A recursive definition of quantum polynomial time computability (extended abstract).

R. Hayakawa, Tomoyuki Morimae

Tomoyuki Yamakami

Abstract Quantum computing has been studied over the past four decades based on two computational models of quantum circuits and quantum Turing machines. To capture quantum polynomial-time computability, a new recursion-theoretic approach was taken lately by Yamakami [J. Symb. Logic 80, pp. 1546–1587, 2020] by way of recursion schematic definition, which constitutes six initial quantum functions and three construction schemes of composition, branching, and multi-qubit quantum recursion. By taking a similar approach, we look into quantum polylogarithmic-time computability and further explore the expressing power of elementary schemes designed for such quantum computation. In particular, we introduce an elementary form of the quantum recursion, called the fast quantum recursion, and formulate $EQS$ (elementary quantum schemes) of “elementary” quantum functions. This class $EQS$ captures exactly quantum polylogarithmic-time computability, which forms the complexity class BQPOLYLOGTIME. We also demonstrate the separation of BQPOLYLOGTIME from NLOGTIME and PPOLYLOGTIME. As a natural extension of $EQS$ , we further consider an algorithmic procedural scheme that implements the well-known divide-and-conquer strategy. This divide-and-conquer scheme helps compute the parity function, but the scheme cannot be realized within our system $EQS$ .

Tomoyuki Yamakami

Abstract In the past four decades, the notion of quantum polynomial-time computability has been mathematically modeled by quantum Turing machines as well as quantum circuits. This paper seeks the third model, which is a quantum analogue of the schematic (inductive or constructive) definition of (primitive) recursive functions. For quantum functions mapping finite-dimensional Hilbert spaces to themselves, we present such a schematic definition, composed of a small set of initial quantum functions and a few construction rules that dictate how to build a new quantum function from the existing ones. We prove that our schematic definition precisely characterizes all functions that can be computable with high success probabilities on well-formed quantum Turing machines in polynomial time, or equivalently uniform families of polynomial-size quantum circuits. Our new, schematic definition is quite simple and intuitive and, more importantly, it avoids the cumbersome introduction of the well-formedness condition imposed on a quantum Turing machine model as well as of the uniformity condition necessary for a quantum circuit model. Our new approach can further open a door to the descriptional complexity of quantum functions, to the theory of higher-type quantum functionals, to the development of new first-order theories for quantum computing, and to the designing of programming languages for real-life quantum computer

Atul Mantri

It is not known whether a classical polynomial-time verifier can verify arbitrary polynomial-time quantum computations with an efficient quantum honest prover without computational assumptions. We show that, relative to a classical oracle, such verification is possible for a language beyond the polynomial hierarchy that is decided exactly in quantum polynomial time. The protocol is public-coin, with perfect completeness and soundness error at most 1/4. To our knowledge, no language with such a proof system was previously proved to lie outside AM relative to an oracle. For every unbounded polynomial-time computable function, the oracle can be chosen so that the message count and the oracle query counts of the quantum algorithm, verifier, and honest prover are bounded by this function of the input length. The message count is optimal up to a constant factor, since for a suitable oracle every interactive proof system for the language uses at least a constant fraction of the messages of our protocol on infinitely many lengths, with a constant depending on its verifier. The language is a variant of recursive Fourier sampling in which code tables encode secrets that address data tables. The tables are evaluated recursively, and the oracle stores only the leaves. The quantum algorithm recovers every secret exactly, and the verifier checks the prover's secrets recursively against random code entries. The recursion restricts the input in a correlated way, as the same secret fixes the code bits and chooses the free coordinate, so Håstad's switching lemma does not apply. We prove a switching theorem for these restrictions, which may be of independent interest. For instances of sufficient height, no constant-depth circuit of size $2^{\mathrm{poly}(n)}$ changes its output with probability above $2^{-Ω(n)}$ when the answer of a random instance is flipped by changing a single leaf.

Tomoyuki Yamakami

Nolan J. Coble, Matthew Coudron

We present a classical algorithm that, for any 3D geometrically-local, constant-depth quantum circuit $C$, and any bit string $x \in \{0,1\}^n$, can compute the quantity $| |^2$ to within any inverse-polynomial additive error in quasi-polynomial time. It is known that it is $\#P$-hard to compute this same quantity to within $2^{-n^2}$ additive error [Mov20]. The previous best known algorithm for this problem used $O(2^{n^{1/3}}\text{poly}(1/\epsilon))$ time to compute probabilities to within additive error $\epsilon$ [BGM20]. Notably, the [BGM20] paper included an elegant polynomial time algorithm for the same estimation task with 2D circuits, which makes a novel use of 1D Matrix Product States carefully tailored to the 2D geometry of the circuit in question. Surprisingly, it is not clear that it is possible to extend this use of MPS to address the case of 3D circuits in polynomial time. This raises a natural question as to whether the computational complexity of the 3D problem might be drastically higher than that of the 2D problem. In this work we address this question by exhibiting a quasi-polynomial time algorithm for the 3D case. In order to surpass the technical barriers encountered by previously known techniques we are forced to pursue a novel approach: Our algorithm has a Divide-and-Conquer structure, constructing a recursive sub-division of the given 3D circuit using carefully designed block-encodings, each creating a 3D-local circuit on at most half the number of qubits as the original. This division step is then applied recursively, expressing the original quantity as a weighted sum of smaller and smaller 3D-local quantum circuits. A central technical challenge is to control correlations arising from the entanglement that may exist between the different circuit pieces produced this way.