The reason for the higher efficiency exhibited by some quantum algorithms over their classical counterparts is examined by considering the interplay between the reversible actions required to prepare the computer registers in an entangled state before measurement (the "initial actions"), and the final measurement action -- whereas measurement is interpreted in a new way, particularly suited to a problem solving context. This unification shows that the computation process, comprising the measurement outcome, is significantly influenced by both the initial actions and the need to satisfy the constraints set by the final measurement action. Reviewing the existing quantum algorithms in the light of this dual influence, yields new valuable insight in the nature of the quantum computation speed up.
The reason for the higher efficiency exhibited by some quantum algorithms over their classical counterparts is examined by considering the interplay between the reversible actions required to prepare the computer registers in an entangled state before measurement (the "initial actions"), and the final measurement action -- whereas measurement is interpreted in a new way, particularly suited to a problem solving context. This unification shows that the computation process, comprising the measurement outcome, is significantly influenced by both the initial actions and the need to satisfy the constraints set by the final measurement action. Reviewing the existing quantum algorithms in the light of this dual influence, yields new valuable insight in the nature of the quantum computation speed up.
The higher than classical efficiency exhibited by some quantum algorithms is here ascribed to their non-mechanistic character, which becomes evident by joining the notions of entanglement and quantum measurement. Measurement analogically sets a (partial) constraint on the output of the computation of a hard-to-reverse function. This constraint goes back in time along the reversible computation process, computing the reverse function, which yields quantum efficiency. The evolution, comprising wave function collapse (here a revamped notion), is non-mechanistic as it is driven by both an initial condition and a final constraint. It seems that the more the output is constrained by measurement, the higher can be the efficiency. Setting a complete constraint, by means of a special Zeno effect, yields (speculatively) NP-complete=P.
I provide an alternative way of seeing quantum computation. First, I describe an idealized classical problem solving machine that, thanks to a many body interaction, reversibly and nondeterministically produces the solution of the problem under the simultaneous influence of all the problem constraints. This requires a perfectly accurate, rigid, and reversible relation between the coordinates of the machine parts - the machine can be considered the many body generalization of another perfect machine, the bounching ball model of reversible computation. The mathematical description of the machine, as it is, is applicable to quantum problem solving, an extension of the quantum algorithms that comprises the physical representation of the problem-solution interdependence. The perfect relation between the coordinates of the machine parts is transferred to the populations of the reduced density operators of the parts of the computer register. The solution of the problem is reversibly and nondeterministically produced under the simultaneous influence of the state before measurement and the quantum principle. At the light of the present notion of simultaneous computation, the quantum speed up turns out to be "precognition" of the solution, namely the reduction of the initial ignorance of the solution due to backdating, to before running the algorithm, a time-symmetric part of the state vector reduction on the solution; as such, it is bounded by state vector reduction through an entropic inequality. PACS numbers: 03.67.Lx, 01.55.+b, 01.70.+w
In former work, quantum computation has been shown to be a problem solving process essentially affected by both the reversible dynamics leading to the state before measurement, and the logical-mathematical constraints introduced by quantum measurement (in particular, the constraint that there is only one measurement outcome). This dual influence, originated by independent initial and final conditions, justifies the quantum computation speed-up and is not representable inside dynamics, namely as a one-way propagation. In this work, we reformulate von Neumann's model of quantum measurement at the light of above findings. We embed it in a broader representation based on the quantum logic gate formalism and capable of describing the interplay between dynamical and non-dynamical constraints. The two steps of the original model, namely (1) dynamically reaching a complete entanglement between pointer and quantum object and (2) enforcing the one-outcome-constraint, are unified and reversed. By representing step (2) right from the start, the same dynamics of step (1) yields a probability distribution of mutually exclusive measurement outcomes. This appears to be a more accurate and complete representation of quantum measurement. PACS: 03.67.-a, 03.67.Lx, 03.65.Bz
We analyse the quantum speed-up: the fact that some quantum algorithms are faster than their classical counterparts. The classical formulation of problem solving must be modified. In general a problem statement defines the solution only implicitly, giving a property that a solution must possess, and does not represent a process that constructs the solution explicitly. A solution algorithm must then specify explicitly a finite step-by-step procedure that constructs the solution through a dynamical development. In the quantum speed-up, however, an implicit definition directly determines the solution through an extra-dynamical irreversible non-algorithmic quantum measurement. While in an algorithmic solution, the initial conditions determine the solution in a one-way manner, in a solution by measurement, the initial conditions and a final condition influence the solution jointly. The quantum computation is as implicit as the problem statement. Implicit determination with joint influence is richer than explicit one-way dynamical determination, as it yields the speed-up. Quantum problem solving with speed-up is essentially extra-dynamical and irreversible. Other forms of implicit quantum problem solving may exploit extra-dynamical projections for speed-up.
The usual representation of quantum algorithms, limited to the problem solving process, is incomplete as it lacks the initial measurement. Completing it involves three steps: (i) representing the problem setting process, (ii) relativizing this representation to Alice, the problem solver, to whom the problem setting must be concealed, (iii) representing the reversibility of the computation process by symmetrizing it for time reversal. The latter step shows that all is as if Alice, before beginning her problem solving action, knew half of the information that specifies the solution of the problem she will read in the future. An optimal quantum algorithm is a sum over classical histories in each of which Alice knows in advance one of the possible halves of the solution and performs the computation steps (oracle queries) necessary to identify the missing half. This explains the quantum computational speedup and allows to predict the speedup offered by the optimal quantum solution of any oracle problem.
We investigate the reason for the quantum speedup (quantum algorithms require fewer computation steps than their classical counterparts). We extend the representation of the quantum algorithm to the process of setting the problem, namely choosing the function computed by the black box. The initial measurement selects a setting at random, Bob (the problem setter) unitarily changes it into the desired one. With reference to the observer dependent quantum states of relational quantum mechanics, this representation is with respect to Bob and any external observer, it cannot be with respect to Alice (the problem solver). It would tell her the function computed by the black box, which to her should be hidden. To Alice, the projection of the quantum state due to the initial measurement is retarded at the end of her problem solving action, so that the algorithm input state remains one of complete ignorance of the setting. By black box computations, she unitarily sends it into the output state that, for each possible setting, encodes the corresponding solution, acquired by the final measurement. Mathematically, we can ascribe to the final measurement the selection of any fraction R of the random outcome of the initial measurement. This projects the input state to Alice on one of lower entropy where she knows the corresponding fraction of the problem setting. Given the appropriate value of R, the quantum algorithm is a sum over classical histories in each of which Alice, knowing in advance one of the R-th parts of the setting, performs the black box computations still required to identify the solution. Given a quantum algorithm, this retrocausality model provides the value of R that explains its speed up; in the major quantum algorithms, R is 1/2 or slightly above it. Conversely, given the problem, R=1/2 always yields the order of magnitude of the number of black box computations required to solve it in an optimal quantum way. Quanta 2016; 5: 34–52.