Computational devices may be supplied with external sources of information (oracles). Quantum oracles may transmit phase information which is available to a quantum computer but not a classical computer. One consequence of this observation is that there is an oracle which is of no assistance to a classical computer but which allows a quantum computer to solve undecidable problems. Thus useful relativized separations between quantum and classical complexity classes must exclude the transmission of phase information from oracle to computer.
Computational devices may be supplied with external sources of information (oracles). Quantum oracles may transmit phase information which is available to a quantum computer but not a classical computer. One consequence of this observation is that there is an oracle which is of no assistance to a classical computer but which allows a quantum computer to solve undecidable problems. Thus useful relativized separations between quantum and classical complexity classes must exclude the transmission of phase information from oracle to computer.
This paper continues the study of the power of oracles to separate quantum com.plexity classes from classical (including probabilistic and nondeterministic) complexity classes, which we initiated in an earlier paper [5]. The new results are that, under appropriate oracles, (1) there is a decision problem that can be solved in polynomial time on the quantum computer, which would classically require exponential time in the worst case even on a nondeterministic computer (in particular QP g NP relative to this oracle), and (2) there is a decision problem that can be solved in exponential time on the quantum computer for which any deterministic classical computer would require double exponential time on all but finitely many instances.
Recent works have demonstrated that quantum oracles have subtle behavior, as access to inverse, conjugate or controlled queries can exponentially change the query complexity of certain tasks. Inspired by these works, we introduce the notion of meta-complexity of quantum relativization. We ask: for any two quantum complexity classes, under which "types" of quantum oracles are they equal or separated? Different pantheons of oracles (or quantum oracle types, e.g. unitary vs state, poly- vs superpoly-dimensional, closed under inverse or not) form a partially ordered set based on their power in separating complexity classes. Moreover, two oracle pantheons A and B are separated if there exists a pair of complexity classes that are separated under an oracle from pantheon A but yet the complexity classes are equivalent under all oracles from pantheon B.
We show that this meta-complexity can be nontrivial by giving a complete classification, within the family of oracle pantheons defined in this paper, of which models can separate the complexity classes PostBQP and PreciseBQP, the exponentially precise variant of BQP. Both classes equal PP in the unrelativized setting. Within our taxonomy, they remain equal relative to real or polynomial-dimensional unitary oracles and whenever inverse or conjugate access is supplied. In contrast, we give a separation relative to forward-only complex diagonal unitaries of superpolynomial dimension, as well as a separation relative to single-qubit state-preparation oracles. We view this as a test case for the meta-complexity of oracles which underscores the subtlety inherent to the relativization of quantum complexity classes.
An important theoretical problem in the study of quantum computation, that is also practically relevant in the context of near-term quantum devices, is to understand the computational power of hybrid models, that combine poly-time classical computation with short-depth quantum computation. Here, we consider two such models: CQ_d which captures the scenario of a polynomial-time classical algorithm that queries a d-depth quantum computer many times; and QC_d which is more analogous to measurement-based quantum computation and captures the scenario of a d-depth quantum computer with the ability to change the sequence of gates being applied depending on measurement outcomes processed by a classical computation. Chia, Chung & Lai (STOC 2020) and Coudron & Menda (STOC 2020) showed that these models (with d=log^O(1) (n)) are strictly weaker than BQP (the class of problems solvable by poly-time quantum computation), relative to an oracle, disproving a conjecture of Jozsa in the relativised world. We show that, despite the similarities between CQ_d and QC_d, the two models are incomparable, i.e. CQ_d ⊈ QC_d and QC_d ⊈ CQ_d relative to an oracle. In other words, there exist problems that one model can solve but not the other and vice versa. We do this by considering new oracle problems that capture the distinctions between the two models and by introducing the notion of an intrinsically stochastic oracle, an oracle whose responses are inherently randomised, which is used for our second result. While we leave showing the second separation relative to a standard oracle as an open problem, we believe the notion of stochastic oracles could be of independent interest for studying complexity classes which have resisted separation in the standard oracle model. Our constructions also yield simpler oracle separations between the hybrid models and BQP, compared to earlier works.
In this thesis we address the question, what is the resource, or property, that enables the advantage of quantum computers?The theory of quantum computers dates back to the eighties, so one would think there already is an answer to this question.There are several proposed solutions, but to this date, there is no consensus on an answer.Primarily, the advantage of quantum computers is characterized by a speedup for certain computational problems.This speedup is measured by comparing quantum algorithms with the bestknown classical algorithms.For some algorithms we assume access to an object called oracle.The oracle computes a function, and the complexity of the oracle is of no concern.Instead, we count the number of queries to the oracle needed to solve the problem.Informally, the question we ask using an oracle is: if we can compute this function efficiently, what else could we then compute.However, using oracles while measuring a quantum speedup, we assume access to vastly different oracles residing in different models of computation.For our investigation of the speedup, we introduce a classical simulation framework that imitates quantum algorithms.The simulation suggests that the property enabling the potential quantum speedup is the ability to store, process, and retrieve information in an additional degree of freedom.We then theoretically verified that this is true for all problems that can be efficiently solved with a quantum computer.In parallel to this, we also see that quantum oracles sharply specify the information we can retrieve from the additional degree of freedom, while regular oracles do not.A regular oracle does not even allow for an extra degree of freedom.We conclude that comparing quantum with classical oracle query complexity bounds does not provide conclusive evidence for a quantum advantage.I want to thank Jan-Åke Larsson, without whom this research would not have been possible.If there is any magic in the world, it shows through Jan-Åke's ability to simultaneously be a good supervisor, colleague, and friend.I would also like to thank my co-supervisor Peter Jonsson, for many valuable discussions -not only restricted to theoretical computer science.To my friends and collogues that have challenged me to become better, especially Jonathan Jogenfors
In recent years, the quantum oracle model introduced by Aaronson and Kuperberg (2007) has found a lot of use in showing oracle separations between complexity classes and cryptographic primitives. It is generally assumed that proof techniques that do not relativize with respect to quantum oracles will also not relativize with respect to classical oracles. Aaronson (2009) showed that this is not the case by showing a complexity class containment that relativizes with respect to classical oracles but not with quantum oracles. However, their result only works for zero-error quantum complexity classes and they leave open the problem for bounded-error complexity classes. We show that there is a quantum oracle problem that is contained in the class QMA, but not in a class we call polyQCPH. However, with respect to classical oracles, QMA is contained in polyQCPH, because polyQCPH is equal to PSPACE with respect to classical oracles. Our result works for polyQCPH, which is a bounded-error complexity class, thus it resolves the open problem from Aaronson (2009). We also show that the same separation holds relative to a distributional oracle, which is a model introduced by Natarajan and Nirkhe (2024). We believe our findings show the need for some caution when using these non-standard oracle models, particularly when showing separations between quantum and classical resources.
It is a long-standing open question in quantum complexity theory whether the definition of non-deterministic quantum computation requires quantum witnesses (QMA) or if classical witnesses suffice (QCMA). We make progress on this question by constructing a randomized classical oracle separating the respective computational complexity classes. Previous separations [Aaronson-Kuperberg (CCC'07), Fefferman-Kimmel (MFCS'18)] required a quantum unitary oracle. The separating problem is deciding whether a distribution supported on regular un-directed graphs either consists of multiple connected components (yes instances) or consists of one expanding connected component (no instances) where the graph is given in an adjacency-list format by the oracle. Therefore, the oracle is a distribution over n-bit boolean functions.