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arXiv:2408.05812

3-SAT solver for two-way quantum computers

Jarek Duda

Published Aug 11, 2024• 0 citations
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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.

DOI
10.48550/arxiv.2408.05812

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While quantum computers assume existence of state preparation process ∣0⟩|0\rangle∣0⟩, CPT symmetry of physics says that performing such process in CPT symmetry perspective, e.g. reversing used EM impulses (V(t)→V(−t)V(t)\to V(-t)V(t)→V(−t)), we should get its symmetric analog ⟨0∣\langle 0|⟨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⟩|0\rangle∣0⟩ and ⟨0∣\langle 0|⟨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.
Two-way quantum computers (2WQC) are proposed extension of standard 1WQC: adding conjugated state preparation operation ⟨0∣\langle 0|⟨0∣ similar to postselection $|0\ra \langle 0|$, by performing a process which from perspective of CPT symmetry is the original state preparation process, for example by reversing EM impulses used for state preparation. As there were concerns that this extension might violate no-cloning theorem for example for attacks on quantum cryptographic protocols like BB84, here we extend the original proof to show this theorem still holds for 2WQC and postselection.
Standard one-way quantum computers (1WQC) combine time symmetric unitary evolution, with asymmetric treatment of boundaries: state preparation allows to enforce a chosen initial state, however, for the final state measurement chooses a random value instead. As e.g. pull/push, negative/positive pressure, stimulated emission/absorption causing deexcitation/excitation are CPT analogs, and one can be used for state preparation, the second should allow for its CPT analog, referred here as CPT(state preparation) - allowing for additional chosen enforcement of the final state, its more active treatment than measurement. It should act similarly to postselection, but through applied physical constraints (instead of running multiple times). Like pumped to ∣1⟩|1\rangle∣1⟩ prepared state vs its "unpumped" ⟨0∣\langle 0|⟨0∣ CPT analog, hopefully allowing to construct two-way quantum computers (2WQC) e.g. hydrodynamical, and hopefully photonic: seen as ⟨Φfinal∣Uquantum gates∣Φinitial⟩\langle Φ_{\textrm{final}}|U_{\textrm{quantum gates}}|Φ_{\textrm{initial}}\rangle⟨Φfinal​∣Uquantum gates​∣Φinitial​⟩ like for scattering matrix, with influenced both initial and final states. If possible, for example for an instance of 3-SAT problem on nnn variables, we could prepare ensemble of 2n2^n2n inputs with Hadamard gates, calculate 3-SAT alternatives for them, and use CPT(state preparation) to enforce outcomes of all these alternatives to '1'. This way hopefully restricting this ensemble to satisfying given 3-SAT problem: ∑a:SAT(a)∣a⟩\sum_{a:\textrm{SAT}(a)} |a\rangle∑a:SAT(a)​∣a⟩, in theory allowing to attack NP problems by simultaneously pushing and pulling information through the system for better control.
Standard one-way quantum computers (1WQC) combine time symmetric unitary evolution, with asymmetric treatment of boundaries: state preparation allows to enforce chosen initial state, however, for the final state measurement chooses a random value instead. As e.g. pull/push, negative/positive pressure, stimulated emission/absorption causing deexcitation/excitation are CPT analogs, and one can be used for state preparation, there should also exist its CPT analog, referred here as CPT(state preparation) - allowing for additional chosen enforcement of the final state, its more active treatment than measurement. It should act similarly to postselection, but through applied physical constraints (instead of running multiple times). Like pumped to |1> prepared state vs its "unpumped" <0| CPT analog, hopefully allowing to construct two-way quantum computers (2WQC) e.g. hydrodynamical, and hopefully photonic, with influenced both initial and final states. If possible, for example for an instance of 3-SAT problem on n variables, we could prepare ensemble of 2^n inputs with Hadamard gates, calculate 3-SAT alternatives for them, and use CPT(state preparation) to enforce outcomes of all these alternatives to '1'. This way hopefully restricting this ensemble to satisfying given 3-SAT problem, in theory allowing to attack NP problems by simultaneously pushing and pulling information through the system for better control.
Satisfiability (SAT) is a central problem in computer science, and advances in SAT-solving algorithms have a far-reaching impact across many fields. Recent works have proposed quantum SAT solvers based on Grover's algorithm, a quantum search technique. However, Grover-based approaches face a key limitation: they typically require prior knowledge of the number of satisfying assignments of the target Boolean formula. This information is unavailable in most practical settings. Quantum counting can be used to estimate this quantity, but it incurs a computational overhead that is several orders of magnitude higher than Grover search. In this paper, we propose a novel quantum SAT solver based on entanglement and equivalence checking. Our method does not assume prior knowledge of the number of solutions and is computationally more efficient than quantum counting. Although the worst case time complexity is inevitably exponential, we prove that the expected time complexity of our approach is only constant time O(1) over random Boolean functions. Experimental results also support our theoretical claim.
Standard one-way quantum computers (1WQC) combine time symmetric unitary evolution, with asymmetric treatment of boundaries: state preparation allows to enforce chosen initial state, however, for the final state measurement chooses a random value instead. As e.g. pull/push, negative/positive pressure, stimulated emission/absorption causing deexcitation/excitation are CPT analogs, and one can be used for state preparation, there should also exist its CPT analog, referred here as CPT(state preparation) - allowing for additional chosen enforcement of the final state, its more active treatment than measurement. It should act similarly to postselection, but through applied physical constraints (instead of running multiple times). Like pumped to |1> prepared state vs its “unpumped” <0| CPT analog, hopefully allowing to construct two-way quantum computers (2WQC) e.g. hydrodynamical, and hopefully photonic, with influenced both initial and final states. If possible, for example for an instance of 3-SAT problem on n variables, we could prepare ensemble of 2^n inputs with Hadamard gates, calculate 3-SAT alternatives for them, and use CPT(state preparation) to enforce outcomes of all these alternatives to ‘1’. This way hopefully restricting this ensemble to satisfying given 3-SAT problem, in theory allowing to attack NP problems by simultaneously pushing and pulling information through the system for better control.
Although quantum error correction is widely believed to be necessary for impactful applications of quantum computers, the design of quantum error correction codes is largely done by hand, without respect to problem or hardware constraints. In this work, we present a highly general and flexible framework for the computational design of both physics-inspired and hardware-aware quantum codes. To do so, we formulate code design as a Boolean satisfiability (SAT) problem and show how to incorporate all required error correction criteria. We prove that code design is NP-complete, ruling out any efficient algorithm for designing codes in general. Nonetheless, we show that state-of-the-art SAT solvers are able to effectively find solutions for many practical problems. Notably, we are able to design physics-inspired codes with up to 100 physical qubits in minutes, and we design new hardware-aware codes for biased noise which have a lower logical error rate than state-of-the-art surface codes.