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A SAT Solver + Computer Algebra Attack on the Minimum Kochen-Specker Problem

Zhengyu Li, Curtis Bright, Vijay Ganesh

Published Jun 23, 2023• 3 citations
Publisher

Abstract

Abstract One of the foundational results in quantum mechanics is the Kochen--Specker (KS) theorem, which states that any theory whose predictions agree with quantum mechanics must be contextual, i.e., a quantum observation cannot be understood as revealing a pre-existing value. The theorem hinges on the existence of a mathematical object called a KS vector system. While many KS vector systems are known to exist, the problem of finding the minimum KS vector system has remained stubbornly open for over 55 years, despite significant attempts by leading scientists and mathematicians. In this paper, we present a new method based on a combination of a SAT solver and a computer algebra system (CAS) to address this problem. Our approach improves the lower bound on the minimum number of vectors in a KS system from 22 to 24, and is about 35,000 times more efficient compared to the previous best computational methods. The increase in efficiency derives from the fact we are able to exploit the powerful combinatorial search-with-learning capabilities of a SAT solver together with the isomorph-free exhaustive generation methods of a CAS. The quest for the minimum KS vector system is motivated by myriad applications such as simplifying experimental tests of contextuality, zero-error classical communication, dimension witnessing, and the security of certain quantum cryptographic protocols. To the best of our knowledge, this is the first application of a novel SAT+CAS system to a problem in the realm of quantum foundations.

DOI
10.21203/rs.3.rs-2279231/v2

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A SAT Solver + Computer Algebra Attack on the Minimum Kochen-Specker Problem

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Abstract One of the most fundamental results in the foundations of quantum mechanics is the Kochen--Specker (KS) theorem, a `no-go' theorem which states that contextuality is an essential feature of any hidden-variable theory. The theorem hinges on the existence of a mathematical object called a KS vector system. Although the existence of a KS vector system was first established by Kochen and Specker, the problem of the minimum size of such a system has stubbornly remained open for over 50 years. In this paper, we present a new method based on a combination of a SAT solver and a computer algebra system (CAS) to address this problem. We improve the lower bound on the minimum number of vectors in a KS system from 22 to 23 and improve the efficiency of the search by a factor of over 1000 when compared to the most recent computational methods. Finding a minimum KS system would simplify experimental tests of the KS theorem and have direct applications in quantum information processing, specifically in the security of quantum cryptographic protocols based on complementarity, zero-error classical communication, and dimension witnessing.

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One of the fundamental results in quantum foundations is the Kochen–Specker (KS) theorem, which states that any theory whose predictions agree with quantum mechanics must be contextual, i.e., a quantum observation cannot be understood as revealing a pre-existing value. The theorem hinges on the existence of a mathematical object called a KS vector system. While many KS vector systems are known, the problem of finding the minimum KS vector system in three dimensions (3D) has remained stubbornly open for over 55 years. To address the minimum KS problem, we present a new verifiable proof-producing method based on a combination of a Boolean satisfiability (SAT) solver and a computer algebra system (CAS) that uses an isomorph-free orderly generation technique that is very effective in pruning away large parts of the search space. Our method shows that a KS system in 3D must contain at least 24 vectors. We show that our sequential and parallel Cube-and-Conquer (CnC) SAT+CAS methods are significantly faster than SAT-only, CAS-only, and a prior CAS-based method of Uijlen and Westerbaan. Further, while our parallel pipeline is somewhat slower than the parallel CnC version of the recently introduced Satisfiability Modulo Theories (SMS) method, this is in part due to the overhead of proof generation. Finally, we provide the first computer-verifiable proof certificate of a lower bound to the KS problem with a size of 40.3 TiB in order 23.

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On Searching for Small Kochen-Specker Vector Systems (extended version)

Felix Arends, Joël Ouaknine, Charles W. Wampler

Kochen-Specker (KS) vector systems are sets of vectors in R^3 with the property that it is impossible to assign 0s and 1s to the vectors in such a way that no two orthogonal vectors are assigned 0 and no three mutually orthogonal vectors are assigned 1. The existence of such sets forms the basis of the Kochen-Specker and Free Will theorems. Currently, the smallest known KS vector system contains 31 vectors. In this paper, we establish a lower bound of 18 on the size of any KS vector system. This requires us to consider a mix of graph-theoretic and topological embedding problems, which we investigate both from theoretical and practical angles. We propose several algorithms to tackle these problems and report on extensive experiments. At the time of writing, a large gap remains between the best lower and upper bounds for the minimum size of KS vector systems.

arXiv:1111.3301•Nov 14, 2011•1 citation
Abstract One of the foundational results in quantum mechanics is the Kochen--Specker (KS) theorem, which states that any theory whose predictions agree with quantum mechanics must be contextual, i.e., a quantum observation cannot be understood as revealing a pre-existing value. The theorem hinges on the existence of a mathematical object called a KS vector system. While many KS vector systems are known to exist, the problem of finding the minimum KS vector system has remained stubbornly open for over 55 years, despite significant attempts by leading scientists and mathematicians. In this paper, we present a new method based on a combination of a SAT solver and a computer algebra system (CAS) to address this problem. Our approach improves the lower bound on the minimum number of vectors in a KS system from 22 to 24, and is about 35,000 times more efficient compared to the previous best computational methods. The increase in efficiency derives from the fact we are able to exploit the powerful combinatorial search-with-learning capabilities of a SAT solver together with the isomorph-free exhaustive generation methods of a CAS. The quest for the minimum KS vector system is motivated by myriad applications such as simplifying experimental tests of contextuality, zero-error classical communication, dimension witnessing, and the security of certain quantum cryptographic protocols. To the best of our knowledge, this is the first application of a novel SAT+CAS system to a problem in the realm of quantum foundations.
Abstract One of the most fundamental results in the foundations of quantum mechanics is the Kochen--Specker (KS) theorem, a `no-go' theorem which states that contextuality is an essential feature of any hidden-variable theory. The theorem hinges on the existence of a mathematical object called a KS vector system. Although the existence of a KS vector system was first established by Kochen and Specker, the problem of the minimum size of such a system has stubbornly remained open for over 50 years. In this paper, we present a new method based on a combination of a SAT solver and a computer algebra system (CAS) to address this problem. We improve the lower bound on the minimum number of vectors in a KS system from 22 to 23 and improve the efficiency of the search by a factor of over 1000 when compared to the most recent computational methods. Finding a minimum KS system would simplify experimental tests of the KS theorem and have direct applications in quantum information processing, specifically in the security of quantum cryptographic protocols based on complementarity, zero-error classical communication, and dimension witnessing.
One of the fundamental results in quantum foundations is the Kochen–Specker (KS) theorem, which states that any theory whose predictions agree with quantum mechanics must be contextual, i.e., a quantum observation cannot be understood as revealing a pre-existing value. The theorem hinges on the existence of a mathematical object called a KS vector system. While many KS vector systems are known, the problem of finding the minimum KS vector system in three dimensions (3D) has remained stubbornly open for over 55 years. To address the minimum KS problem, we present a new verifiable proof-producing method based on a combination of a Boolean satisfiability (SAT) solver and a computer algebra system (CAS) that uses an isomorph-free orderly generation technique that is very effective in pruning away large parts of the search space. Our method shows that a KS system in 3D must contain at least 24 vectors. We show that our sequential and parallel Cube-and-Conquer (CnC) SAT+CAS methods are significantly faster than SAT-only, CAS-only, and a prior CAS-based method of Uijlen and Westerbaan. Further, while our parallel pipeline is somewhat slower than the parallel CnC version of the recently introduced Satisfiability Modulo Theories (SMS) method, this is in part due to the overhead of proof generation. Finally, we provide the first computer-verifiable proof certificate of a lower bound to the KS problem with a size of 40.3 TiB in order 23.
The problem of finding the minimum three-dimensional Kochen–Specker (KS) vector system, an important problem in quantum foundations, has remained open for over 55 years. We present a new method to address this problem based on a combination of a Boolean satisfiability (SAT) solver and a computer algebra system (CAS). Our approach improved the lower bound on the size of a KS system from 22 to 24. More importantly, we provide the first computer-verifiable proof certificate of a lower bound to the KS problem with a proof size of 41.6 TiB for order 23. The efficiency is due to the powerful combination of SAT solvers and CAS-based orderly generation.
Recently, quantum contextuality has been proved to be the source of quantum computation's power. That, together with multiple recent contextual experiments, prompts improving the methods of generation of contextual sets and finding their features. The most elaborated contextual sets, which offer blueprints for contextual experiments and computational gates, are the Kochen-Specker (KS) sets. In this paper, we show a method of vector generation that supersedes previous methods. It is implemented by means of algorithms and programs that generate hypergraphs embodying the Kochen-Specker property and that are designed to be carried out on supercomputers. We show that vector component generation of KS hypergraphs exhausts all possible vectors that can be constructed from chosen vector components, in contrast to previous studies that used incomplete lists of vectors and therefore missed a majority of hypergraphs. Consequently, this unified method is far more efficient for generations of KS sets and their implementation in quantum computation and quantum communication. Several new KS classes and their features have been found and are elaborated on in the paper. Greechie diagrams are discussed.
As quantum contextuality proves to be a necessary resource for universal quantum computation, we present a general method for vector generation of Kochen-Specker (KS) contextual sets in the form of hypergraphs. The method supersedes all three previous methods: (i) fortuitous discoveries of smallest KS sets, (ii) exhaustive upward hypergraph-generation of sets, and (iii) random downward generation of sets from fortuitously obtained big master sets. In contrast to previous works, we can generate master sets which contain all possible KS sets starting with nothing but a few simple vector components. From them we can readily generate all KS sets obtained in the last half a century and any specified new KS sets. Herewith we can generate sufficiently large sets as well as sets with definite required features and structures to enable varieties of different implementations in quantum computation and communication.
Kochen-Specker (KS) vector systems are sets of vectors in R^3 with the property that it is impossible to assign 0s and 1s to the vectors in such a way that no two orthogonal vectors are assigned 0 and no three mutually orthogonal vectors are assigned 1. The existence of such sets forms the basis of the Kochen-Specker and Free Will theorems. Currently, the smallest known KS vector system contains 31 vectors. In this paper, we establish a lower bound of 18 on the size of any KS vector system. This requires us to consider a mix of graph-theoretic and topological embedding problems, which we investigate both from theoretical and practical angles. We propose several algorithms to tackle these problems and report on extensive experiments. At the time of writing, a large gap remains between the best lower and upper bounds for the minimum size of KS vector systems.