Anticoncentration Theorems for Schemes Showing a Quantum Speedup

Anticoncentration theorems for schemes showing a quantum speedup

D. Hangleiter, J. Bermejo-Vega, M. Schwarz, and J. Eisert

Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany May 16, 2018

One of the main milestones in quantum in- a lack of imagination, such a quantum speedup is usu- formation science is to realise quantum de- ally meant to refer to schemes for which the speedup vices that exhibit an exponential computa- can be related to a notion of computational complex- tional advantage over classical ones without ity. For a quantum speedup scheme to be physically being universal quantum computers, a state realisable in principle in the absence of quantum error of affairs dubbed quantum speedup, or some- correction, it is crucial that the hardness of the task is times “quantum computational supremacy”. robust under physically realistic errors. To have any The known schemes heavily rely on mathe- hope of realising such a scheme in the near term one matical assumptions that are plausible but un- would moreover wish for the resources required for an proven, prominently results on anticoncentra- implementation of the architecture in the intractable tion of random prescriptions. In this work, we regime to be achievable with present-day (or near- aim at closing the gap by proving two anticon- term) technology. centration theorems and accompanying hard- There are only very few quantum speedup archi- ness results, one for circuit-based schemes, tectures that are robust against physically realistic the other for quantum quench-type schemes constant total-variation distance errors [5–12]. Even for quantum simulations. Compared to the fewer of those are physically realistic when it comes to few other known such results, these results an implementation in present-day technology in that give rise to a number of comparably simple, they require only nearest-neighbour interactions and physically meaningful and resource-economical are feasible in the available experimental platforms schemes showing a quantum speedup in one such as linear optics [5], superconducting qubits [7,8], and two spatial dimensions. At the heart of ion traps or cold atoms in optical lattices [10]. The the analysis are tools of unitary designs and computational task that is solved in all of these pro- random circuits that allow us to conclude that posals is a sampling task, in particular, the task of universal random circuits anticoncentrate as sampling from the output distribution of a certain well as an embedding of known circuit-based random time-evolution. That random time evolution schemes in a 2D translation-invariant architec- may take the form of a Haar-random unitary applied ture. to a bosonic state [5], a random circuit from a gate set [8], IQP circuits [7] applied to an all-zero state, or even a translation-invariant nearest-neighbour Ising Hamil- 1 Introduction tonian that is applied to a random product state [10]. In addition to this discussion, there is the question to Realising a quantum device that computationally out- what extent schemes showing a quantum speedup can performs state-of-the art classical supercomputers for be certified in their correctness [9, 10, 12–16]. a certain task that is provably intractable classically has become a key milestone in the field of quantum The central ingredient of all existing quantum simulation and computing. This goal is often referred speedup proofs is Stockmeyer’s algorithm [17] that arXiv:1706.03786v4 [quant-ph] 15 May 2018 to as “quantum (computational) supremacy” [1] or implies a collapse of the Polynomial Hierarchy if sam- quantum speedup. Such a quantum speedup is not pling from the output distribution of the respective merely meant in the sense of quantum dynamics being circuits is #P-hard on average. In order for this hard- no longer tractable on classical supercomputers using ness argument to be valid, one crucially requires so- the best known algorithms to date, for which there is called anticoncentration bounds for the output prob- evidence already today [2–4]. Instead, to make sure ability distribution of the respective random circuits that the inefficient classical simulation is not victim of [18]. Indeed, it has been shown that one can efficiently D. Hangleiter: [email protected] classically sample from output distributions of cer- J. Bermejo-Vega: [email protected] tain circuit families, including IQP circuits, the out- M. Schwarz: [email protected] put distribution of which concentrates on a polynomi- J. Eisert: [email protected] ally small subset of the sample space [19]. This shows

Accepted in Quantum 2018-05-09, click title to verify 1 that concentrated output distributions are in many hence, linearly in a one-dimensional architecture [8]. relevant cases simulable, rendering anticoncentration Still, to the best of our knowledge there is no rigorous a necessary condition for classical hardness for these proof for anticoncentration of universal random cir- cases. cuits. In contrast, in the light of this intuition the sec- Despite of their central role in the hardness ar- ond result is quite surprising: It has even been argued gument of quantum speedup proposals, only few [18] that it cannot be expected to reach anticoncen- proofs of anticoncentration bounds are known so far trating output distributions in constant depth, retain- [6,7, 11]. In all other speedup architectures – bo- ing its classical intractability. We conjecture the scal- son sampling [5], universal random circuits [8], and ing of both results to be optimal in the settings con- translation-invariant Ising models [9, 10] – there exists sidered (unstructured circuits in one dimension and none or merely numerical evidence for anticoncentra- highly structured circuits in two dimensions). tion of the respective circuit families and the validity To prove the first result (Theorem5) we make of the anticoncentration assumption needs to be con- use of properties of approximate unitary 2-designs jectured. This still gives rise to plausible schemes, but and apply the Paley-Zygmund inequality. In apply- in order to complete the program of realising quan- ing this result to show that several schemes anticon- tum schemes showing a quantum speedup, these gaps centrate we use the fact that all these schemes form must necessarily be closed. Rigorous anticoncentra- approximate 2-designs. This includes, universal ran- tion results for classically intractable circuit families dom circuits as in [8], Clifford circuits acting on in- are therefore both of crucial importance to corrobo- put product magic states [20], and certain models of rate the validity of those existing speedup proposals, diagonal unitaries [21]. What is more, we provide as well as to shine light on the conditions of them new complexity-theoretic evidence for the hardness coming about which are highly debated in the litera- of classically simulating these random-circuit fami- ture. lies in the approximate sampling sense. In doing so, In this work, we provide rigorous anticoncentration we focus on universal random circuits, which attain results for two types of such quantum architectures this property already in linear O(n) depth [22, 23]. that are at the same time not classically simulable. For this case, we derive a matching classical-hardness First, we show that random circuits drawn from a upper bound by proving that a broad class of 1D unitary 2-design anticoncentrate. We then apply this random quantum circuits are hard to simulate clas- result to both show that random circuits comprised sically already in linear O(n) circuit depth, in the of nearest-neighbour gates that are drawn from a uni- strong simulation sense (Lemma8). The novel as- versal gate set containing inverses anticoncentrate in pects of our work are the generality of these results linear depth, and propose two new schemes based and the minimal linear-depth requirements for achiev- on this insight. Second, we prove that the output ing both classical hardness and anticoncentration on distribution of a particular nearest-neighbour quan- a 1D nearest-neighbour architecture. This drastically tum quench architecture based on the time evolution improves over prior work [8] on universal random cir- of product states under certain translation-invariant cuits, which gave only numerical√ evidence for anti- Ising models anticoncentrate in constant depth. Thus, concentration (in depth O( n), and in 2D), and no we consider two types of architectures tailored to- matching classical-hardness depth bound. Prior to wards different kinds of experimental platforms in the us, analogous depth results were only available for following sense. In platforms in which achieving large nearest-neighbour IQP circuit models supplemented√ numbers of qubits is expensive and local control fea- by SWAP gates: for these, Ref. [7] gave O( n log n) sible, circuit-based schemes such as universal random depth bounds for anticoncentration and hardness in circuits can be reasonably implemented. A paradig- 2D architectures, and Ref. [10] proved linear O(n) matic example of such a platform might be consti- ones in 1D layouts. tuted of superconducting qubits. In contrast, there For the second result (Theorem9) and Corollar- are physically most natural settings of quantum sim- ies 10–12 thereof, we use the facts that IQP circuits ulators in which local control on the level of individual anticoncentrate [7] and can be implemented in a 1D gates is difficult to achieve, but for which extremely architecture in linear depth [10]. Our technical con- large numbers of local constituents can be reached. tribution is to provide an embedding of such IQP Cold atoms in optical lattices, in which 104 − 105 circuits in the constant-time evolution of a product atoms are readily reachable, provide the most promi- state under a translation-invariant Ising model on a nent example of such an architecture. Our quench- two-dimensional lattice. Via this embedding we are type architecture is tailored toward such settings. thus able to show the first anticoncentration bound The application of our first result to universal ran- for translation-invariant constant-depth schemes that dom circuits complies with the intuition that due to exhibit a quantum speedup [9, 10]. the ballistic spread of correlations anticoncentration This work is structured as follows: First, in Sec.2, will generically arise in depth that scales linearly with we will introduce the formal statement of anticon- the diameter of the system under consideration, and centration and show how it is used in the Stock-

Accepted in Quantum 2018-05-09, click title to verify 2 meyer hardness proof. In Sec.3 we will both state multiplicative errors in the average case. This needs and prove the anticoncentration result for approxi- to be conjectured for all quantum speedup schemes1, mate unitary 2-designs and then apply this result to preferably in terms of a universal quantity such as relevant examples, most importantly, universal ran- the imaginary-time partition function of Ising models dom circuits in linear depth. In Sec.4 we will then [6], the permanent [5], or the Jones polynomial [29]. prove the anticoncentration result for the constant- Together, (ii) and (iii) permit a reduction from hard- time evolution of a random product state under a ness of strong simulation up to multiplicative error to certain nearest-neighbour translation-invariant Ising hardness of weak simulation up to an additive error Hamiltonian. Finally, we discuss the implications of using Stockmeyer’s algorithm [17] in the third level our results in the context of the timely literature in of the Polynomial Hierarchy. As a result, very often Sec.5, and conclude in Sec.6. three conjectures need to be made when proving a quantum speedup using this technique:

2 Preliminaries: Anticoncentration C1 The Polynomial Hierarchy cannot collapse to its and quantum speedups 3rd level [30–32].

Throughout this work, we consider quantum systems C2 If it is #P-hard to approximate the output prob- consisting of n qubits (with obvious generalisation ability of a circuit drawn from F up to a constant to d-dimensional local constituents). To start with, relative error, then the same problem is #P-hard 2 let us make precise, what is meant by anticoncentra- for a constant fraction of the instances . tion of the output distribution of a unitary U drawn from a certain measure µ. We call the distribution of C3 The output distribution of F anticoncentrates. probabilities |hx|U|0i|2 of obtaining x ∈ {0, 1}n when In the following, we will prove the anticoncentra- applying a unitary U ∈ U(N), N = 2n, to an ini- tion conjecture C3 in the sense of equation (1) both tial state vector |0i := |0i⊗n and measuring in the for certain circuit-based schemes, in fact, those that computational basis, the output distribution. We say form an approximate 2-design, and a quantum-quench that this output distribution anticoncentrates if there architecture in the mindset of Ref. [10]. exist universal constants α, β > 0 such that for any x ∈ {0, 1}n, the probabilities |hx|U|0i|2 of this unitary anticoncentrate, 3 Anticoncentration of circuit-based α Pr |hx|U|0i|2 ≥ > β . (1) schemes U∼µ N In this section we will begin by introducing and prov- We can interpret this probability as the probability ing our first result, namely, that approximate unitary that an arbitrarily chosen entry x of the first col- 2-designs anticoncentrate, and then apply this result umn of a µ-randomly chosen U is larger than α/N. to three relevant examples of circuit-based schemes, Throughout this work, we say that a quantity X is most prominently, universal random circuits. approximated by a quantity X˜ with multiplicative error c if X/c ≤ X˜ ≤ cX, with relative error r if (1 − r)X ≤ X˜ ≤ (1 + r)X, and with additive error a 3.1 Anticoncentration of unitary 2-designs if kX − Xk∗ ≤ a for some norm k · k∗. In the commonly used proof technique for quantum Unitary k-designs approximate the uniform (Haar) speedups [24, 25] Stockmeyer’s algorithm [17] is ap- measure on the unitary group (see App. A.1) in the plied to show a collapse of the polynomial hierarchy if sense that the first k moments of a unitary k-design for an arbitrary such x the amplitude |hx|U|0i|2 is #P- and the Haar measure match (exactly or approxi- hard to approximate multiplicatively. Anticoncentra- mately). The definition of a k-design is motivated by tion comes into this proof when hardness is shown the fact that in experiments samples from a unitary not up to multiplicative but up to an additive error k-design are much easier to realise than samples from in total-variation distance. More specifically, to prove the full Haar measure. In order to define the notion a quantum speedup with constant total-variation dis- of a k design, we need the notion of the kth-moment tance errors for sampling from the output distribution operator that acts as a unitary twirl with respect to of a circuit family F using the argument developed in some measure µ on the unitary group maps on an Refs. [5,6] one requires three ingredients: (i) The out- operator. put distribution of F anticoncentrates in the sense of Eq. (3). (ii) postF = postBQP. By the result of Refs. 1In the exact case, one can even prove average-case results [26, 27] the output probabilities are then #P-hard to for the permanent [5] and random-circuit based schemes [28]. approximate up to relative error 1/4. (iii) The output 2This conjecture may be regarded as a qubit analogue of the probabilities of F are #P-hard to approximate up to “permanent-of-Gaussians” conjecture of Ref. [5].

Accepted in Quantum 2018-05-09, click title to verify 3 th k Definition 1 (k -moment operator). Let Mµ be the application to an arbitrary reference state. Also note k-th moment operator on L(H⊗k) with respect to a that the fact that µ is a relative -approximate 1- distribution µ on U(N), N = 2n = dim H defined as design (cf. App. A.3) is crucial for the bound (3) to become non-trivial. If instead µ was an additive de- k ⊗k † ⊗k X 7→ Mµ (X) := Eµ U X(U ) sign the lower bound would asymptotically tend to Z zero as 1/N and hence not stay larger than a con- ⊗k † ⊗k (2) = U X(U ) µ(U). stant. However, the 1-design condition holds even ex- U(N) actly for many distributions µ, although for the higher We can now define unitary k-designs [22, 33]. moments it may only hold approximately. Definition 2 (Unitary k-design). Let µ be a distribu- Proof of Theorem5. Our proof of the anticoncentration on the unitary group U(N). Then µ is an exact tion bound (3) has two steps and relies on two ingre- unitary k-design if dients: In the first step, we prove anticoncentration of a single but fixed entry of Haar random unitaries. To M k = M k . µ µHaar this end we make use of the Paley-Zygmund inequality and an explicit expression of the distribution of ma- In all of what follows, we will need to relax this no- trix elements of Haar-random unitaries. In the second tion to the notion of an approximate unitary k-design. step, we extend this result to full anticoncentration of In such a definition we can allow for both relative and all output probabilities in the sense of equation (3). additive errors on the equality (2) [23, 34]: The Paley-Zygmund inequality is a lower-bound Definition 3 (Approximate unitary k-designs). Let analogue of Markov-type tail bounds and can be µ be a distribution on the unitary group U(N). Then stated as follows. If Z ≥ 0 is a random variable with µ is finite variance, and if 0 ≤ α ≤ 1 [Z]2 1. an additive -approximate unitary k-design if (Z > α [Z]) ≥ (1 − α)2 E . (4) P E [Z2] k k E kMµ − Mµ k ≤ , Haar That is, it lower bounds the probability that a positive 2. a relative -approximate unitary k-design if random variable is small in terms of its mean and variance. (1 − )M k ≤ M k ≤ (1 + )M k . Now let µ be a relative -approximate unitary 2- µHaar µ µHaar design. Then for l = 2, 4, it holds that

Since the former definition is much more common l l (1 − )EU∼Haar |ha|U|bi| ≤ EU∼µ |ha|U|bi| in the literature, let us remark that the two definitions (5) l are closely related via the following Lemma of Ref. [23] ≤ (1 + )EU∼Haar |ha|U|bi| . in which, however, a factor of the dimension enters. This is due to the fact that for any unitary k- Lemma 4 (Additive and relative approximate de- design µ the expectation value of an arbitrary signs). If µ is a relative -approximate unitary k- polynomial P of degree 2 in the matrix elements design then kM k − M k k ≤ 2. Conversely, if † µ µHaar of both U and U over µ equals the same ex- kM k − M k k ≤ , then µ is a relative N 2k- pectation value but taken over the Haar measure µ µHaar approximate unitary k-design. up to a relative error > 0 [35]. To see this, observe that averaging a monomial in the matrix We are now ready to state our first anticoncentra- elements of U over the k-design µ can be expressed tion result on unitary 2-designs. k 0 0 0 0 as hi1, . . . , ik|Mµ (|j1, . . . , jkihj1, . . . , jk|)|i1, . . . , iki. k k Theorem 5 (Anticoncentration of unitary 2-designs). Hence, if Mµ = MHaar, “then any polynomial of Let µ be a relative -approximate unitary 2-design degree k in the matrix elements of U will have the on the group U(N). Then the output probabilities same expectation over both distributions” [35]. This |hx|U|0i|2 for x ∈ {0, 1}n of a µ-random unitary U ∈ gives rise to

U(N) anticoncentrate in the sense that for 0 ≤ α ≤ 1 2 2 PU∼µ(|hx|U|0i| > α(1 − )EU∼Haar[|hx|U|0i| ]) α(1 − ) (1 − α)2(1 − )2 2 2 2 ≥ PU∼µ(|hx|U|0i| > αEU∼µ[|hx|U|0i| ]) PU∼µ |hx|U|0i| > ≥ . N 2(1 + ) 2 2 2 EU∼µ[|hx|U|0i| ] (6) (3) ≥ (1 − α) 4 EU∼µ[|hx|U|0i| ] We point out that Theorem5 also holds in ex- 2 2 2 2 (1 − ) EU∼Haar[|hx|U|0i| ] actly the same way for relative -approximate state ≥ (1 − α) 4 . (1 + )EU∼Haar[|hx|U|0i| ] 2-designs. This is a weaker condition than the unitary design condition since any (approximate) unitary 2- Lemma 6 (Marginal output distribution). The dis- design generates an (approximate) state 2-design via tribution of the marginal output probabilities p =

Accepted in Quantum 2018-05-09, click title to verify 4 |hx|U|0i|2 of Haar random unitaries U and arbitrary (a) but fixed x is given by 0 | i N−2 N1 PHaar(p) = (N − 1)(1 − p) −−−→ N exp(−Np). 0 (7) | i In particular, P ’s first and second moments are 0 Haar | i given by 0 | i 1 2 2 EHaar[p] = , EHaar[p ] = . (8) 0 N N(N + 1) | i (b) We prove this lemma in App.B. Inserting the 2 expressions Eqs. (8) for EU∼Haar[|hx|U|0i| ] and 0 4 | i EU∼Haar[|hx|U|0i| ], we find U0 U0 U2 U2 U3 0 | i U4 U4 U2 U0 U0 2 α(1 − ) PU∼µ |hx|U|0i| > 0 N | i U4 U1 U3 U4 U3 N(N + 1) (1 − )2 (1 − )2 0 ≥ (1 − α)2 ≥ (1 − α)2 , | i 2N 2 (1 + ) 2(1 + ) U4 U0 U1 U0 U0 0 | i which completes the proof. Figure 1: Layout of the parallel random circuit families. Note that the moments (8) can alternatively be ob- In each step either the even or odd configuration of parallel tained for both state and unitary 2-designs exploiting two-qubit unitaries is applied with probability 1/2. Every two- qubit gate is chosen from the respective measure on U(4) – Schur-Weyl duality. This yields an explicit expression th k (a) the Haar measure, (b) the uniform distribution on the of the k -moment operators Mµ (X) as the projec- gate set G. Here we depict a five-qubit random instance of tor onto the span of the symmetric group on k tensor depth 10 where in (a) the colour choice represents different copies of the Hilbert space H. Moreover, the moments gates, and in (b) the gate set consists of 5 two-qubit unitaries of the output probabilities of state 2-designs are also G = {U0,U2,...,U4}. given by (8). This can be seen similarly using the fact that the expectation value over a state k-design is given by the projection onto the k-partite symmet- gates, most√ √ prominently, the gate set GBIS = ric subspace of H⊗k [36]. We note that the output {CZ,H, X, Y,T } studied by Boixo et al. [8, 39]. distribution (7) of a Haar random unitary asymp- In presenting this example we put particular emphasis totically approaches the exponential (Porter-Thomas) on the circuit depth required to reach a scheme that distribution. This behaviour has already been ob- shows a provable quantum speedup. As a first step (i) served numerically in many different contexts involv- of the general strategy, the following Corollary estab- ing pseudo-random operators [22, 37], non-adaptive lishes that the output distribution of a random circuit measurement-based quantum computation [38], and formed in a particular fashion from GBIS anticoncen- universal random circuits [8]. trates. This holds already in linear depth.

Corollary 7 (Universal random circuits anticoncen- 3.2 Applications: a “recipe” for quantum trate). The output probabilities of universal random speedups circuits in one dimension from the following two cir- Our anticoncentration theorem5 for approximate uni- cuit families (illustrated in Fig.1) anticoncentrate in tary (and state) 2-designs leads to a generic “recipe” a depth that scales as O(n log(1/)) in the sense of for the identiﬁcation of quantum circuit families and Eq. (3). input states that are hard to simulate classically un- • Parallel local random circuits: In each step ei- der plausible complexity-theoretic conjectures, build- ther the unitary U1,2 ⊗ U3,4 ⊗ · · · ⊗ Un−1,n or the ing upon the approach of Refs. [5,6]. The strategy unitary U ⊗ U ⊗ · · · ⊗ U is applied goes in three steps parallel to ingredients (i-iii) of the 2,3 4,5 n−2,n−1 (each with probability 1/2), with Uj,j+1 indepen- proof based on Stockmeyer’s algorithm introduced in dent unitaries drawn from the Haar measure on Sec.2. In the following, we apply this general strat- U(4). (This assumes n is even.) egy to a few examples of random circuits, most prominently, universal random circuits. m • Universal gate sets: Let G := {gi}i=1 with each gi ∈ U(4) be a universal gate set containing in- Universal random circuits. The ﬁrst exam- verses with elements composed of algebraic iden- ple that we also focus on are random quan- tities, i.e., a gate set G such that the group tum circuits constructed from single- and two-qubit generated by G is dense in U(4) and satisfying

Accepted in Quantum 2018-05-09, click title to verify 5 −1 gi ∈ G ⇒ gi ∈ G. In each step either the Proof of Lemma8. We begin by showing that both unitary U1,2 ⊗ U3,4 ⊗ · · · ⊗ Un−1,n or the uni- given target gate sets can exactly implement sub- tary U2,3 ⊗ U4,5 ⊗ · · · ⊗ Un−2,n−1 is applied (each groups of the 2-qubit dense IQP circuits of Ref. with probability 1/2), with Uj,j+1 independent [6]. Specifically, the first gate set gives us the π π unitaries drawn uniformly from G. group G1 generated by exp i 8 Xi , exp i 4 XiXj Proof of Corollary7. The central ingredient of our gates acting on a complete graph, while the second proof of Corollary7 is the result of Ref. [23]. There, gives the group generated by arbitrary long range exp i π X X gates. In both cases, long range in- the authors show that the two random circuit families 8 i j are relative -approximate unitary k-designs on U(2n) teractions are obtained via the available SWAPs. in depth poly(k) · O(n log(1/)) (Corollary 6 and 7 in Next, we show that, like the circuits in Ref. [6], Ref. [23]). both G1 and G2 are universal under post-selection. Hence, in particular, these random circuits are Indeed, both can adaptively implement a single- relative -approximate unitary 2-designs in depth qubit Hadamard via gate teleportation [47] (see also O(n log(1/)), i.e., linear in the number of qubits and [24, 48]), and non-adaptively, if we can post-select. logarithmic in 1/. Applying Theorem5 to the output The claim follows from the universality of known gate probabilities |hx|C|0i|2 of a random circuit C applied sets [40, 41]. to an initial all-zero state yields the claimed anticon- Last, due to Refs. [27, 49], the output proba- centration bound for the output probabilities of such bilities of post-selected universal quantum circuits circuits. are #√P-hard to approximate up to multiplicative error 2 (relative error 1/4). The previous fact im- To prove a quantum speedup using the Stock- plies that this holds for the dense IQP circuits in meyer technique the second required ingredient is #P- G1 and G2. Furthermore, n-qubit dense IQP cir- hardness of strong classical simulation of the output cuit can be exactly implemented in O(n) depth on a probabilities (ii). Indeed, since the gate set GBIS is 1D nearest-neighbour architecture using SWAP gates universal, the postF =postBQP connection is im- [10, Lemma 6]. It follows that the output probabilities mediate. Boixo et al. [8] moreover showed that the of linear-depth circuits in G1 or G2 are #P-hard to ap- output probabilities can be expressed in terms of the proximate. This readily extends to any circuit family imaginary-time partition function of a random Ising that can exactly synthesise either of the former, since model, suggesting that the average-case conjecture for this process only introduces a constant depth over- random circuits is a natural one (iii). It remains to head. be shown that random universal circuits are both not classically strongly simulable and anticoncentrate in We do not know whether Lemma8 extends to ar- linear depth in a one-dimensional setting. Lemma8 bitrary gate sets since applying some Solovay-Kitaev (below) establishes this is indeed the case for a large type gate synthesis algorithms [41, 50, 51] should in- class of finite gate sets with efficiently-computable troduce a polynomial overhead factor in depth. This matrix entries (so that they cannot artificially encode is because due to Chernoff-Hoeffding’s bound #P- solutions to hard problems). It is an open question hard-to-approximate quantum probabilities need to whether this can be improved to square-root depth in be (at least) super-polynomially small, for other- a two-dimensional setting such as that of Refs. [8, 39]. wise they could be inferred in quantum polynomial Given two O(1)-local gate sets A and B, we say time by mere sampling, which is not believed possible [52, 53]. To approximate such small probabili- that A exactly synthesises B if every gate V ∈ B can α be exactly implemented via a polynomial-time com- ties via the Solovay-Kitaev algorithm requires Ω(n ) putable constant-size circuit of gates in A. overhead for some α > 0 assuming the counting exponential time hypothesis [54]. These issues are closely Lemma 8 (Hardness of strong classical simulation). related to the open question of whether or not the Let G be any finite universal gate set with alge- power of post-selected quantum circuits is gate set braic efficiently computable matrix entries that can independent given some O˜(nα) depth bound [26]. i π X i π X⊗X exactly synthesise either the {e 8 , e 4 , SWAP} i π X⊗X or {e 8 , SWAP}. Then, approximating the out- Commuting circuits. As a second example, we put probabilities of O(n)-depth circuits of G nearest- consider circuits of diagonal unitaries composed of neighbour gates in one dimension up to relative error controlled-phase type one- and two-qubit gates of the 1/4 is #P-hard. form diag(1, 1, 1, eiφ), and an input state |+i⊗n. By Let us highlight that Lemma8 applies to many well- the result of Ref. [21] this gate set yields a state studied universal gate sets, including GBIS, the√ ubiqui- 2-design if the phases are picked from discrete sets tous Clifford+T [40], Hadamard+controlled- Z [41], ({0, π} for the two-qubit gates, and {0, 2π/3, 4π/3} Hadamard+Toffoli [42, 43] and others [44–46]. Inter- for the single qubit gates), and thus satisfies anticon- estingly, Lemma8 holds also for non-universal gate centration in the sense of Eq.3 (i). Adding the S- sets, though the latter may not always anticoncen- gate to the gate set and measuring all qubits in the trate. We now prove Lemma8. X-basis we obtain postF = postBQP by Refs. [6, 27]

Accepted in Quantum 2018-05-09, click title to verify 6 Circuit family Input state (State) 2-design Worst-case hardness Average-case conjecture in

F |ψ0i property (postF = postBQP) terms of universal quantity ⊗n GBIS |0i [23] [8, 40] Ising part. func./Jones polyn. Diagonal unitaries |+i⊗n [21] [26, 27] Ising partition function Cliﬀord circuits (T |0i)⊗n [33] [20] Ising partition function

Table 1: Examples of random circuit families that exhibit a provable quantum speedup up to total-variation distance errors. as IQP circuits are an instance of diagonal unitaries U := e−iH under a nearest-neighbour translation- (ii). Here, we have used the fact that adding the S- invariant Ising Hamiltonian gate and post-selection gives us access to the universal X X gate set Clifford + π/12 [55, 56]. Again, the average- H := Ji,jZiZj − hiZi, (9) case conjecture can be phrased in terms of an Ising (i,j)∈E i∈V partition function (iii). Last, the circuits can be im- and, third, measuring all qubits in the X basis. Ref. plemented in linear depth if either long-range interac- [10] proved that quantum simulations of this form tions or nearest-neighbour SWAPs are allowed [10]. cannot be efficiently classically sampled from up to constant total-variation distance assuming variants of Clifford circuits with product-state inputs. A the average-case and anticoncentration conjecture. As similar argument can be applied to Clifford circuits supporting evidence for the anticoncentration conjec- which are known to be an exact 2-design [33, 57] ap- ture, Ref. [10] built a link to the anticoncentration plied to magic input states. By the result of Ref. [58] of certain families of universal random circuits and an arbitrary element of the Clifford group in 2n di- provided numerical data. mensions can be decomposed into O(n3) elementary We note that these architectures are closely related Clifford gates. The result of Ref. [57] even achieves to that of Gao et al. [9]. The similarities and differ- an exact 2-design using only quasi-linearly many one- ences are spelled out in Ref. [10]. and two-qubit Clifford gates. The postF =postBQP We now introduce a new variant of such a quan- for this case is due to Ref. [20]. We summarise these tum quench architecture, named Qac and illustrated examples in Table1. in Fig.2 that produces provably hard-to-approximate anticoncentrated distributions which cannot be classically sampled from if an average-case conjecture holds 4 Anticoncentration of quenched and the Polynomial Hierarchy does not collapse. The architecture picks a uniformly-random input prod- many-body dynamics uct state from a finite family Sac = {|ψβi}β and lets it evolve under a nearest-neighbour translation- In this section, we investigate anticoncentration of invariant Hamiltonian Hac (defined below). a particular type of quantum simulation scheme ex- Specifically, the architecture uses n(m) := m(2m + hibiting a quantum speedup based on the architec- 1) qubits, arranged in an m-row (2m + 1)-column ture recently introduced in Ref. [10] (specifically, ar- square lattice. Boundary qubits on even-rows are ini- chitectures I-II therein). These implement quenched tialised on |0i or |1i uniformly at random. The re- (constant-time) dynamical evolutions [59, 60] under maining ones are divided in two groups, named “blue” many-body Ising Hamiltonians on the square lattice and “yellow”, using a lattice 2-colouring that places no whose graph we denote by L = (V,E). More specifi- blue qubit on the top-left and top-right√ columns. Blue cally, we consider a particular variant of such a setting qubits are initialised on (|0i+|1i)/ 2; yellow ones on −ikiπZi/8 and prove both anticoncentration for its output dis- e |+i with uniformly-random ki ∈ {0, 1, 2, 3}. tribution and the hardness of strongly classically sim- Next, the prepared state evolves under a ulating it. In virtue of the Stockmeyer-type argument translation-invariant Ising Hamiltonian Hac. Letting presented in Sec.2 this gives rise to a new quantum [i, j] denote the qubit on the i-th row and j-th col- speedup result for this architecture. umn lattice (in left-to-right top-to-bottom order), the latter reads

X π k,l X π 4.1 A new quantum quench architecture Hac= 4 δi,j Z[i,j]Z[k,l] − 4 degI (v)Zv, i≤k, j≤l v∈V Let us start by reviewing the idea of the quench archi- ([i,j],[k,l])∈E tectures introduced in Ref. [10]. There, the computa- (10) tion in the circuit model amounts to, ﬁrst, preparing  √ 0 if (i 6= k) ∧ (j = 0 mod 4) ∧ ([i, j] is blue), N iβi  a product state vector |ψβi = i∈V (|0i + e |1i)/ 2 k,l  with β chosen randomly from a ﬁnite set of an- δi,j := 0 if (i 6= k) ∧ (j = 2 mod 4) ∧ ([i, j] is yellow), i  gles; second, implementing a constant-time evolution 1 otherwise.

Accepted in Quantum 2018-05-09, click title to verify 7 for some xL, β-dependent m-qubit dense IQP cir-

cuit VxL,β ∈ GIQP such that Pr(xL,β)∼qac (VxL,β) =

µIQP(VxL,β). Before we turn to proving this theorem, let us state the two Corollaries important for us, namely, that both the anticoncentration result and the simulability result proven for dense IQP circuits in Ref. [6] carry Figure 2: Quantum quench architecture. Circles denote qubits, lines denote interactions. Blue sites are initialised over to the quench architecture Qac. on |+i; pink ones on |0i or |1i at random; yellow ones on Corollary 10 (Anticoncentration from quenched dy- e−ikiπZ/8|+i with random k ∈ {0, 1, 2, 3}. The Hamilto- i namics). The distribution qac (11) of the quench nian evolution (10) implements CZ gates on connected qubit architecture Qac described below satisfies qac(β) = pairs. Qubits are measured in the X basis. . 1/|Sac| and k,l Above, δi,j is the indicator function of the edge set 1 1 Pr q (x|β) ≥ ≥ . (12) EI of a (4,2)-periodic interaction sub-lattice I = β∼qac ac 2N 12 (V,EI ) ⊂ L (Fig.2) and degI (v) is the degree of v ∈ V in I. It is easily seen that I is a brick- Corollary 11 (Hardness of approximation). Approx- work pattern of 2-square-cells with closed boundaries imating either qac(x, β) or qac(x|β) up to relative error (Fig.2). The net effect of the dynamics is to imple- 1/4 is #P-hard. ment a controlled-Z gate on every pair of neighbouring Corollary 10 proves anticoncentration for Qac’s out- qubits in I before all qubits are measured in the X put probabilities, while Corollary 11 shows that the basis. latter are #P-hard to approximate. As an application In what follows the central quantity will be the of these two technical statements, a quantum-speedup probability distribution defined by the probabilities result follows from a Stockmeyer argument. This re-

2 sult is based on the average case conjecture C1*: ⊗n −iHac qac(x, β) := hx|H e |ψβi /|Sac| (11) P C1* Let Hv := H + i∈V viZi be the random Ising model derived from (9) by adding uniformly ran- of measuring the outcomes x after picking |ψβi and dom on-site fields v Z with random angles v = evolving it under Hac for unit time. We also let xR be i i i x’s sub-string of rightmost column’s outcomes in the (βi + xi)/2, where βi are the phases of the input state |ψ i and x are the measurement out- square lattice, and xL := x − xR be its set-theoretic β i complement. comes. The conjecture states that, if its #P-hard to approximate the imaginary temperature partition function tr [exp(iHv)] up to a constant rela- 4.2 Anticoncentration and classical hardness tive error, then the same problem is #P-hard for a of Qac constant fraction of the instances—intuitively, because random Ising models have no visible struc- Our second main result from which anticoncentra- ture making this problem easier in average (see tion and classical hardness follow, shows that this also Refs. [6–8]). quantum quench architecture is closely related to the “dense” IQP circuit family of Ref. [6] consisting of cir- Corollary 12 (Intractability of classical sampling). cuits of eiθiπXi , eiθi,j πXiXj gates acting at arbitrary If conjectures C1-C1* hold, then a classical computer pairs of qubits, with θi, θi,j chosen fully and uniformly cannot sample from the outcome distribution of archi- at random from {kπ/8 : k ∈ {0,..., 7}}. Dense IQP tecture Qac up to `1-error 1/192 in time O(poly(n)). circuits form a commutative finite group under mul- The significance of Corollary 12 is that, as shown tiplication GIQP with Haar measure µIQP. The im- below, architecture Qac defines a resource-wise plausi- plementation of a dense IQP circuit proposed in [6] ble, certifiable experiment for demonstrating a quan- requires a fully-connected architecture, and the enact- 2 tum speedup with experimental demands competitive ment of Θ(m ) long-range gates for m-qubits in aver- to those in Ref. [10]. However, unlike the latter, the age. Here, we show that our constant-depth nearest- speedup of Qac relies only on a natural average-case neighbour architecture Qac implements exact sam- hardness conjecture about Ising models and a Polyno- pling over dense IQP circuits with a linear (2m + 1) mial Hierarchy collapse. Hence, we believe this corol- overhead-factor in qubit number. lary should help in the analysis of quantum speedups Theorem 9 (Quantum quench architecture). For in near-term quantum devices. n(m) = m(2m + 1) qubits, the output probability distribution qac of architecture Qac fulfils 4.3 Proof of Theorem9 and its Corollaries 1 Proof of Theorem9. We make use of two circuit gad- q (x |β) = , q (x |x , β) = |hx |V |0i|2 ac L 2n−m ac R L R xL,β gets, named “odd” and “even”, illustrated next,

Accepted in Quantum 2018-05-09, click title to verify 8 Odd gadget Even gadget ues of xL. Qubits are initialised on |0i, followed by a “blue” Hadamard (resp. a “pink” uniformly random {Ii,Xi}) gate on odd (resp. even) rows. Even qubit lines are measured on the Z basis (preceded by “pink” identity gates in the figure); and odd ones on the X which represent quantum circuit identities modulo πk i Zi terminal Pauli operator corrections. Vertical links basis preceded by a e 8 gate. Straight-line random represent CZ gates. Crossing qubit lines perform blocks are mutually uncorrelated (terminal “dashed” SWAP gates. Blue “H” blocks implement Hadamard ones are not). Dashed CZs are “gauge gates” that x can be included or removed from by inserting CNOT gates preceded by uniformly-random Zi , x ∈ {0, 1}, 0 gates at predetermined input/output locations and single-qubit gates. Yellow “H” blocks, H e−iπkZi/8Zx i i reinterpreting the measurement outcomes. As before, gates with uniformly-random 0 ≤ k ≤ 3, x0 ∈ {0, 1}, 0 we assume H block’s by-product Pauli operators are where e−ikπZi/8Zx is a uniformly-random power of i w.l.o.g. conjugated to the end. e−iπZi/8 up to a global phase since the latter has order Next, we apply our odd/even gadgets to the bulk 8 and Z ∝ e−i4πZi/8. Analogously, yellow “Z” (resp. i of our network to rewrite the full quantum circuit in “ZX” blocks) perform uniformly-random powers of −i π Z −i π Z X an m-layered brickwork normal form e 8 i (resp. e 8 i i+1 ). The correctness of the identities is easily verified using the stabiliser formalism [51]. Pauli corrections correspond to “by-product” Zs in blue blocks, which we can propagate to the end of the circuit by flipping some of the e−ikπZi/8 gates’ angles in yellow blocks, which leaves them invariant. We next show that the computation carried out by Qac is equivalent to a 1D circuit of our odd and even gadgets composed in a brickwork layout. We where odd layers execute random gates of the form begin by reminding the reader of the properties of X-teleportation circuits [47], namely, that given an Y h −i πai Z X −i πbi Z i H H SWAP e 8 i i+1 e 8 i , (r +1)-qubit state vector |ψi|+i, the effect of measur- i i+1 i,i+1 odd i ing the i-th qubit of |ψi in the D†XD basis after en- tangling it with |+i via a CZ gate is, first, to produce with ai, bi∈Z8, followed by random-gate even layers a uniformly-random bit x; second, teleport the value of the form of the former qubit onto the latter; and, third, imple- h πdi πci i x Y −i Xi −i ZiXi+1 ment a single-qubit gate Hr+1Zr+1Dr+1 on site r +1. e 8 SWAPi,i+1e 8 Hi−1Hi , Next, note that pink sites in Fig.2 can be eliminated even i from the lattice by introducing uniformly-random si- multaneous Zi rotations on their neighbouring qubits. where ci, di∈Z8 and we define Hj = Zj = Xj = Combining these three facts and using induction, we SWAPk,k+1 = 1 for j, k < 1 and j, k + 1 > n. Trailing obtain that Qac can be simulated exactly by an al- Hadamard gates in odd layers cancel out with their gorithm that first generates a uniformly-random clas- counterparts in even-layers. By a parity-counting ar- n−m sical bit-string xL ∈ {0, 1} and then draws xR gument, it follows that SWAP gates move qubits ini- from the output of the following network of random tially on odd (resp. even) rows travel down (resp. up) 1D nearest-neighbour quantum gates, the circuit; the latter first undergo Z-type (resp. X- type) interactions, meet an odd number of H gates when they reach the bottom (resp. top) qubit line, and then undergo the opposite process. By propagat- ing all Hadamards in the full circuit to the measurement step, we are left only with a bulk of n brickwork −i πai X X −i πbi X layers of uniformly-random e 8 i i+1 , e 8 i and SWAPs, and some additional IQP gates and random Pauli by-products in the preparation/measurement which we draw for m = 4 and explicate next. The steps. It was shown in Ref. [10] that all pairs of “bulk” of this network (white area) contains an m- qubits in a bulk circuit of the given form meet exactly layered brickwork layout of odd and even gadgets once, hence, the network implements exact sampling with boundaries connected by pairs of blue and yellow over dense IQP circuits (crucially, due to their lack blocks. Blue/yellow blocks act as before. n−m out of of temporal structure). Furthermore the remaining these are placed in the bulk; their associated random gates are either also dense IQP gates, which leave xLi Zi gates originally correspond to the by-product the Haar measure µIQP invariant, or terminal Pauli rotations introduced via X-teleportation, and are ac- Z gates, which do not affect the final measurement tivated by the algorithm depending distinct bits val- statistics.

Accepted in Quantum 2018-05-09, click title to verify 9 We now exploit the mapping in Lemma9 between the depth t as a function of n required for the classical Qac’s and IQP circuits’ output statistics to prove hardness of generic circuits could be brought down to Corollaries 10 and 12. sub-linear, this would violate the counting exponential time hypothesis [62] and is therefore considered Proof of Corollary 10. Recall that m-qubit dense highly unlikely. IQP circuits fulfil Second, we highlight that the anticoncentration result for the two-dimensional quenched-dynamics set- Pr |hx|V |0i|2 ≥ 1 ≥ 1 , ∀x∈{0, 1}m. V ∼µIQP 2m+1 12 ting provably achieves the optimal asymptotic scaling (13) of depth, namely, constant in the number of qubits. Since V is drawn according to µ in Lemma9, xL,β IQP This is due to the highly specific structure of the we get dynamical evolution and not believed to hold in an 1 1 approach that relies on sampling random gates such Pr(xL,β)∼qac qac(xR|xL, β) ≥ 2m+1 ≥ 12 . (14) as Refs. [√7,8, 63]. Indeed, in such settings a scal- n−m ing as Θ( n) is expected to be necessary and suffi- Since qac(xL|β) = 1/2 , we derive (12). Last, cient for an average-case hardness result and hence qac(β) = 1/|Sac| by definition. for anticoncentration. Again, this is due to the bal- Proof of Corollary 12. The proof of Corollary 12 is listic spreading of correlations in the system. Last, analogous to those of Ref. [10, Theorem 1] and Ref. [6, the discussed connections between 2D quenches and Theorem 7], noting that X-measurements on qubits one-dimensional random circuits lead us to conjec- prepared in states |0i or |1i in Qac are equivalent to ture that the required lattice width in our result, the Z-measurements on qubits prepared in the |+i- m × (2m + 1) ∈ O(m2), is also asymptotically op- state of architecture III in Ref. [10]. Then, the same timal. argument as in Ref. [10] shows that the output probabilities qac(xL, xR|β) are proportional to an Ising partition function as in conjecture C1*. 6 Conclusion The only remaining difference with the proof of Theorem 1 in Ref. [10] is that we employ a different In summary, we have presented two anticoncentra- anticoncentration bound. Here, we use Eq. (12) of tion theorems for quantum speedup schemes that are Theorem 10, which is the same bound used in Ref. [6, based on simple nearest-neighbour interactions and Theorem 7]. As a result, we obtain a bound of 1/192 hence realisable with plausible physical architectures, for the allowed sampling error identical to that of The- filling a significant gap in the literature. We contrast orem 7 in [6]. the anticoncentration property of random circuits in one dimension that are sampled from a universal gate set with anticoncentration of the output distribu- 5 Implications and discussion tion of quenched constant-time evolution of product states under translation-invariant nearest-neighbour We now discuss the implications of our two main an- Ising models. In the former setting the depth required ticoncentration results and discuss possible improve- to achieve classical hardness and at the same time an- ments in both settings. ticoncentration of the output distribution scales with First, we conjecture that the linear-circuit-depth the diameter of the system size. In the latter set- scaling in our anticoncentration result for universal ting a similar hardness and anticoncentration result random circuits in one dimension is optimal. Indeed, is achieved after evolution for constant time. We ar- on the one hand, this result is in agreement with gue that both results are optimal for the respective the intuition that anticoncentration arises as soon as setting. We hope that this kind of endeavour sig- correlations have spread across the entire system, a nificantly contributes to the quest of realising quan- process that occurs ballistically and thus scales with tum devices that outperform classical supercomput- the diameter of the system. On the other hand, for ers, equipped with strong complexity-theoretic claims. one-dimensional random universal circuits to be intractable classically, the depth needs to be polynomial in the number of qubits. Hence, our result only 7 Acknowledgements leaves room for a sub-linear improvement, since for circuits of poly-logarithmic depth there is a quasi- We are grateful to Richard Kueng for pointing us polynomial time classical simulation based on matrix- to the application of our result to diagonal unitary product states. However, as is argued in Refs. [7, 18], circuits. Moreover, we thank Richard Kueng and it would seem counter-intuitive that one can achieve Emilio Onorati for insightful discussions and com- sub-linear depth. Indeed, standard tensor network ments on the draft, Andreas Elben for discussions on contraction techniques would allow any output prob- Haar random matrices, Tomoyuki Morimae for com- abilities of a circuit of depth t in one dimension to be ments on the draft, and the EU Horizon 2020 (640800 computed in a time scaling as O(2t) [61]. Hence, if AQuS), the ERC (TAQ), the Templeton Foundation,

Accepted in Quantum 2018-05-09, click title to verify 10 the DFG (CRC 183, EI 519/7-1, EI 519/14-1), and the [20] R. Jozsa and M. V. d. Nest, Quant. Inf. Comp Alexander-von-Humboldt Foundation for support. 14, 0633–0648 (2014), arXiv:1305.6190. [21] Y. Nakata, M. Koashi, and M. Murao, New J. Phys. 16, 053043 (2014), arXiv:1311.1128. References [22] D. Gross, K. Audenaert, and J. Eisert,J. Math. Phys. 48, 052104 (2007), arXiv:quant- [1] J. Preskill, Bull. Am. Phys. Soc. 58 (2013), ph/0611002. arXiv:1203.5813. [23] F. G. S. L. Brandão, A. W. Harrow, and [2] S. Trotzky, Y.-A. Chen, A. Flesch, I. P. McCul- M. Horodecki, Commun. Math. Phys. 346, 397 loch, U. Schollwöck, J. Eisert, and I. Bloch, Na- (2016). ture Phys. 8, 325 (2012), arXiv:1101.2659. [24] M. J. Bremner, R. Jozsa, and D. J. [3] J.-Y. Choi, S. Hild, J. Zeiher, P. Schauß, Shepherd, Proc. Roy. Soc. 467, 2126 (2010), A. Rubio-Abadal, T. Yefsah, V. Khemani, D. A. arXiv:1005.1407. Huse, I. Bloch, and C. Gross, Science 352, 1547 [25] B. M. Terhal and D. P. DiVincenzo, Quant. Inf. (2016), arXiv:1604.04178. Comp. 4, 134 (2004), arXiv:quant-ph/0205133. [4] S. Braun, M. Friesdorf, S. S. Hodgman, [26] G. Kuperberg, Theory of Computing 11, 183 M. Schreiber, J. P. Ronzheimer, A. Riera, M. del (2015). Rey, I. Bloch, J. Eisert, and U. Schneider, Proc. Natl. Ac. Sc. 112, 3641 (2015), arXiv:1403.7199 [27] K. Fujii and T. Morimae, New J. Phys. 19, . 033003 (2017), arXiv:1311.2128. [5] S. Aaronson and A. Arkhipov, Th. Comp. 9, 143 [28] A. Bouland, B. Feﬀerman, C. Nirkhe, and (2013), arXiv:1011.3245. U. Vazirani, (2018), arXiv:1803.04402. [6] M. J. Bremner, A. Montanaro, and D. J. [29] R. L. Mann and M. J. Bremner, Shepherd, Phys. Rev. Lett. 117, 080501 (2016), arXiv:1711.00686. arXiv:1504.07999. [30] S. Aaronson, “P6=NP?” in Open problems in [7] M. J. Bremner, A. Montanaro, and D. J. Shep- mathematics (Springer, 2016). herd, Quantum 1, 8 (2017). [31] L. Fortnow, in Proceedings of the Thirty-seventh [8] S. Boixo, S. V. Isakov, V. N. Smelyanskiy, Annual ACM Symposium on Theory of Comput- R. Babbush, N. Ding, Z. Jiang, M. J. Bremner, ing, STOC ’05 (ACM, 2005). J. M. Martinis, and H. Neven, Nature Physics , [32] R. M. Karp and R. J. Lipton, in Proceedings of 1 (2018), arXiv:1608.00263. the Twelfth Annual ACM Symposium on Theory [9] X. Gao, S.-T. Wang, and L.-M. Duan, Phys. of Computing, STOC ’80 (1980). Rev. Lett. 118, 040502 (2017), arXiv:1607.04947 [33] C. Dankert, R. Cleve, J. Emerson, and E. Livine, . Phys. Rev. A 80, 012304 (2009), arXiv:quant- [10] J. Bermejo-Vega, D. Hangleiter, M. Schwarz, ph/0606161. R. Raussendorf, and J. Eisert, Phys. Rev. X 8, [34] E. Onorati, O. Buerschaper, M. Kliesch, 021010 (2018), arXiv:1703.00466. W. Brown, A. H. Werner, and J. Eis- [11] T. Morimae, Phys. Rev. A 96, 040302 (2017), ert, Commun. Math. Phys. 355, 905 (2017), arXiv:1704.03640. arXiv:1606.01914. [12] J. Miller, S. Sanders, and A. Miyake, Phys. Rev. [35] A. W. Harrow and R. A. Low, Commun. Math. A 96, 062320 (2017), arXiv:1703.11002. Phys. 291, 257 (2009), arXiv:0802.1919. [13] C. Gogolin, M. Kliesch, L. Aolita, and J. Eisert, [36] H. Zhu, R. Kueng, M. Grassl, and D. Gross, “Boson sampling in the light of sample complex- (2016), arXiv:1609.08172. ity,” arXiv:1306.3995. [37] J. Emerson, Y. S. Weinstein, M. Saraceno, [14] S. Aaronson and A. Arkhipov, “BosonSampling S. Lloyd, and D. G. Cory, Science 302, 2098 is far from uniform,” arXiv:1309.7460. (2003). [15] D. Hangleiter, M. Kliesch, M. Schwarz, and [38] W. G. Brown, Y. S. Weinstein, and L. Viola, J. Eisert, Quantum Sci. Technol. 2, 015004 Phys. Rev. A 77, 040303 (2008), arXiv:0802.2675 (2017), arXiv:1602.00703. . [16] T. Kapourniotis and A. Datta, (2017), [39] C. Neill, P. Roushan, K. Kechedzhi, S. Boixo, arXiv:1703.09568. S. V. Isakov, V. Smelyanskiy, R. Barends, [17] L. Stockmeyer, Proceedings of the Fifteenth An- B. Burkett, Y. Chen, and Z. Chen, (2017), nual ACM Symposium on Theory of Computing, arXiv:1709.06678. STOC ’83, 118 (1983). [40] P. O. Boykin, T. Mor, M. Pulver, V. Roychowd- [18] A. P. Lund, M. J. Bremner, and T. C. Ralph, hury, and F. Vatan, Inf. Proc. Lett. 75, 101 npj Quant. Inf. 3, 15 (2017), arXiv:1702.03061. (2000), arXiv:quant-ph/9906054. [19] M. Schwarz and M. V. den Nest, (2013), [41] A. Kitaev, A. Shen, and M. Vyalyi, Classical arXiv:1310.6749. and Quantum Computation, Graduate studies in

Accepted in Quantum 2018-05-09, click title to verify 11 mathematics (American Mathematical Society, [68] M. Pozniak, K. Zyczkowski, and M. Kus, 2002). J. Phys. A 31, 1059 (1998), arXiv:chao- [42] Y. Shi, Quant. Inf. Comp. 3, 84 (2003), dyn/9707006. arXiv:quant-ph/0205115. [69] F. Haake, Quantum signatures of chaos, Springer [43] A. Paetznick and B. W. Reichardt, Phys. Rev. Series in Synergetics, Vol. 54 (Springer Berlin Lett. 111, 090505 (2013), arXiv:1304.3709. Heidelberg, Berlin, Heidelberg, 2010). [44] P. W. Shor, in Proc. of 37th Conf. Found. Comp. Sci. (1996) pp. 56–65, arXiv:quant-ph/9605011. [45] E. Knill, R. Laflamme, and W. Zurek, (1996), A Some facts on random matrix the- arXiv:quant-ph/9610011. [46] E. Knill, R. Laflamme, and W. H. Zurek, Proc. ory, the Haar measure, and unitary de- Roy. Soc. A 454, 365 (1998). signs [47] D. Gottesman and I. L. Chuang, Nature 402, 390 (1999). A.1 The Haar measure [48] R. Raussendorf, D. E. Browne, and H. J. Briegel, Phys. Rev. A 68, 022312 (2003), arXiv:quant- In this appendix, we give a precise definition of the ph/0301052. Haar measure on the unitary group. To do so, let us [49] L. A. Goldberg and H. Guo, (2014), first define a Radon measure. arXiv:1409.5627. Definition 13 (Radon measure). Let (X, T ) be a [50] A. Y. Kitaev, Russ. Math. Surv. 52, 1191 (1997). topological space and B its Borel algebra. A Radon [51] M. A. Nielsen and I. L. Chuang, Quantum com- measure on X is a measure µ : B → [0, +∞] such putation and quantum information, Cambridge that Series on Information and the Natural Sciences (Cambridge University Press, 2000). i for any compact set K ⊂ X, µ(K) < ∞ [52] C. H. Bennett, E. Bernstein, G. Brassard, and ii for any B ∈ B, µ(B) = inf{µ(V ): B ⊂ U. Vazirani, SIAM J. Comp. 26, 1510 (1997), V and V open} arXiv:quant-ph/9701001. [53] S. Aaronson, Proc. Roy. Soc. A 461, 2063 (2005), iii for any open set V ⊂ X, µ(V ) = sup{µ(K): arXiv:quant-ph/0412187. K ⊂ V and K compact}. [54] H. Dell, T. Husfeldt, D. Marx, N. Taslaman, and Definition 14 (Haar measure on the unitary group). M. Wahlén, ACM Trans. Algorithms 10, 21:1 The Haar measure is the unique (up to a strictly (2014). positive scalar factor) Radon measure which is non- [55] S. X. Cui and Z. Wang, J. Math. Phys. 56, zero on non-empty open sets and is left- and right- 032202 (2015). translation invariant, i.e. [56] A. Bocharov, M. Roetteler, and K. M. Svore,

Phys. Rev. A 91, 052317 (2015), arXiv:1409.3552 µHaar(U) > 0 for any non-empty open set U ⊂ U . (15) [57] R. Cleve, D. Leung, L. Liu, and C. Wang, Quant. and Inf. Comp. 16, 0721 (2016), arXiv:1501.04592. µHaar(B) = µHaar(uB) = µHaar(Bu) (16) [58] R. Koenig and J. A. Smolin, J. Math. Phys. 55, for any u ∈ U and Borel set B of U, where the left- 122202 (2014), arXiv:1406.2170. and right-translate of B with respect to u is given by [59] J. Eisert, M. Friesdorf, and C. Gogolin, Nature Phys 11, 124 (2015), arXiv:1408.5148. uB = {u b : b ∈ B} and Bu = {b u : b ∈ B}. [60] A. Polkovnikov, K. Sengupta, A. Silva, and (17) M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011), arXiv:1007.5331. A.2 Random matrix ensembles [61] R. Jozsa, (2006), arXiv:quant-ph/0603163. [62] R. Impagliazzo and R. Paturi, in Proc. XIV IEEE For the calculation of the distribution matrix elements Conf. Comp. Compl. (1999) pp. 237–240. of Haar-random unitaries it is instructive to introduce [63] S. Aaronson and L. Chen, (2016), a few important ensembles of random matrices. In arXiv:1612.05903. this appendix we do so from a rather hands-on per- [64] M. Ozols, How to generate a random unitary ma- spective. trix (Mar, 2009). G(N) (Ginibre Ensemble): The set of matrices Z [65] F. Mezzadri, (2006), arXiv:math-ph/0609050. • [66] Y. S. Weinstein and C. S. Hellberg, Phys. Rev. with complex Gaussian entries. A 72, 022331 (2005). G(N) is characterised by the measure dµG(Z) := 2 [67] K. Zyczkowski and M. Kus, J. Phys. A 27, 4235 π−N exp(− tr(Z†Z)dZ, i.e., each individual en- 2 (1994). try zi,j is distributed as exp(−|zi,j| )/π.

Accepted in Quantum 2018-05-09, click title to verify 12 GUE(N) (Gaussian Unitary Ensemble): The set Proof . Let µ be a unitary k design. That means that • of N ×N Hermitian matrices with complex Gaus- it holds sian entries, i.e., H ∈ GUE ⇔ H = D + R + R†, ⊗k † ⊗k ⊗k † ⊗k where D is a diagonal matrix with real Gaussian Eµ U X(U ) = EHaar U X(U ) (18) entries and R is an upper triangular matrix with ⊗k complex Gaussian entries. for all operators X acting on L(H ). Choose X = Y ⊗ id with Y being an arbitrary operator on GUE(N) is characterised by the measure ⊗k−1 −1 2 L(H ). Then dµGUE = ZGUE(N) exp(−Ntr(H )/2)dH on the space of Hermitian matrices. ⊗k−1 † ⊗k−1 ⊗k † ⊗k Eµ U Y (U ) = Eµ U X(U ) (19) CUE(N) (Circular Unitary Ensemble): The set • i.e., µ is a unitary (k − 1)-design. of Haar-random N × N unitary matrices. Corollary 16 (Approximate k − 1 designs from ap- CUE(N) is characterised by the Haar measure proximate k designs). Let µ be an (additive or rela- dµHaar. tive) approximate unitary k-design. Then µ is also an approximate unitary (k − 1)-design, i.e., All of dµGUE, dµG, and dµHaar are left- and right k k k−1 k−1 invariant under the action of U(N). There are two kMµ − MHaark ≤ ⇒ kMµ − MHaark ≤ (20) ways of constructing Haar-random matrices. and likewise for relative errors. 1. Draw a Gaussian matrix Z ∈ G(N), and perform the unique QR decomposition such that Z = QR, with an orthogonal matrix Q and R is required to have positive diagonal entries. Setting U = Q, B Matrix elements of Haar-random yields a Haar-random unitary [64, 65]. unitaries 2. Draw a GUE matrix Z ∈ GUE. Since Z is Let us now derive the distribution of the amplitudes Hermitian, the eigenvectors v , i = 1,...,N of i |ha|U|bi|2 of the matrix elements a Haar-random uni- Z are orthonormal. Multiplying each eigen- tary U [66–69]. To this end we apply knowledge about vector v by a random phase eφi we can con- i the distribution of entries of eigenvectors of GUE ma- struct a Haar-random unitary matrix U = trices and their relation to Haar-random unitaries (see (eφ1 v eφ2 v ··· eφn v ) writing those eigenvec- 1 2 N App. A.2). We follow Ref. [69], Chapter 4.9. tors into the columns of U [66]. The eigenvectors vi of a given operator H ∈ GUE(N) have N complex components ck and unit norm kvik2 = 1. Since every eigenvector can be uni- A.3 Unitary designs tarily transformed into an arbitrary vector of unit norm, the only invariant characteristic of those eigen- It is a simple exercise to show that if µ is a unitary vectors is the norm itself. Thus, the joint probability k-design, all up to the kth moments of µ equal the for its components {c } must read moments of the Haar measure. k N ! Lemma 15 (k − 1 designs from k designs). Let µ X P ({c }) = const · δ 1 − |c |2 , (21) be a distribution on the unitary group U(N) that is GUE k k k=1 an exact unitary k-design. Then µ is also a (k − 1)- design. where the constant is ﬁxed by normalisation.

Assuming real entries for now (we can always go to complex ones by doubling N) we can calculate that normalisation by evaluating the integral on the N-dimensional unit sphere

Z ∞ N ! N ! Y X 2 const = dci δ 1 − |ck| (22) −∞ i=1 k=1 Z Z ∞ = dωN−1 dRRN−1δ(1 − R2) (23) 0 Z Z ∞ 1 = dωN−1 dRRN−1 [δ(1 − R) + δ(1 + R)] (24) 0 2R = πN/2/Γ(N/2) . (25)

Accepted in Quantum 2018-05-09, click title to verify 13 Similarly, we can calculate the marginal distribution

Z ∞ N ! N ! (N,l) −N/2 Y X 2 P (c1, . . . , cl) = π Γ(N/2) dci δ 1 − |ck| (26) −∞ i=l+1 k=1 Z Z ∞ l N−l−1 N−l−1 2 X 2 = dω dRR δ(1 − R − |ck| ) (27) 0 k=1 l !(N−l−2)/2 Γ(N/2) X = π−l/2 1 − |c |2 . (28) Γ((N − l)/2) k k=1

For the GUE we then obtain the probability density 2 2 for the amplitude y = x1+x2 of a single complex entry x1 + ix2 of an eigenvector to be the twofold integral over real and imaginary part Z (2N,2) 2 2 PGUE(y) = dx1dx2P (x1, x2)δ(y − x1 − x2) = (N − 1)(1 − y)N−2. (29)

Since the eigenvectors of a GUE matrix are identically distributed (up to a global phase) as the columns of a CUE matrix, we obtain the same distribution as (29) for the amplitudes of the matrix elements of a CUE matrix [66]. Notably, as N becomes much larger than 1, we obtain

N−2 N1 PHaar(p) = (N − 1)(1 − p) −−−→ N exp(−Np) . (30) The ﬁrst and second moments of PCUE are then given by 1 [p] = , (31) EHaar N 2 [p2] = . (32) EHaar N(N + 1)

Accepted in Quantum 2018-05-09, click title to verify 14