

Error Correction and Optimization

A significant portion of the paper deals with strategies to address quantum errors, chiefly aiming to attenuate error probabilities through strategic use of multiple gate copies and innovative error correction mechanisms, such as proposed equalizers. The technique of mid-circuit syndrome enforcement through these equalizer gadgets offers an additional error suppression method, striving to stabilize superpositions relevant to 3-SAT solutions.

Moreover, this approach attends to the trade-off between algorithmic depth and fault-tolerance, suggesting both serial and parallel settings for optimum performance, hinging on advances in quantum technology.

Implications and Future Research Directions

The implications, as discussed, hint at transformative potential in quantum computing, especially concerning solving NP-complete problems like 3-SAT with practically feasible quantum resources. The expanded error correction capabilities offered by 2WQC could have ramifications beyond 3-SAT, potentially extending to other classes of NP problems after polynomial-time transformations, thus inviting substantial exploration into broader NP problem domains.

Future work will likely involve refinement of the discussed imperfection models, practical optimization of gate operations, simulations to portray real-world feasibility, and assessments on the potential impact on postquantum cryptography. Intriguing directions also include exploring PSPACE complexity class problems and investigating real-world applications, contingent on scaling quantum computing infrastructures.

Overall, while the theoretical underpinnings of 2WQC and its advantages over conventional quantum computing models are thoroughly articulated, significant experimental and theoretical work remains to harness the full capability of this paradigm in solving computationally complex problems. The paper serves as an enticing roadmap for subsequent advancements in the quantum computation landscape.