Abstract
Inferring algorithmic structure in data is essential for discovering causal generative models. In this research, we present a quantum computing framework using the circuit model, for estimating algorithmic information metrics. The canonical computation model of the Turing machine is restricted in time and space resources, to make the target metrics computable under realistic assumptions. The universal prior distribution for the automata is obtained as a quantum superposition, which is further conditioned to estimate the metrics. Specific cases are explored where the quantum implementation offers polynomial advantage, in contrast to the exhaustive enumeration needed in the corresponding classical case. The unstructured output data and the computational irreducibility of Turing machines make this algorithm impossible to approximate using heuristics. Thus, exploring the space of program-output relations is one of the most promising problems for demonstrating quantum supremacy using Grover search that cannot be dequantized. Experimental use cases for quantum acceleration are developed for self-replicating programs and algorithmic complexity of short strings. With quantum computing hardware rapidly attaining technological maturity, we discuss how this framework will have significant advantage for various genomics applications in meta-biology, phylogenetic tree analysis, protein-protein interaction mapping and synthetic biology. This is the first time experimental algorithmic information theory is implemented using quantum computation. Our implementation on the Qiskit quantum programming platform is copy-left and is publicly available on GitHub.
Highlights
The evaluation of metrics such as the algorithmic complexity and algorithmic probability of finite sequences are key in scientific inference
The quantum advantage of using QPULBA for experimental algorithmic information theory (EAIT) cannot be in the time complexity of a single execution as it takes the same scaling of resources of quantum and classical gates to run a automata for a specified number of cycles
The Algorithmic Complexity of Short Strings (ACSS) database is constructed by approximating the output frequency distribution for Turing machines with 5 states and 2 symbols generating the algorithmic complexity of strings of size ≤ 12 over ≈ 500 iterations
Summary
The evaluation of metrics such as the algorithmic complexity and algorithmic probability of finite sequences are key in scientific inference. The uncomputable nature of these metrics is approximated in practice by restricting the resources available to the computational models, such as time/cycles and space/memory Such approaches remain intractable except for the simplest of cases. Quantum search-based approaches require high qubit multiplicity and quality as compared to those available in the NISQ era, these search approaches cannot be dequantized for this use case of the unstructured and computationally irreducible database of program-output relations. That makes this particular algorithm an ideal candidate for demonstrating quantum supremacy with widespread application.
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