The Quantum Computer Puzzle

Abstract

Q uantum computers are hypothetical devices, based on quantum physics, which would enable us to perform certain computations hundreds of orders of magnitude faster than digital computers. This feature is coined "quantum supremacy", and one aspect or another of such quantum computational supremacy might be seen by experiments in the near future: by implementing quantum error-correction or by systems of noninteracting bosons or by exotic new phases of matter called anyons or by quantum annealing, or in various other ways. We concentrate in this paper on the model of a universal quantum computer that allows the full computational potential for quantum systems, and on the restricted model, called "BosonSampling", based on noninteracting bosons.A main reason for concern regarding the feasibility of quantum computers is that quantum systems are inherently noisy. We will describe an optimistic hypothesis regarding quantum noise that will allow quantum computing and a pessimistic hypothesis that won't. The quantum computer puzzle is to decide between these two hypotheses. We list some remarkable consequences of the optimistic hypothesis, giving strong reasons for the intensive efforts to build quantum computers, as well as good reasons for suspecting that this might not be possible. For systems of noninteracting bosons, we explain how quantum supremacy achieved without noise is replaced, in the presence of noise, by a very low yet fascinating computational power. 1 Finally, we describe eight predictions about quantum physics and computation from the pessimistic hypothesis. 2 Are quantum computers feasible? Is quantum supremacy possible? My expectation is that the pessimistic hypothesis will prevail, leading to a negative answer. Rather than regarding this possibility as an unfortunate failure that impedes the progress of humanity, I believe that the failure of quantum supremacy itself leads to important consequences for quantum physics, the theory of computing, and mathematics. Some of these will be explored here. Gil Kalai is Henry and Manya Noskwith Professor of Mathematics at the Hebrew University of Jerusalem and is also affiliated with A Brief SummaryHere is a brief summary of the author's pessimistic point of view as explained in the paper: understanding quantum computers in the presence of noise requires consideration of behavior at different scales. In the small scale, standard models of noise from the mid-90s are suitable, and quantum evolutions and states described by them manifest a very low-level computational power. This small-scale behavior has far-reaching consequences for the behavior of noisy quantum systems at larger scales. On the one hand, it does not allow reaching the starting points for quantum fault tolerance and quantum supremacy, making them both impossible at all scales. On the other hand, it leads to novel implicit ways for modeling noise at larger scales and to various predictions on the behavior of noisy quantum systems. 508

The Argument Against Quantum Computers

“…Correlation is part of this picture since, as is well known, the value of the threshold becomes smaller when there is a larger positive correlation for 2-qubit gated errors. We note (Kalai 2016) that Prediction (f) implies that the quality of logical qubits via quantum error-correction is limited by the quality of gates. See also Chubb and Flammia (2018) for an interesting analysis of the effect of correlated errors on surface codes.…”

“…It is an interesting challenge to give a satisfactory mathematical description of what "time dependent" means. We proposed in Kalai (2016) to base a lower bound on the rate of noise on a measure of the non-commutativity of unitary operations. This measure of non-commutativity can also serve as an "internal" clock of the noisy quantum evolution.…”

“…This measure of non-commutativity can also serve as an "internal" clock of the noisy quantum evolution. We also proposed in Kalai (2016) a restricted class of noisy evolutions called "smoothed Lindblad evolutions" defined via a certain smoothing operation that mathematically expresses the idea that in the absence of quantum error-correction, noise accumulates. An important consequence of Principle 2 is that there are inherent lower bounds on the quality of qubits, or, more concretely, a universal upper bound on the number of non-commuting Pauli operators you can apply to a qubit before it get destroyed.…”

“…Implicitly, these notions already play a role inPreskill (1998). The importance of multi-scale analysis of the quantum computer puzzle is emphasized inKalai (2016).…”

Quantum Footprints of Symplectic Rigidity

“…We refer to a survey by Bush, Lahti and Werner [5] for a discussion of noise in quantum measurements, and to Kalai's article [23] for an intriguing link between quantum noise production and non-commutativity in the context of quantum computing.…”