A celebrated important result due to (2002 Commun. Math. Phys. 227 605–22) states that providing additive approximations of the Jones polynomial at the kth root of unity, for constant k=5 and k≥7, is BQP-hard. Together with the algorithmic results of (2005) and (2002 Commun. Math. Phys. 227 587–603), this gives perhaps the most natural BQP-complete problem known today and motivates further study of the topic. In this paper, we focus on the universality proof; we extend the result of (2002) to ks that grow polynomially with the number of strands and crossings in the link, thus extending the BQP-hardness of Jones polynomial approximations to all values to which the AJL algorithm applies ( 2005), proving that for all those values, the problems are BQP-complete. As a side benefit, we derive a fairly elementary proof of the density result, without referring to advanced results from Lie algebra representation theory, making this important result accessible to a wider audience in the computer science research community. We make use of two general lemmas we prove, the bridge lemma and the decoupling lemma, which provide tools for establishing the density of subgroups in SU(n). Those tools seem to be of independent interest in more general contexts of proving the quantum universality. Our result also implies a completely classical statement, that the multiplicative approximations of the Jones polynomial, at exactly the same values, are #P-hard, via a recent result due to Kuperberg (2009 arXiv:0908.0512). Since the first publication of those results in their preliminary form (Aharonov and Arad 2006 arXiv:quant-ph 0605181), the methods we present here have been used in several other contexts (Aharonov and Arad 2007 arXiv:quant-ph 0702008; Peter and Stephen 2008 Quantum Inf. Comput. 8 681). The present paper is an improved and extended version of the results presented by Aharonov and Arad (2006) and includes discussions of the developments since then.
Since the first publication of the results presented here (in preliminary form) @cite_26 , they were already used in several contexts: Shor and Jordan @cite_15 built on the methods we develop here to prove universality of a variant of the Jones polynomial approximation problem, in the model of quantum computation with one clean qubit. In the extension of the AJL algorithm @cite_21 to the Potts model @cite_36 , Aharonov build on those methods to prove universality of approximating the Jones polynomial in many other values, and even in values which correspond to non-unitary representations. We hope that the method we present here will be useful is future other contexts as well.
@cite_36 @cite_15 @cite_26 @cite_21
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In the first 36 pages of this paper, we provide polynomial quantum algorithms for additive approximations of the Tutte polynomial, at any point in the Tutte plane, for any planar graph. This includes as special cases the AJL algorithm for the Jones polynomial, the partition function of the Potts model for any weighted planer graph at any temperature, and many other combinatorial graph properties. In the second part of the paper we prove the quantum universality of many of the problems for which we provide an algorithm, thus providing a large set of new quantum-complete problems. Unfortunately, we do not know that this holds for the Potts model case; this is left as an important open problem. The main progress in this work is in our ability to handle non-unitary representations of the Temperley Lieb algebra, both when applying them in the algorithm, and, more importantly, in the proof of universality, when encoding quantum circuits using non-unitary operators. To this end we develop many new tools, that allow proving density and applying the Solovay Kitaev theorem in the case of non-unitary matrices. We hope that these tools will open up new possibilities of using non-unitary reps in other quantum computation contexts. It is known that evaluating a certain approximation to the Jones polynomial for the plat closure of a braid is a BQP-complete problem. That is, this problem exactly captures the power of the quantum circuit model[13, 3, 1]. The one clean qubit model is a model of quantum computation in which all but one qubit starts in the maximally mixed state. One clean qubit computers are believed to be strictly weaker than standard quantum computers, but still capable of solving some classically intractable problems [21]. Here we show that evaluating a certain approximation to the Jones polynomial at a fifth root of unity for the trace closure of a braid is a complete problem for the one clean qubit complexity class. That is, a one clean qubit computer can approximate these Jones polynomials in time polynomial in both the number of strands and number of crossings, and the problem of simulating a one clean qubit computer is reducible to approximating the Jones polynomial of the trace closure of a braid. Freedman, Kitaev, and Wang (2002), and later Aharonov, Jones, and Landau (2009), established a quantum algorithm to "additively" approximate the Jones polynomial V(L;t) at any principal root of unity t. The strength of this additive approximation depends exponentially on the bridge number of the link presentation. Freedman, Larsen, and Wang (2002) established that the approximation is universal for quantum computation at a non- lattice, principal root of unity. We show that any value-distinguishing approximation of the Jones polynomial at these non-lattice roots of unity is #P-hard. Given the power to decide whetherjV(L;t)j b for fixed constants 0 1, T(G; x; y) is #P-hard to approximate within a factor of c even for planar graphs G. Along the way, we clarify and generalize both Aaronson's theorem and the Solovay- Kitaev theorem. In the near future, there will likely be special-purpose quantum computers with 40--50 high-quality qubits. This paper lays general theoretical foundations for how to use such devices to demonstrate "quantum supremacy": that is, a clear quantum speedup for some task, motivated by the goal of overturning the Extended Church-Turing Thesis as confidently as possible. First, we study the hardness of sampling the output distribution of a random quantum circuit, along the lines of a recent proposal by the Quantum AI group at Google. We show that there's a natural average-case hardness assumption, which has nothing to do with sampling, yet implies that no polynomial-time classical algorithm can pass a statistical test that the quantum sampling procedure's outputs do pass. Compared to previous work -- for example, on BosonSampling and IQP -- the central advantage is that we can now talk directly about the observed outputs, rather than about the distribution being sampled. Second, in an attempt to refute our hardness assumption, we give a new algorithm, inspired by Savitch's Theorem, for simulating a general quantum circuit with n qubits and depth d in polynomial space and dO(n) time. We then discuss why this and other known algorithms fail to refute our assumption. Third, resolving an open problem of Aaronson and Arkhipov, we show that any strong quantum supremacy theorem -- of the form "if approximate quantum sampling is classically easy, then the polynomial hierarchy collapses"-- must be non-relativizing. This sharply contrasts with the situation for exact sampling. Fourth, refuting a conjecture by Aaronson and Ambainis, we show that there is a sampling task, namely Fourier Sampling, with a 1 versus linear separation between its quantum and classical query complexities. Fifth, in search of a "happy medium" between black-box and non-black-box arguments, we study quantum supremacy relative to oracles in P poly. Previous work implies that, if one-way functions exist, then quantum supremacy is possible relative to such oracles. We show, conversely, that some computational assumption is needed: if SampBPP = SampBQP and NP ⊆ BPP, then quantum supremacy is impossible relative to oracles with small circuits.