We use the class of commuting quantum computations known as IQP (Instantaneous Quantum Polynomial time) to strengthen the conjecture that quantum computers are hard to simulate classically. We show that, if either of two plausible average-case hardness conjectures holds, then IQP computations are hard to simulate classically up to constant additive error. One conjecture relates to the hardness of estimating the complextemperature partition function for random instances of the Ising model; the other concerns approximating the number of zeroes of random low-degree polynomials. We observe that both conjectures can be shown to be valid in the setting of worst-case complexity. We arrive at these conjectures by deriving spin-based generalisations of the Boson Sampling problem that avoid the so-called permanent anticoncentration conjecture. Quantum computers are conjectured to outperform classical computers for a variety of important tasks ranging from integer factorisation [1] to the simulation of quantum mechanics [2] . However, to date there is relatively little rigorous evidence for this conjecture. It is well established that quantum computers can yield an exponential advantage in the query and communication complexity models. But in the more physically meaningful model of time complexity, there are no proven separations known between quantum and classical computation. This can be seen as a consequence of the extreme difficulty of proving bounds on the power of classical computing models, such as the famous P vs. NP problem. Given this difficulty, the most we can reasonably hope for is to show that quantum computations cannot be simulated efficiently classically, assuming some widely believed complexity-theoretic conjecture. For example, any set of quantum circuits that can implement Shor's algorithm [1] provides a canonical example, with the unlikely consequence of efficient classical simulation of this class of quantum circuits being the existence of an efficient classical factoring algorithm. However, one could hope for the existence of other examples that have wider-reaching complexity-theoretic consequences. With this in mind, in both [3] and [4] it was shown that the existence of an efficient classical sampler from a distribution that is close to the output distribution of an arbitrary quantum circuit, to within a small multiplicative error in each output probability, would imply that post-selected classical computation is equivalent to post-selected quantum computation. This consequence is considered very unlikely as it would collapse the infinite tower of complexity classes known as the Polynomial Hierarchy [5] to its third level. In both works this was proven even for non-universal quantum circuit families: commuting quantum circuits in the case of [3] , and linear-optical networks in [4] . These non-universal families are of physical interest because they are simpler to implement, and easier to analyse because of the elegant mathematical structures on which they are based. However, in physically realistic scenar- * michael.bremner@uts.edu.au ios where the quantum computer and its corresponding classical simulator are allowed to be accurate up to a small additive error, these results have little meaning. One important recent step to addressing this was proposed by Aaronson and Arkhipov [4], who gave a sophisticated argument based on counting complexity that approximately sampling from the output probability distribution of a network of noninteracting photons (a problem known as Boson Sampling) should be classically hard, even up to a reasonable additive error bound. This major breakthrough rests on two tantalising but as yet unproven conjectures: the so-called permanent anticoncentration conjecture and the permanentof-Gaussians conjecture. In this paper we propose a generalisation of the Boson Sampling argument of [4] that is native to spin systems, specifically to the class of commuting quantum circuits known as IQP (Instantaneous Quantum Polynomial time), which was introduced in [6] and [3] . In our opinion this leads to a mathematically simpler setting, while still apparently retaining the essential complexity-theoretic ingredients. This simplicity allows us to prove the IQP analogues of the permanent anticoncentration conjecture. The only remaining conjecture to prove is an IQP analogue of the permanent-of-Gaussians conjecture, of which we find two natural examples. Informally (see [3] for the formal definition) an n-qubit IQP circuit C is a quantum circuit which takes as input the state |0 ⊗n , whose gates are diagonal in the Pauli-X basis, and whose n-qubit output is measured in the computational basis. We say that a classical sampler of an IQP circuit is accurate up to error in 1 norm if its output probability distribution has total variation distance at most /2 from that of C. We can now state the main result of this paper: Theorem 1. Assume either Conjecture 2 or 3 below is true. If it is possible to classically sample from the output probability distribution of any IQP circuit C in polynomial time, up to an error of 1/192 in 1 norm, then any problem in #P can be solved in FBPP NP . Hence the Polynomial Hierarchy collapses to its third level. Loosely speaking, the complexity class #P appearing in this theorem is the class of problems reducible to counting the number of solutions of an NP problem [5] 2 class of problems that can be solved by randomised classical polynomial-time computation equipped with an oracle that can solve any problem in NP. Theorem 1 is based around two natural conjectures, one of which is native to computer science, and other common in condensed-matter physics. Each conjecture straightforwardly relates to a family of IQP circuits. Essentially the result states that if we assume either of our conjectures, then there is no way of efficiently simulating these families of quantum circuits without a major reevaluation of the existing status-quo of complexity theory. The first conjecture is based on the complexity of one of the most commonly studied models of statistical physics, the Ising model. Consider the partition function 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: and [7-11]). Our conjecture is: Conjecture 2. Let Z(ω) be the partition function of a random instance of the Ising model, picked by choosing uniformly random weights from the set {0, . . . , 15} on the vertices and edges of the complete graph on n vertices, evaluated at ω. Then it is #P-hard to approximate |Z(e iπ/8 )| 2 up to multiplicative error 1/4 + o(1) for a 1/24 fraction of instances over the choice of vertex and edge weights. 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. Our second conjecture is based on the hardness of computing the gap of degree-3 polynomials over F 2 , f : {0, 1} n → {0, 1}, which are expressible (up to an additive constant) as n . Then we have the following conjecture: Conjecture 3. Let f : {0, 1} n → {0, 1} be a uniformly random degree-3 polynomial over F 2 . Then it is #P-hard to approximate ngap(f ) 2 up to a multiplicative error of 1/4 + o(1) for a 1/24 fraction of polynomials f . It has been known for some time that ngap(f ) is #P-hard to compute exactly in the worst case [12] . We show in Appendix D, using IQP techniques, that this worst-case hardness still holds for approximating ngap(f ) 2 up to multiplicative error less than 1/2. Just as with Conjecture 2, what remains is to lift this worst-case hardness result to average-case hardness. We remark that an additional piece of evidence provided in [4] for the average-case hardness of multiplicative approximations to Boson Sampling (the permanent-of-Gaussians conjecture) was a direct proof that exact simulation of Boson Sampling probability distributions is hard on average. This was based on average-case hardness results for computation of the permanent, for which we do not know IQP analogues. However, currently known techniques do not seem sufficient to extend these exact average-case hardness results for Boson Sampling to approximate hardness results [4, Section 9.2]. As with the case of Boson Sampling, the worst-case hardness of multiplicative approximations to both Z(ω) and ngap(f ) ([11] and Appendix D) implies via standard results on random-self-reducibility [13] that there exists some distribution over the choices of these functions that is #P-hard on average -but not necessarily the distributions that we require for Conjectures 2 and 3. There are a number of technical ingredients of Theorem 1 which will be discussed in full below. The basic idea is that, for the class of problems underlying Conjectures 2 and 3, any classical sampler that is accurate up to a good additive error bound in the worst case, is forced to also be accurate to within a reasonable multiplicative error on average. This is shown by using a classic result of complexity theory, the so-called Stockmeyer counting algorithm [14] . Another essential ingredient is anticoncentration results for ngap(f ) (for Conjecture 3) and the partition function of the random Ising model (for Conjecture 2). That such anticoncentration results can be proven is a consequence of the elegant mathematical structures upon which IQP circuits are based. We first mention that recent independent work of Fefferman and Umans [15] has explored an alternative way to generalise the ideas of Aaronson and Arkhipov [4]. This work uses Quantum Fourier Sampling to construct states whose corresponding probability distributions are hard to sample from classically, under similar conjectures to [4] . An appealing aspect of the construction of [15] is that it shows that there are specific, and rather simple, quantum states which are hard to simulate classically, assuming an anticoncentration conjecture holds. However, constructing these states appears to require the full power of quantum computation, unlike the results described here and in [4] . We now describe our results. We assume basic knowledge of standard complexity classes [5, 16] such as P, #P and BPP. A crucial complexity-theoretic tool we will use is a result of Stockmeyer on approximate counting: There exists an FBPP NP machine which, for any boolean function f : {0, 1} n → {0, 1}, can approximate to within a multiplicative factor of (1 + ) for all = Ω(1/ poly(n)), given oracle access to f . We refer to [3, 6] for formal definitions of the IQP model. We commonly consider IQP circuits which consist of an nfold tensor product of a Hadamard gate H ⊗n , a component 3 whose gates are diagonal in the Pauli-Z basis, and a further application of H ⊗n . For the main results of this paper we will consider IQP circuits whose diagonal-Z component consists of either Z, CZ (Controlled-Z, aka Controlled-Phase), and CCZ (Controlled-Controlled-Z) gates or Z, √ CZ, and gates. Both of these sets of gates would be universal if we could also perform Hadamard (H) gates at any point in the circuit -which we cannot do in IQP because this gate does not commute with the Z gate. However, if we allow the unphysical resource of postselection, these Hadamard gates can effectively be implemented [3] , allowing IQP circuit amplitudes y|C|x to express any quantum circuit amplitude (up to a known constant). See Appendix C for a description of this construction. A consequence is that exact classical computation of such amplitudes is #P-hard, even up to constant multiplicative error [3] or exponentially small additive error [9] . Approximation of general IQP circuits -We first prove a key technical ingredient, which relates approximate sampling from the output distributions of IQP circuits to approximating individual output probabilities. This is essentially the same argument as used in [4] for the permanent, although we believe it becomes substantially simpler in the setting of IQP. The intuition behind this result is that adding random X gates to an IQP circuit randomly permutes the output probabilities. This allows the user of a sampler which is accurate for all circuits to obfuscate from the sampler which one of the output probabilities the user is interested in. Lemma 5. Let C be an arbitrary IQP circuit on n qubits. Let C x , for x ∈ {0, 1} n , be the circuit produced by appending an X gate to C for each i such that x i = 1. Assume there exists a classical polynomial-time algorithm A which, for any IQP circuit C , can sample from a probability distribution which approximates the output probability distribution of C up to additive error in 1 norm. Then, for any δ such that 0 < δ < 1, there is a FBPP NP algorithm which, given access to A, approximates | 0|C x |0 | 2 up to additive error with probability at least 1 − δ (over the choice of x). Proof. For each x, y ∈ {0, 1} n , set where in both cases the probability is taken over the algorithm's internal randomness. For any y of our choice, we can apply Theorem 4 to A to estimate q 0y (where we write 0 for 0 n ). This gives us an FBPP NP algorithm which produces an estimate q y such that | q y − q 0y | ≤ q 0y / poly(n). | q y − p 0y | ≤ | q y − q 0y | + |q 0y − p 0y | ≤ q 0y / poly(n) + |q 0y − p 0y | ≤ (p 0y + |q 0y − p 0y |)/ poly(n) + |q 0y − p 0y | = p 0y / poly(n) + |q 0y − p 0y |(1 + 1/ poly(n)). As A approximates the output probability distribution of C 0 up to 1 error , it follows from Markov's inequality that for any 0 < δ < 1, where y is picked uniformly at random. Thus | q y − p 0y | ≤ p 0y / poly(n) + (1 + 1/ poly(n)) 2 n δ with probability at least 1 − δ, over the choice of y. But This completes the proof. If | 0|C x |0 | 2 = Ω(2 −n ), the algorithm of Lemma 5 gives a good approximation -i.e. a multiplicative approximation up to a 1 + 1/ poly(n) factor. We encapsulate this formally as the following corollary. Corollary 6. Let F be a family of IQP circuits on n qubits. Pick a random circuit C by choosing a circuit from F uniformly at random, then appending X gates on a uniformly random subset of the qubits. Assume that there exist universal constants α, p > 0 such that Further assume there exists a classical polynomial-time algorithm A which, for any IQP circuit C , can sample from a probability distribution which approximates the output probability distribution of C up to additive error = αp/8 in 1 norm. Then there is a FBPP NP algorithm which, given access to A, approximates | 0|C|0 | 2 up to multiplicative error 1/4 + o(1) on at least a p/2 fraction of circuits C. Proof. With probability at least p over the choice of C, | 0|C|0 | 2 ≥ α · 2 −n . Setting = αp/8, δ = p/2 in Lemma 5, there is a FBPP NP algorithm which, given access to A, approximates | 0|C|0 | 2 up to additive error O((1 + o(1))α · 2 −n−2 +| 0|C|0 | 2 / poly(n)) with probability at least 1−p/2 (over the choice of C). Therefore, with probability at least p/2, the algorithm approximates | 0|C|0 | 2 up to multiplicative error 1/4 + o(1). We next show that the first precondition of Corollary 6 is indeed satisfied for two interesting families of IQP circuits. Anticoncentration bounds -Fix a family F of IQP circuits. We would like to show that | 0|C|0 | 2 is likely to be high for a circuit C formed by picking a random circuit D from F, then appending X gates on a uniformly random subset S of the qubits. We will use the following fact: Fact 7 (Paley-Zygmund inequality). If R is a non-negative random variable with finite variance, then for any 0 < α < 1, We will apply Fact 7 to the random variable R = | 0|C|0 | 2 , first observing that where in the second expectation x is picked uniformly at random from {0, 1} n . This deals with the numerator; to handle the denominator, we need to upper-bound E[| 0|C|0 | 4 ]. The first family of circuits we consider corresponds to polynomials over F 2 . We prove in Appendix E that for uniformly random degree-3 polynomials f , E f [ngap (f ) 4 ] ≤ 3 · 2 −2n . Based on this, and the tight connection between IQP circuits over the gate set {Z, CZ, CCZ} and degree-3 polynomials, we have the following result: Theorem 8. Assume there exists a classical polynomial-time algorithm A which, for any IQP circuit C, can sample from a probability distribution which approximates the output probability distribution of C up to additive error 1/192 in 1 norm. Then there is an FBPP NP algorithm which, given access to A, approximates ngap(f ) 2 up to multiplicative error 1/4+o(1) on at least a 1/24 fraction of degree-3 polynomials f : {0, 1} n → {0, 1}. Proof. Combining Fact 7 and the bound on The claim then follows by inserting these parameters into Corollary 6. We next consider the Ising model, where we are interested in evaluating the partition function Z(ω) of a randomly weighted graph at ω = e iπ/8 (see (1)). Each edge of the complete graph has a weight w ij , and each vertex has a weight v k , each picked uniformly at random from the set {0, . . . , 15}. We show in Appendix E that 0|C|0 = Z(ω)/2 n for an IQP circuit C consisting of gates from the set {diag(1, 1, 1, i), diag(1, e iπ/4 )}; and further that we can consider a random circuit of this form as being chosen by picking a random circuit using this gate set, then following it by a random choice of X gates. In addition, E w,v |Z(ω)| 4 ≤ 3 · 2 2n . This implies the following result, whose proof is essentially the same as that of Theorem 8: Theorem 9. Let Z(ω) be the partition function of a random instance of the Ising model, picked by choosing random weights from the set {0, . . . , 15} on the vertices and edges of the complete graph on n vertices, evaluated at ω. Assume there exists a classical polynomial-time algorithm A which, for any IQP circuit C, can sample from a probability distribution which approximates the output probability distribution of C up to additive error 1/192 in 1 norm. Then there is a FBPP NP algorithm which, given access to A, approximates |Z(e iπ/8 )| 2 up to multiplicative error 1/4 + o(1) with probability at least 1/24 (over the choice of weights). Combining Theorems 8 and 9 gives our main result, Theorem 1. Outlook -In our opinion, the most compelling question is how best to physically implement the classes of IQP circuits discussed in this paper, as they are arguably among the simplest classes of quantum computations that are unlikely to be classically simulable. For both the Ising model and degree-3 polynomial case the most obvious experimental implementations require non-local gates, which is quite challenging. In terms of the Ising model case 2-qubit interactions might need to be generated between any two qubits in a system. Such interactions do not arise naturally in lattice geometries and as such will need to be engineered. From an experimental perspective there has already been a large amount of progress in these directions. The dynamics of the Ising model with local interactions have been digitally simulated in ion traps [17, 18] and very recently non-local interactions have been utilised in the digital simulation of fermionic systems with superconductors [19] . As technologies such as cavity buses for superconducting systems [20] become more reliable, we expect that an increasing number of systems will be able to implement IQP circuits in a regime that is likely not to be classically simulable. Theoretically there are a number of natural questions that remain to be answered, the most obvious of which is whether or not Conjectures 2 and 3 are true. To the best of our knowledge there are no techniques that are capable of lifting the known worst-case hardness results for the Ising model and gaps of degree-3 polynomials to average-case results. However, recent breakthroughs [21] [22] [23] in categorising the complexity of statistical mechanical systems via the underlying interaction graph properties give some hope that these conjectures can be resolved. As well as the connections used here between IQP, the Ising model and low-degree polynomials, it is known that IQP circuits are closely related to Tutte polynomials and weight enumerator polynomials of binary linear codes [7] . It is a compelling question as to whether the ideas developed here can be extended to these models. It would also be very interesting to explore the connection between nonadaptive measurementbased quantum computing and IQP [24, 25] in order to find subclasses of IQP that are physically straightforward to implement, and for which one can prove conditional hardness results similar to those shown here. ACKNOWLEDGMENTS
While these results do not hold for the hardness conjectures made in REF , they are a clear incentive to determine to what extent the IQP sampling argument can be applied to Ising models on sparse graphs.
Average-case complexity versus approximate simulation of commuting quantum computations
Phys. Rev. Lett. 117, 080501 (2016)
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