Commuting quantum circuits and complexity of Ising partition functions

Abstract

Instantaneous quantum polynomial-time (IQP) computation is a class of quantum computation consisting only of commuting two-qubit gates and is not universal. Nevertheless, it has been shown that if there is a classical algorithm that can simulate IQP efficiently, the polynomial hierarchy collapses to the third level, which is highly implausible. However, the origin of the classical intractability is still less understood. Here we establish a relationship between IQP and computational complexity of calculating the imaginary-valued partition functions of Ising models. We apply the established relationship in two opposite directions. One direction is to find subclasses of IQP that are classically efficiently simulatable by using exact solvability of certain types of Ising models. Another direction is applying quantum computational complexity of IQP to investigate (im)possibility of efficient classical approximations of Ising partition functions with imaginary coupling constants. Specifically, we show that a multiplicative approximation of Ising partition functions is #P-hard for almost all imaginary coupling constants even on planar lattices of a bounded degree.Aaronson and Arkhipov introduced BOSONSAMPLING [15], a sampling problem according to the probability distribution of n bosons scattered by linear optical unitary operations. The probability distribution is given by the permanent of a complex matrix, which is determined by the linear optical unitary operations. Calculation of the permanent of complex matrices is known to be #P-hard [16,17]. Since a polynomial-time machine with an oracle for #P can solve all problems in the PH according to Toda's theorem [18], an exact classical simulation (in the strong sense [19,20] meaning a calculation of the probability distribution of the output) of BOSONSAMPLING is highly intractable in a classical computer. They showed under assumptions of plausible conjectures that if there exists an efficient classical approximation of BOSONSAMPLING (classical simulation in the weak sense [19, 20] meaning a sampling according the probability distribution of the output), the PH collapses to the third level, which is unlikely to occur. (The detailed notions of classical simulation are provided in section 3.) This result brings a novel perspective on linear optical quantum computation and drives many researchers into the recent proof-of-principle experiments [21][22][23][24][25][26][27][28].Another subclass of quantum computation of this kind is instantaneous quantum polynomial-time computation (IQP) proposed by Shepherd and Bremner [29]. IQP consists only of commuting unitary gates, such as q  Î [ ] Z exp i k S k . Here q p Î [ ) 0, 2 is a rotational angle, Z k indicates the Pauli operator on the kth qubit, and S indicates a set of qubits on which the commuting gate acts. (A detailed definition will be provided in the next section.) The input is given by +ñ Ä | n with +ñ º ñ + ñ | (| | ) 0 1 2, and the output qubits are measured in the X-basis. Since all unitary operations are commutabl...

Average-Case Complexity Versus Approximate Simulation of Commuting Quantum Computations

“…It is known that the family of partition functions Z(ω) parametrised as above is #P-hard to compute in the worst case up to the above multiplicative error bound [10,11]. Conjecture 2 thus states that this worst-case hardness result can be improved to an average-case hardness result.…”

“…where the exponentiated sum is over the complete graph on n vertices, w ij and v k are real edge and vertex weights, and ω ∈ C. Then, for any ω = e iθ , Z(ω) arises straightforwardly as an amplitude of some IQP circuit C I (ω): 0| ⊗n C I (ω)|0 ⊗n = Z(ω)/2 n (see Appendix A and [7][8][9][10][11]). For our purposes it is sufficient to restrict to the case where ω = e iπ/8 and the weights are picked by choosing uniformly at random from the set {0, .…”

Quantum sampling problems, BosonSampling and quantum supremacy

“…12,47,48 The hardness of IQP-sampling to within additive errors follows from the observation that Stockmeyer's algorithm combined with sufficiently accurate classical additive simulation returns a very precise estimate to the probability p 0 = |〈0| ⊗n C y |0〉 ⊗n | 2 for a wide range of randomly chosen circuits C y . A multiplicative approximation to p 0 can be delivered on a large fraction of choices of y when both: (a) for a random bitstring x, the circuit n i¼1 X xi is a hidden subset of the randomly chosen circuits C y ; and (b) p 0 anticoncentrates on the random choices of circuits C y .…”

Verification of hypergraph states

“…For example, measurement-based quantum computing [1], which is nowadays one of the standard quantum computing models, enables universal quantum computing with only adaptive single-qubit measurements on certain quantum states, such as graph states [1] and other condensed-matter-physically motivated states including the AKLT state [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Furthermore, not only adaptive but also non-adaptive singlequbit measurements on graph states can demonstrate a quantumness which cannot be classically efficiently simulated: it is known that if probability distributions of nonadaptive sequential single-qubit measurements on graph states are classically efficiently sampled, then the polynomial hierarchy collapses to the third level [18][19][20] or the second level [21]. The polynomial hierarchy is a hierarchy of complexity classes generalizing P and NP, and it is not believed to collapse in computer science.…”

“…For example, the hypergraph states naturally induced from the IQP circuits corresponding to the non-adaptive Union Jack state measurement-based quantum computing [39] can be used for that purpose. Since the non-adaptive Union Jack state measurement-based quantum computing is universal with postselections, a multiplicative error calculation of its output probability distribution is #P-hard [20]. If we assume the worst case hardness can be lifted to the average case one, we can show the hardness of the classical constant L1-norm error sampling.…”