Counterfactual Computation

Suppose that we are given a quantum computer programmed ready to perform a computation if it is switched on. Counterfactual computation is a process by which the result of the computation may be learnt without actually running the computer. Such processes are possible within quantum physics and to achieve this effect, a computer embodying the possibility of running the computation must be available, even though the computation is, in fact, not run. We study the possibilities and limitations of general protocols for the counterfactual computation of decision problems (where the result r is either 0 or 1). If p(r) denotes the probability of learning the result r ``for free'' in a protocol then one might hope to design a protocol which simultaneously has large p(0) and p(1). However we prove that p(0)+p(1) never exceeds 1 in any protocol and we derive further constraints on p(0) and p(1) in terms of N, the number of times that the computer is not run. In particular we show that any protocol with p(0)+p(1)=1-epsilon must have N tending to infinity as epsilon tends to 0. These general results are illustrated with some explicit protocols for counterfactual computation. We show that "interaction-free" measurements can be regarded as counterfactual computations, and our results then imply that N must be large if the probability of interaction is to be close to zero. Finally, we consider some ways in which our formulation of counterfactual computation can be generalised.Comment: 19 pages. LaTex, 2 figures. Revised version has some new sections and expanded explanation

Does the solar system compute the laws of motion?

The counterfactual account of physical computation is simple and, for the most part, very attractive. However, it is usually thought to trivialize the notion of physical computation insofar as it implies ‘limited pancomputationalism’, this being the doctrine that every deterministic physical system computes some function. Should we bite the bullet and accept limited pancomputationalism, or reject the counterfactual account as untenable? Jack Copeland would have us do neither of the above. He attempts to thread a path between the two horns of the dilemma by buttressing the counterfactual account with extra conditions intended to block certain classes of deterministic physical systems from qualifying as physical computers. His theory is called the ‘algorithm execution account’. Here we show that the algorithm execution account entails limited pancomputationalism, despite Copeland’s argument to the contrary. We suggest, partly on this basis, that the counterfactual account should be accepted as it stands, pancomputationalist warts and all

Loss tolerance in one-way quantum computation via counterfactual error correction

We introduce a scheme for fault tolerantly dealing with losses (or other "leakage" errors) in cluster state computation that can tolerate up to 50% qubit loss. This is achieved passively using an adaptive strategy of measurement - no coherent measurements or coherent correction is required. Since the scheme relies on inferring information about what would have been the outcome of a measurement had one been able to carry it out, we call this "counterfactual" error correction.Comment: Published version - much revised and with a new title. Here we now focus solely on the general aspects of the protocol - a much expanded and improved discussion of its application in linear optical quantum computation can now be found in quant-ph/070204

Understanding counterfactuality:A review of experimental evidence for the dual meaning of counterfactuals

Cognitive and linguistic theories of counterfactual language comprehension assume that counterfactuals convey a dual meaning. Subjunctive-counterfactual conditionals (e.g., ‘If Tom had studied hard, he would have passed the test’) express a supposition while implying the factual state of affairs (Tom has not studied hard and failed). The question of how counterfactual dual meaning plays out during language processing is currently gaining interest in psycholinguistics. Whereas numerous studies using offline measures of language processing consistently support counterfactual dual meaning, evidence coming from online studies is less conclusive. Here, we review the available studies that examine online counterfactual language comprehension through behavioural measurement (self-paced reading times, eye-tracking) and neuroimaging (electroencephalography, functional magnetic resonance imaging). While we argue that these studies do not offer direct evidence for the online computation of counterfactual dual meaning, they provide valuable information about the way counterfactual meaning unfolds in time and influences successive information processing. Further advances in research on counterfactual comprehension require more specific predictions about how counterfactual dual meaning impacts incremental sentence processing