arXiv:1103.2500v1 [quant-ph] 13 Mar 2011 How much of quantum mechanics is really needed to defy Extended Church-Turing Thesis? Leonid Gurvits 1 , Vwani Roychowdhury 2 , Sudhir Kumar Singh 2 , and Farrokh Vatan 3 1. Los Alamos National Laboratory, Los Alamos, NM 87545 2. Electrical Engineering Department, UCLA, Los Angeles, CA 90095 3. Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109 (Dated: February 14, 2019) The Church-Turing Thesis[1, 2] states that intuitive no- tion of algorithms is equal to the Turing machine algo- rithms i.e. Turing machines provide us with the most gen- eral concept for computability. The modern strenthening of this thesis called Extended Church-Turing Thesis(ECT) takes efficiency in to consideration and assert that any rea- sonable (physically realizable) model of computation can be simulated by a BPP algorithm (i.e. an efficient classi- cal probabilistic algorithm). Several fundamental ques- tions arise when we take ECT seriously as a statement not just about mathematics but about the physical real- ity [3, 4]. In particular, what can we say about the valid- ity of ECT when we take quantum mechanics aboard in our notion of efficient computability? We have a BQP al- gorithm (i.e. an efficient quantum algorithm) for Abelian Hidden Subgroup Problem (HSP) but we do not know of any BPP algorithm for the same (Factoring and Discrete Log problems are special cases of Abelian HSP)[5–9]. We can list three possibilities all of which are fundamentally surprising-(1) ECT is false (2) HSP is in BPP (3) Quantum mechanics is false. An interesting point to note is that at least one of these possibilities is true. So if we take quan- tum mechanics to be true it is the HSP which is respon- sible for refuting ECT. An immediate question arises that whether there are other natural candidate problems po- tentially tenable to refute ECT. Recent results of Freed- man et al.[10, 11] and Aharonov et al.[12] shows that ad- ditively approximating Jones polynomials is exactly what BQP is capable of and therefore provides such a potential example. But do we really need the full power of quantum computers (BQP) for ECT to be false? Is there a natu- ral subclass of BQP with such a candidate problem, and further narowing it down, is there a natural subclass of BQP which is not even universal for classical computation yet has such a candidate problem? We show that the an- swer to this is affirmative and linear optics based proposal for building quantum computers naturally provides such a subclass of BQP. We call this subclass LOBQP, which characterizes the notion of efficient computation with lin- ear optics circuits consisting entirely of beam-splitters and phase-shifters and without any non-linear media (such as Kerr-media) or feedback from photo-detectors[13]. We prove that a problem called Multilinear Term Weight Prob- lem (LTW) is complete for LOBQP and it also contains the additive approximation of permanent of complex ma- trices as a special case. We obtain several results on the complexity of LTW and the class LOBQP seems to be a slice in the complexity hierarchy as depicted in Figure 1. As a by-product of our quantum formalism we also obtain PP NP coNP BQP BPP P PSPACE LOBQP LTW FIG. 1: LOBQP: A Slice Through the Complexity Hierarchy a non-trivial upper bound on the permanent of complex matrices. We also argue how the additive approximation of #P − hard problems lies at the heart of quantumness. Besides, the above mentioned foundational motivations there are other good reasons to study a subclass of BQP such as LOBQP. We have Gottesman-Knill theorem which states that a class of quantum computations known as stabilizer cir- cuits, which involves just CNOT , Hadamard, phase and pauli gates and pauli measurements, can be simulated efficiently by classical computers and infact unlikely to be universal for classical computation [14] but yet we do not know of any nat- ural subclass of BQP not simulable by classical computers. Secondly, we know that photons do not naturally interact with each other, and in order to implement two-qubit gates such as CNOT such interactions are essential and therefore, to facil- itate universal quantum computation, non-linearities such as cross-Kerr media or feedback from photo-detectors becomes inevitable, however acheiving interactions of photons through such techniques is not easy [15]. Therefore, it is natural to ask that how powerful this model of computation remains if we do not use any kind of non-linearity or feedback from photo- detectors. Thirdly, in the optical based quantum computation, a qubit is usually represented by two photon modes as a dual- rail where |0〉 and |1〉 are respectively represented by an ab- sence or presence of the single photon in the first mode. We need not necessarily think in terms of a photon as a qubit in this usual dual-rail representation and should allow starting with more than single photons in a mode which in general can provide an exponentially larger hilbert space. Moreover, in principle we can always create polynomially many photons in a mode by using polynomially many ancilla modes start-