This thesis explores the relationship between quantum computation and combinatorial structures, with the goal of improving our understanding of the complexity of quantum computation. We begin by studying the case when the complexity of combinatorial structures can be used to provide evidence for the hardness of classically simulating quantum computations. To this end, we show that the complexity of evaluating multiplicative-error approximations of Jones polynomials can be used to bound the classical complexity of simulating random quantum computations. We then proceed by studying the contrary case, that is, when do the combinatorial structures allow for an efficient classical simulation of quantum computations? We establish an efficient deterministic approximation algorithm for the Ising model partition function with complex parameters when the interactions and external fields are absolutely bounded close to zero. This provides an efficient classical algorithm for simulating a class of quantum computations with bounded interactions between the qubits.
In the second part of this thesis, we present some independent results on the efficient preparation of Fock states with a high number of photons from a resource of single photons. These Fock states are a fundamental resource in many quantum information protocols.
Classical approximation complexity asks what solution quality can be guaranteed with polynomial-time computation. The classes APX, PTAS, and FPTAS distinguish a fixed approximation ratio, approximation to any fixed accuracy, and approximation schemes whose running time is also polynomial in inverse accuracy. Their randomized counterparts are R-APX, R-PTAS, and R-FPTAS. We define bounded-error quantum counterparts BQ-APX, BQ-PTAS, and BQ-FPTAS. Membership requires a uniform quantum algorithm that, on every input, returns a feasible classical solution achieving at least the claimed approximation ratio with probability at least 2/3. Scores (objective values) must be efficiently classically computable. Running time includes parameter selection, preparation, measurement, decoding, and repetition. Many quantum optimization methods are used heuristically, and high benchmark scores alone do not establish these guarantees. We further establish a conditional hierarchy for logarithmic, polynomial, and exponential approximation factors.
Assuming NP
⊈ BQP, the quantum classes form a strict hierarchy. Problems based on prime factorization and discrete logarithms give conditional quantum-classical separations. Certified Maximum Order has an exact quantum algorithm, while any randomized polynomial-time algorithm guaranteeing at least an inverse-polynomial approximation ratio on every input would yield efficient factoring. Discrete-Logarithm Fitting has an exact quantum algorithm and a deterministic one-half approximation, but any fixed improvement over one half would give a randomized polynomial-time algorithm for the safe-prime discrete logarithm problem. Our results show, under explicit complexity assumptions, that quantum computation can improve worst-case approximation guarantees. A quantum-classical gap for common problems such as MaxCut or MaxSAT remains open.
Motivated by the result that an ‘approximate’ evaluation of the Jones polynomial of a braid at a 5th root of unity can be used to simulate the quantum part of any algorithm in the quantum complexity class BQP, and results relating BQP to the counting class GapP, we introduce a form of additive approximation which can be used to simulate a function in BQP. We show that all functions in the classes #P and GapP have such an approximation scheme under certain natural normalizations. However, we are unable to determine whether the particular functions we are motivated by, such as the above evaluation of the Jones polynomial, can be approximated in this way. We close with some open problems motivated by this work.
We use the class of commuting quantum computations known as IQP (instantaneous quantum polynomial time) to strengthen the conjecture that quantum computers are hard to simulate classically. We show that, if either of two plausible average-case hardness conjectures holds, then IQP computations are hard to simulate classically up to constant additive error. One conjecture relates to the hardness of estimating the complex-temperature partition function for random instances of the Ising model; the other concerns approximating the number of zeroes of random low-degree polynomials. We observe that both conjectures can be shown to be valid in the setting of worst-case complexity. We arrive at these conjectures by deriving spin-based generalizations of the boson sampling problem that avoid the so-called permanent anticoncentration conjecture.
We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al. that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-cornplete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a #P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.
We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al. that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-complete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a #P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.