We present characterisations of "exact" gap-definable classes, in terms of indeterministic models of computation which slightly modify the standard model of quantum computation. This follows on work of Aaronson [arXiv:quant-ph/0412187], who shows that the counting class PP can be characterised in terms of bounded-error "quantum" algorithms which use invertible (and possibly non-unitary) transformations, or postselections on events of non-zero probability. Our work considers similar modifications of the quantum computational model, but in the setting of exact algorithms, and algorithms with zero error and constant success probability. We show that the gap-definable counting classes [J. Comput. Syst. Sci. 48 (1994), p.116] which bound exact and zero-error quantum algorithms can be characterised in terms of "quantum-like" algorithms involving nonunitary gates, and that postselection and nonunitarity have equivalent power for exact quantum computation only if these classes collapse.
We present characterisations of "exact" gap-definable classes, in terms of indeterministic models of computation which slightly modify the standard model of quantum computation. This follows on work of Aaronson [arXiv:quant-ph/0412187], who shows that the counting class PP can be characterised in terms of bounded-error "quantum" algorithms which use invertible (and possibly non-unitary) transformations, or postselections on events of non-zero probability. Our work considers similar modifications of the quantum computational model, but in the setting of exact algorithms, and algorithms with zero error and constant success probability. We show that the gap-definable counting classes [J. Comput. Syst. Sci. 48 (1994), p.116] which bound exact and zero-error quantum algorithms can be characterised in terms of "quantum-like" algorithms involving nonunitary gates, and that postselection and nonunitarity have equivalent power for exact quantum computation only if these classes collapse.
In 2004, Aaronson introduced the complexity class PostBQP (BQP with postselection) and showed that it is equal to PP. Following their line of work, we introduce two new complexity classes. The first, CorrBQP, is a modification of BQP which has the power to perform correlated measurements, i.e. measurements that output the same value across a partition of registers. The second, MajBQP, augments BQP with the ability to collapse a register to its most likely measurement outcome. Specifically, we consider two variants, MajBQP and AdMajBQP, where the latter may perform intermediate measurements. We exactly characterize the computational power of the models, CorrBQP=AdMajBQP=BPPPP and MajBQP=PPP. In fact, we show that other metaphysical modifications of BQP, such as CBQP (i.e. BQP with the ability to clone arbitrary quantum states), are also equal to BPPPP. We show that CorrBQP and MajBQP are self-low with respect to classically-accessible queries. In contrast, if they were self-low under quantumly-accessible queries, the counting hierarchy would collapse. Furthermore, we introduce a variant of rational degree that lower-bounds the query complexity of BPPPP. Lastly, we extend the adversary lower-bounding technique to AdPDQP, BQP with the ability to sample the current state of an algorithm with collapsing it and adapt the computation based on the samples.
From the general difficulty of simulating quantum systems using classical systems, and in particular the existence of an efficient quantum algorithm for factoring, it is likely that quantum computation is intrinsically more powerful than classical computation. At present, the best upper bound known for the power of quantum computation is that , where is a classical complexity class (known to be included in , hence ). This work investigates limits on computational power that are imposed by simple physical, or information theoretic, principles. To this end, we define a circuit-based model of computation in a class of operationally-defined theories more general than quantum theory, and ask: what is the minimal set of physical assumptions under which the above inclusions still hold? We show that given only an assumption of tomographic locality (roughly, that multipartite states and transformations can be characterized by local measurements), efficient computations are contained in . This inclusion still holds even without assuming a basic notion of causality (where the notion is, roughly, that probabilities for outcomes cannot depend on future measurement choices). Following Aaronson, we extend the computational model by allowing post-selection on measurement outcomes. Aaronson showed that the corresponding quantum complexity class, , is equal to . Given only the assumption of tomographic locality, the inclusion in still holds for post-selected computation in general theories. Hence in a world with post-selection, quantum theory is optimal for computation in the space of all operational theories. We then consider whether one can obtain relativized complexity results for general theories. It is not obvious how to define a sensible notion of a computational oracle in the general framework that reduces to the standard notion in the quantum case. Nevertheless, it is possible to define computation relative to a 'classical oracle'. Then, we show there exists a classical oracle relative to which efficient computation in any theory satisfying the causality assumption does not include .
Non-Hermitian (NH) quantum systems demonstrate striking differences from their Hermitian counterparts, leading to claims of NH advantage in areas ranging from metrology to entanglement generation. We show that in the context of quantum computation, any such NH advantage is unlikely to be scalable as an efficient computational resource: If coherent nonunitary evolution with renormalization could be realized with only polynomial overhead, then the resulting model could implement postselection, implying implausibly strong complexity-theoretic power under standard assumptions. We define NHBQP ( U ) as the computational power of polynomial-size quantum circuits that, in addition to a standard universal unitary gate set, may apply a fixed gate U on O ( 1 ) qubits that is not proportional to a unitary, with the state renormalized after each use of U . We prove that this model is powerful enough to decide every language in PostBQP (equivalently PP ). Moreover, in the standard uniform circuit-family model, this characterization is tight: For any fixed such nonunitary gate U , NHBQP ( U ) = PostBQP = PP . PostBQP is believed intractable, so this suggests that any scalable NH computational advantage must come with a compensating cost limiting its efficiency. Additionally, we study simulation complexity of restricted classes of nonunitary systems by purifying them to postselected unitary evolution in a form preserving system locality. Using this framework, we show that unitary gates with postselection can simulate not only evolution under NH Hamiltonians but arbitrary quantum trajectories. Any NH model whose purification lies in a strongly simulable unitary family (e.g., Clifford, matchgate, or low-bond-dimension tensor-network circuits) remains efficiently classically simulable, provided the relevant postselected events occur with probability Ω ( 2 − poly ( n ) ) . Thus, adding non-Hermiticity to a universal unitary system makes it infeasibly computationally powerful, while adding it to a strongly simulable system adds no computational power in this setting.
We consider restricted versions of postBQP where postselection probabilities can be efficiently calculated by classical or quantum computers. We show that such restricted postBQP classes are in AWPP(⊆PP=postBQP). This result suggests that postselecting an event with an exponentially small probability does not necessarily boost BQP to PP. The best upperbound of BQP is AWPP, and therefore restricted postBQP classes are other examples of complexity classes slightly above BQP. In [C. M. Lee and J. Barrett, arXiv:1412.8671], it was shown that the computational capacity of a general probabilistic theory which satisfies the tomographic locality is in AWPP. Our result therefore implies that quantum physics with restricted postselections is an example of a super quantum theory which seems to be outside of their general probabilistic theory but its computational capacity is also in AWPP. Although it is physically natural to expect that such restricted postBQP classes are not equivalent to BQP, we show that UP∩coUP, which is unlikely to be in BQP, is contained in the restricted postBQP classes. Finally, we also consider another restricted class of postBQP, where the postselected probability depends only on the size of inputs. We show that this restricted version is contained in APP, where AWPP⊆APP⊆PP=postBQP.
From the general difficulty of simulating quantum systems using classical systems, and in particular \nthe existence of an efficient quantum algorithm for factoring, it is likely that quantum computation is \nintrinsically more powerful than classical computation. At present, the best upper bound known for \nthe power of quantum computation is that BQP AWPP ⊆ , where AWPP is a classical complexity \nclass (known to be included in PP, hence PSPACE). This work investigates limits on computational \npower that are imposed by simple physical, or information theoretic, principles. To this end, we define \na circuit-based model of computation in a class of operationally-defined theories more general than \nquantum theory, and ask: what is the minimal set of physical assumptions under which the above \ninclusions still hold? We show that given only an assumption of tomographic locality (roughly, that \nmultipartite states and transformations can be characterized by local measurements), efficient \ncomputations are contained in AWPP. This inclusion still holds even without assuming a basic notion \nof causality (where the notion is, roughly, that probabilities for outcomes cannot depend on future \nmeasurement choices). Following Aaronson, we extend the computational model by allowing postselection \non measurement outcomes. Aaronson showed that the corresponding quantum complexity \nclass, PostBQP, is equal to PP. Given only the assumption of tomographic locality, the inclusion in \nPP still holds for post-selected computation in general theories. Hence in a world with post-selection, \nquantum theory is optimal for computation in the space of all operational theories. We then consider \nwhether one can obtain relativized complexity results for general theories. It is not obvious how to \ndefine a sensible notion of a computational oracle in the general framework that reduces to the \nstandard notion in the quantum case. Nevertheless, it is possible to define computation relative to a \n‘classical oracle’. Then, we show there exists a classical oracle relative to which efficient computation in \nany theory satisfying the causality assumption does not include NP.