Computing with a single qubit faster than the computation quantum speed limit

Abstract

The possibility to save and process information in fundamentally indistinguishable states is the quantum mechanical resource that is not encountered in classical computing. I demonstrate that, if energy constraints are imposed, this resource can be used to accelerate information-processing without relying on entanglement or any other type of quantum correlations. In fact, there are computational problems that can be solved much faster, in comparison to currently used classical schemes, by saving intermediate information in nonorthogonal states of just a single qubit. There are also error correction strategies that protect such computations.Introduction. The quantum phase space of a qubit is a sphere (Fig. 1). One can discretize this space into any number of states and then apply field pulses to switch between the chosen states in an arbitrary order. In this sense, a qubit comprises the whole universe of choices for computation. For example, a qubit can work as finite automata [1] when different unitary gates act on this qubit depending on arriving digital words. However, different states of a qubit are generally not distinguishable by measurements. So, if the final quantum state encodes the result of computation, we cannot generally extract this information because we cannot distinguish this state by a measurement from other non-orthogonal possibilities reliably.For such reasons, qubits are believed to provide computational advantage over classical memory only when they are used to create purely quantum correlations, i.e., entanglement or quantum discord [2]. While very powerful algorithms have been designed based on such correlations, the degree of control over the state of many qubits that is needed to implement commercially competitive quantum computing is far from the level of the modern technology.In this note, I will argue that the ability to use nonorthogonal states for computation should be considered as the completely independent resource that is provided by quantum mechanics. With a specific example, I will show that there are computational problems for which the access to just one high quality qubit may provide speed of computation that, fundamentally, cannot be reached by a classical computer under the specified restrictions on raw resources such as memory coupling strength to control fields.The idea of this article is based on the well known observation that time-energy uncertainty relation in quantum mechanics imposes limits on computation speed at fixed power supply for classical schemes of computer operation [3,4]. Such claims are generally justified by the fact that digital computers save information in the form of clearly distinguishable states, such as 0 and 1 that encode one bit of information. Quantum mechanically, distinguishable states must be represented by orthogonal vectors that produce definitely different measurement outcomes. However, the switching time between two orthogonal quantum states is restricted from below by a fundamental computation speed limit T = h/(4∆E), where ∆E is cha...

Quantum computational complexity from quantum information to black holes and back

“…For late times, the part behind the past horizon becomes vanishingly small, so the only contribution comes from the part within the future horizon. 41 The patch extends to the singularity. However, the relevant contribution to the action is finite due to the fact that the sphere shrinks there and there is no need to regularize the singularity.…”

“…6. First, the UV divergent part obeys a volume law, and depends on the cutoff 41 There may be subtleties in this statement, as discussed in [15,103].…”

Complexity is simple!

“…In other words, failure to saturate Lloyd's bound would be a "diagnostic tool" for IR theories which do not have a sensible UV completion and thus lie in the Swampland. Instead, we found what is a well-known result in the quantum information literature [45][46][47] -namely that a generic quantum system will violate Lloyd's bound. Systems that obey the bound can be replaced by a classical computer working at the same energy.…”

“…According to the discussion in subsection 2.4, this means that there is no obstruction to violating the bound (2.16). We are thus in a regime where arbitrarily fast computation is allowed, as described by [45][46][47]. In contrast, we do find evidence that for small black holes the bound can be violated while satisfying the orthogonalizing assumption of Lloyd.…”

“…As a result, one does not expect a bound like (2.16) to hold in general, and in fact one can easily find violations when simple gates are used. This was recently illustrated in the works [45,46], where arbitrarily large complexification rates are achieved in particular examples -in the latter case, in a system with one qubit only. To illustrate this point, assume we have a reference state |0 and any other state |Ψ and suppose we have unitary evolution described by…”