Theory of quasi-exact fault-tolerant quantum computing and valence-bond-solid codes

In this work, we develop the theory of quasi-exact fault-tolerant quantum (QEQ) computation, which uses qubits encoded into quasi-exact quantum error-correction codes (``quasi codes''). By definition, a quasi code is a parametric approximate code that can become exact by tuning its parameters. The model of QEQ computation lies in between the two well-known ones: the usual noisy quantum computation without error correction and the usual fault-tolerant quantum computation, but closer to the later. Many notions of exact quantum codes need to be adjusted for the quasi setting. Here we develop quasi error-correction theory using quantum instrument, the notions of quasi universality, quasi code distances, and quasi thresholds, etc. We find a wide class of quasi codes which are called valence-bond-solid codes, and we use them as concrete examples to demonstrate QEQ computation.

Exact Quantum 1-Query Algorithms and Complexity

Deutsch–Jozsa problem (D–J) has exact quantum 1-query complexity (“exact” means no error), but requires super-exponential queries for the optimal classical deterministic decision trees. D–J problem is equivalent to a symmetric partial Boolean function, and in fact, all symmetric partial Boolean functions having exact quantum 1-query complexity have been found out and these functions can be computed by D–J algorithm. A special case is that all symmetric Boolean functions with exact quantum 1-query complexity follow directly and these functions are also all total Boolean functions with exact quantum 1-query complexity obviously. Then there are pending problems concerning partial Boolean functions having exact quantum 1-query complexity and new results have been found, but some problems are still open. In this paper, we review these results regarding exact quantum 1-query complexity and in particular, we also obtain a new result that a partial Boolean function with exact quantum 1-query complexity is constructed and it cannot be computed by D–J algorithm. Further problems are pointed out for future study.

Partial Boolean Functions With Exact Quantum Query Complexity One

We provide two sufficient and necessary conditions to characterize any n-bit partial Boolean function with exact quantum query complexity 1. Using the first characterization, we present all n-bit partial Boolean functions that depend on n bits and can be computed exactly by a 1-query quantum algorithm. Due to the second characterization, we construct a function F that maps any n-bit partial Boolean function to some integer, and if an n-bit partial Boolean function f depends on k bits and can be computed exactly by a 1-query quantum algorithm, then F(f) is non-positive. In addition, we show that the number of all n-bit partial Boolean functions that depend on k bits and can be computed exactly by a 1-query quantum algorithm is not bigger than an upper bound depending on n and k. Most importantly, the upper bound is far less than the number of all n-bit partial Boolean functions for all efficiently big n.