Spin Models and Boson Sampling

Spin Hamiltonians are as hard as Boson Sampling

Borja Peropadre,1 AlánAspuru-Guzik,1 and Juan JoséGarc´ıa-Ripoll2 1Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, United States 2Instituto de F´ısica Fundamental IFF-CSIC, Calle Serrano 113b, Madrid E-28006, Spain Aaronson and Arkhipov showed that predicting or reproducing the measurement statistics of a general linear optics circuit with a single Fock-state input is a classically hard problem. Here we show that this problem, known as boson sampling, is as hard as simulating the short time evolution of a large but simple spin model with long-range XY interactions. The conditions for this equivalence are the same for efficient boson sampling, namely having small number of photons (excitations) as compared to the number of modes (spins). This mapping allows efficient implementations of boson sampling in small quantum computers and simulators and sheds light on the complexity of time-evolution with critical spin models.

Boson sampling requires (i) an optical circuit with M modes, randomly sampled from the Haar measure; (ii) an input state with N M photons, with at most one photon per mode; (iii) photon counters at the output ports that post-select events with at most one photon per port. Under these conditions, the probability distribution for M 2 any configuration n Z2 p(n1, n2 . . . nM ) = γn , is proportional to the permanent∈ of a complex matrix| whose| computation is #-P hard. This result, combined with some reasonable conjectures [1], implies that linear optics and interferometers have computing power that ex- ceeds that of classical computation, and that the classical simulation of random optical circuits with non-classical inputs likely involves itself an exponential overhead of FIG. 1. (a) A setup that consists of beam splitters and free resources. More recently, boson sampling has been gen- propagation implements boson sampling if the input state has eralized to consider other input states [2, 3], extensions to a fixed number of single photons on each port (red wiggles). Fourier sampling [4] or trapped ion implementations [5]. (b) We can regard those photons as arising from the sponta- Boson sampling has also been related to practical prob- neous emission of two-level systems onto the circuit (up spins), lems, such as the prediction of molecular spectra [6] and which after propagation map onto other two-level systems at quantum metrology [7]. Finally, there are other quantum the end, via the unitary matrix R. models, such as circuits of commuting quantum gates [8] which also establish potential limits of what can be classically simulated. eter [cf. Fig. 1b]. An interpretation of this setup in terms In this article we prove that boson sampling is equiv- of earlier works with qubits in photonic waveguides [9–11] alent to a many-body problem with spins that interact shows it contains a coherent, photon-mediated interac- through a long-range, XY coupling and evolve for a very tions of the form (1), accompanied by collective dissipa- short time, of the order of a single hopping or spin-swap tion terms. In addition of this physical motivation [12], event. The model involves a bipartite set of input and our work introduces a new physical problem that has the output spins same complexity as boson sampling, but which can be reproduced on state-of-the art quantum simulators and arXiv:1509.02703v2 [quant-ph] 20 Mar 2016 M X + small quantum computers. Unlike boson sampling, spin H = σout,jRjiσin,i− + H.c., (1) sampling may be implemented using error correction, an i,j=1 idea that pushes the boundaries of what can be exper- joined by the (unitary) matrix R. We show that the time imentally implemented beyond the limitations of linear evolution of an initial state that has only N excited in- optical circuits. put spins approximates the wavefunction of the boson sampling problem after a finite time t = π/2. All errors in this mapping can be assimilated to bunching of exci- RESULTS tations in the optical circuit, and the mapping succeeds whenever boson sampling actually does. The boson sampling Hamiltonian Where does Hamiltonian (1) come from? It arises from a setup with two-level systems as photo-emitters and de- Before studying the spin model (1), let us develop a tectors at the entrance and exit of an M-port interferom- Hamiltonian for boson sampling. Let R a the unitary 2

(t) transformation implemented by the circuit in Fig. 1a. R | i is sampled from U(M) according to the Haar measure, HBS Free bosons and has associated an equivalent Hamiltonian " | i M M X X HBS = (bj†Rjiai + H.c.) + ω(bj†bj + aj†aj). (2) Q (t) | i i,j=1 j=1 H | i

(0) = (0) (t) for the input and output modes, a and b, of the problem. | i | i | i Evolution with this Hamiltonian transforms the initial Hard core bosons state of boson sampling

(0) = a1† aN† vac | i + ··· + | i φ(0) = a† a† vac , (3) (0) = 01 02M | i 1 ··· N | i | i 1 ··· N | ··· i into the N boson sampling superposition FIG. 2. The distance between the full bosonic state, |φ(t)i and the state approximated with spins, |ψ(t)i, is covered by N two error vectors: one |δi that lives in the hard-core boson N Y X φ(π/2) = ( i) R∗ b† vac , (4) space (red dashed), and another one that covers the distance | i − ji j | i i=1 j between the projected state Q |φ(t)i and the full boson state, |φi (black dashed). with a photon distribution given by the permanents 2 n1 nM 2 γn = vac b1† bM† φ(π/2) , ni 0, 1 . | | | h | ··· | i | ∈ { } Continuing with this line of thought, we now study how well the dynamics of the full bosonic system can be Dilute limit approximated by the hard-core boson model (1). We regard the spin Hamiltonian as the projection of the full bo- The final state in Eq. (4) contains a non-zero proba- son sampling model onto the hard-core-boson subspace, bility of two or more bosons accumulating in the same H = QHBSQ. Using this idea, we show that the boson- mode. It can be split as in Fig. 2 sampling dynamics is reproduced by this spin model at short times, with an error that grows with excitation den- 2 φ = Q φ + ε , (5) sity, and which is bounded by ε . | i | i | i Let us assume that ψ is a hard-corek k boson state that | i where Q φ is the projection onto states with zero or one initially coincides with the starting distribution of the boson per| i site and ε contains bunched states. For the boson sampling problem, ψ(0) = φ(0) . This state | i evolves with the spin model| as i∂i ψ =| QHi Q ψ . Let errors ε to be eliminated in postselection while main- t | i BS | i taining| thei efficiency of the sampling, the number of us introduce the actual sampling error which we make by modes must be larger than the number of excitations. working with spins Formally the limit seems to be M N 5 log2(N), while ' δ = Q φ ψ . (6) M N 2 is the suspected ratio [1] at which sampling | i | i − | i ' becomes efficient, with bounds being tested theoretically A formal solution for this error and experimentally [1, 13]. Z t iQHBS Q(t τ) δ(t) = i e− − QHBS ε(τ) dτ. (7) | i − 0 | i Spin sampling Bounds for the different terms in this integral provide us with the core result (see supplementary material [15]): The assumption of ’diluteness’ of excitations, which is needed for the efficient sampling of bosons, not only en- N 2 δ(t) 2 t , (8) sures that we have a small probability of boson bunch k k ≤ × O √M at the end of the beam-splitter dynamics, but also at all times. More precisely, the probability of bunching of par- which states that the error probability at t = π/2 is negli- 2 1/4 ticles in the state φ(t) , estimated by Qφ(t) 2, roughly gible when N (M ). In that material we also show grows in time and| is boundedi by thek final postselectionk numerical evidence∼ O [15] that this bound can be at least success probability [cf. Ref. [15]]. In other words, bo- improved to N (M 1/3). Moreover, we show that the son sampling dynamics is efficient only when it samples same bounds apply∼ O for the variation distance between states with at most one boson per mode, the so called the probability distributions associated to the quantum hard-core-boson (HCB) subspace or the spin space. In states Q φ and ψ , the measure used by Aaronson and this situation one would expect that the models (1) and Arkhipov| ini Ref.| [1].i (2) become equivalent, with the soft-boson corrections Equation (7) shows that the errors due to using a spin becoming negligible. model for boson sampling feed from bunching events in 3 the original problem, δ ε , which are prevented by this kind of simulations is the D-Wave machine or equiva- the HCB condition. For| i ∝ times | i short enough, these er- lent superconducting processors with long-range tunable rors amount to excitations being “back-scattered” to the interactions [24–26]. These devices can now randomly “in” spins. This means that sampling errors can be effi- sample J from a set of unitaries over a graph that is ciently postselected in any given realization of these ex- a subset of the available connectivity graph. Since the periments, rejecting measurement outcomes where there number of spins is very large, with over 900 good-quality + contain less than N excitations in the σout,j spins. In qubits available, we expect that those simulations would this case, what we characterized as an error becomes a surpass the complexity of the sampling problems that can postselection success probability, P = 1 δ 2. be modeled in state-of-the art linear optics circuits. ok − k k

Quantum computing Complexity theory

Our previous results suggest an efficient implementa- Our mapping of boson sampling to spin evolution tion of boson sampling in a general purpose quantum shows that classically simulating the dynamics of long- computer using the following algorithm: (i) Prepare a range interacting spin models at short times has exactly quantum register with M + M qubits, encoding the in- the same complexity, which if the conjectures in Ref. [1] put and output spins, respectively. (ii) Initialize the reg- hold, is #-P hard. More precisely, if spin sampling were ister to the state ψ(0) = 1 ,..., 1 , 0 ,..., 0 . to be solvable in a classical computer, then we would | i | 1 N N+1 2M i (iii) Implement the unitary U = exp( iHπ/2), where approximate the boson sampling solution with precision − H is given by (1). (iv) Measure the quantum register. poly(N 2/√M), which would imply a collapse of the poly- Postselect experiments where the N qubits are at zero, nomial hierarchy [1]. We have thus established a new recording the resulting state of the output qubits to esti- family of problems that are efficiently simulatable in a mate the sampling probability. Note that step (iii) may quantum computer but not on a classical one. overcome the accuracy limitations of optical devices with This idea connects to earlier results that relate the dif- the use of error correction, allowing scaling to arbitrar- ficulty of classically simulating time-evolution due to very ily large problems. It is worth mentioning that while a fast entanglement growth [27, 28]. It also does not con- general purpose quantum computer might implement bo- tradict the fact that free fermionic problems can be ef- son sampling via the Schwinger representation of bosons, ficiently sampled because model (1) only maps to free this would require larger number of qubit resources, and fermions for a subclass of matrices, R, which are tridiag- a greater complexity in implementing beam-splitting op- onal. erations, while our spin sampling problem requires a There are other remarks to be done about our work smaller Hilbert space, and has a natural implementation and its place in the existing literature. First of all, it can on a small quantum computer. be argued that spins or qubits are the underlying components of a quantum computer whose computation will in general amount to evolution with an effective Hamilto- Quantum simulation nian. This argument is bogus in that the resulting Hamil- tonian will, in general, not be physically implementable, We can use a quantum simulator with spins to imple- involving interactions to an arbitrary number of spins and ment spin sampling. As a concrete application, let us distance. Moreover, even if certain models such 1 are uni- assume that we have a quantum simulator that imple- versal and may encode quantum computations [29, 30], ments the Ising model with arbitrary connectivity and the timescales of our result amount to a single hopping coupling to a transverse magnetic field event, which is scarcely the time to implement a single quantum gate and not an arguably complex computation. 2M 2M Finally, while at least one work has established connec- X x x X z HIsing = Ja,bσa σb + B σa. (9) tions between the collapse of the polynomial hierarchy a,b=1 a=1 and spin models [8], that works builds on the conjecture that the complex partition function of a spin model is In the limit of very large transverse magnetic field, already in #-P, and thus time-evolution of those spin B J , we can map this problem via a rotating wave models is hard to be approximated, which is instead the | | k k approximation to the Hamiltonian (1) where the coupling conclusion of this work. matrix is Ji,j+M = Jj∗+M,i = Rij, (i, j = 1,...,M). Summing up, we have established that boson sampling The Ising interaction (9) is already present in trapped can also be efficiently implemented using spins or qubits ions quantum simulators with phonon-mediated interac- interacting through a rather straightforward XY Hamil- tions [16], a setup which has been repeatedly demon- tonian. This map opens the door to simulating this prob- strated in experiments [17–19], even for frustrated models lem with quantum simulators of spin model, of which we [20, 21], extremely large 2D crystals [22], and in partic- have offered two examples: trapped ions and supercon- ular in the XY limit [23]. Another suitable platform for ducting circuits. Moreover, the same map states that 4 boson sampling can be efficiently simulated in a general As explained above, it shows that the errors in approx- purpose quantum computer. imating the boson sampling with spins result from the accumulation of processes that, through a single application of HBS, undo a pair of bosons from ε, taking this METHODS vector into the hard-core boson sector. We now bound the maximum error probability as an Linear model integral of two norms. For that we realize that out of varepsilon, QHBS cancels all terms that have more than

We build new orthogonal modes cj† = Rjiai†, that one mode with double occupation. Thus, satisfy the appropriate bosonic commutation relations, P Z t [cm, cn† ] = i Rmi∗ Rni = (R†R)m,n = δn,m. These 1/2 canonical modes almost diagonalize the previous Hamil- = δ 2 QHBSP1bpair 2 P1bpair (τ) 2dτ, k k ≤ 0 k k k | i k tonian, which becomes a sum of beam-splitter models (12) P HBS = j(bj†cj + cj†bj). The dynamics of this Hamil- where Q is a projector onto HCB states with N particles tonian involves a swap of excitations from the normal and P is a projector onto the states with N 2 iso- 1bpair − modes ci into the output modes bi, so that after a time lated bosons and 1 pair of b bosons on the same site. As t = π/2 the initial state (3) is transformed into (4). explained in the supplementary material [15], the value It is very important to remark that the state φ(t) can P ε(τ) 2 = P φ(τ) 2 is the probability | i 1bpair 1bpair at all times be written as kof ﬁnding| a singlei k bunchedk pair| ini the k full bosonic state.

N Combining a similar bound by Arkhipov [1] with the ac- X n N n tual structure of the evolved state, we find φ(t) = cos(t) sin(t) − ξ , (10) | i | n,M i n=0 N P1bpair (τ) 2 , (13) where ξn,M is the output state of a Boson-Sampling k | i k ≤ O √M problem| withi n input excitations in M modes. Thus, φ(t) has the bunching statistics of efficiently sampled which works provided that N = (M 3/4). We have also Boson-Sampling| i problems, and it is ruled by the boson shown [15] that the operator normO QH P is birthday paradox [1]. This in in sharp contrast with BS 1bpair 2 strictly smaller than the maximum kinetick energy ofk N actual intermediate states in a random interferometer, bosons in the original model, H , so that which may contain a lot of bunching before the bosons BS exit the circuit, and it is due to the fact that the model QH P N. (14) that we use to recreate the BS output states, H, does not k BS 1bpairk2 ≤ describe those intermediate stages, where the dynamics of individual beam-splitters matters. Combining both bounds we finally end up with (8). Note that because we only work with our bound for times U 2t π/2, the Boson-Sampling error ε con- A. Spin model bounds tains onlyk k some≤ multiply occupied sites at the out| modes,i bj. Moroever, because of the construct in δ, the domi- Equation (7) arises from the simple Schrödingerequa- nant contribution from ε to the error consists on emp- tion tying a single bunched site bj and using it to refill an input boson, ak, leading to a spin configuration where an i∂ δ = QH Q δ + QH ε . (11) excitation has been “back-scattered” to σ . t | i BS | i BS | i in,k

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ACKNOWLEDGMENTS

B.P. and A.A.-G. acknowledge the Air Force Office of Scientific Research for support under the award: FA9550- 12-1-0046. A.A.-G acknowledges the Army Research Of- fice under Award: W911NF-15-1-0256 and the Defense Security Science Engineering Fellowship managed by the 6 Supplementary material

I. BOSON SAMPLING DYNAMICS where only n = 0, 1 ...,N bosons participate and fed into the M output modes, while N,N 1 ..., 0 remain in the − Let us begin with the beam-splitter Hamiltonian input modes. In other words, we have X N 1/2 HBS = (b†cj + c†bj). (15) X N N n n j j φ(t) = cos(t) − sin(t) ξBS,n , (25) j | i n | i n=0 deﬁned in terms of the transformed modes with normalized states ξ . | BS,ni M It is important now to discuss each of these states, X which we may write as cj† = Rjiai†. (16) i=1 X ξBS,n a† (π/2) a† (π/2) (26) | i ∝ k1 ··· kn × We need to study how the evolution of an initial state k,j with N M excitations { } † † aj1 (0) ajN−n (0) 0 , N × ··· | i Y ψ = a† 0 , (17) and which consist on N n excitations that stay in the | i k | i − k=1 input modes (aj(0)) and n excitations that have been fully transferred to a (π/2), which are linear combina- For that we write down the Heisenberg equations for op- k iHt iHt tions of the bk(0) modes. In other words, each of these erators evolving as O(t) = e− Oe instances ξ represent themselves the outcome of a | BS,ni d Boson Sampling experiments with n input excitations b† = i[H, b†] = ic†, (18) and they will have the properties and statistics of bunch- dt j − j − j d ing events of any boson sampling problem of such size [1], cj† = ibj†, (19) a fact that we will use for proving bounds on the distance dt − between φ(t) and the hard-core-boson subspace at all | i which has as solutions times.

bj†(t) = cos(t)bj†(0) i sin(t)cj†(0), (20) − A. Comparison with Optics c†(t) = cos(t)c†(0) i sin(t)b†(0). (21) j j − j Inverting the relation (16), we recover It is very important to understand while the bunching statistics of φ(t) does not contradict the behavior X of actual optical| circuits.i In this work we are only study- ak† (t) = Rjk∗ cj†(t) (22) ing the unitary transformation implemented by a boson = cos(t)R∗ Rjia†(0) i sin(t)R∗ b†(0) sampling circuit. That transformation is generated by jk i − jk j our toy Hamiltonian, HBS = cos(t)a† (0) i sin(t)R∗ b†(0), k − jk j W := exp( iH π/2), (27) where we implicitly assume summation over repeated in- BS − BS dices. Dynamics under Hamiltonian (15) is coherently but intermediate stages have no other physical reality transferring population from the a to the b modes, as in than being an aid in proving our other results regard- N ing the separation between boson and spin transforma- Y tions. In particular, the optical transformation W will φ(t) = cos(t)a† i sin(t)R∗ b† 0 . (23) BS | i k − jk j | i k=1 in general be implemented as a sequence of elementary transformations —beam splitters and phase shifters—, At time t = π/2 all population is transferred Y WBS = Wi, (28) N Y i φ(π/2) = a† (π/2) 0 , with (24) | i k | i k=1 and intermediate stages of those transformations will, M in absolute generality, lead to highly bunch and also X a† (π/2) = ( i) R∗ b†,. highly entangled states which may be useful for other k − jk j j=1 purposes. However, the final state WBS φ(0) can not be highly bunched, or otherwise it would not| fiti the frame- But in-between, φ(t) may be regarded as a coherent su- work of boson-sampling (And indeed it is not, following | i perposition of different boson-sampling instances, ξBS,n Arkhipov’s boson birthday paradox). 7

II. BUNCHING BOUNDS N. Using Eq. (25) and this idea we arrive at N 2 X N 2(N n) 2n Qφ(t) cos(t) − sin(t) p (n, M) If we want Boson Sampling to be efficient, we need to k k2 ' n HCB impose that the number of bunching events in φ(π/2) n=0 remains small with increasing problem size. Such| prop-i N X N 2(N n) 2n erty is guaranteed on average by the random unitaries cos(t) − sin(t) pHCB(N,M) ≥ n U sampled with the Haar measure, as explained by n=0 2 2 N Arkhipov and Kuperberg in the boson-birthday paradox = (cos(t) + sin(t) ) pHCB(N,M) paper [1]. Below we will use the fact that the number = pHCB(N,M). (30) of bunching events in φ(π/2) = ξBS,N indeed upper- bounds the number of bunches| i in each| of itsi constituents, Using the fact that Q φ and ε are orthogonal and thus 2 2 2 | i | i ξBS,n N , and use this idea to draw conclusions on the φ 2 = Qφ 2 + ε 2, we can find a very loose bound for |distance≤ betweeni the true Boson Sampling problem and kthek errork probabilityk k k of single bunching events the HCB spin model. 2 2 P1bpairε 2 ε(t) 2 1 pHCB(N,M). (31) We cannot sufficiently stress the fact that the number k k ≤ k k ≤ − of bunching events in φ(t) is not related to the num- Note that this bound can be translated into an upper ber of bunching events| in thei intermediate stages of a bound of (N 2/M) using the fact that the exponential O linear optics circuit. In order to implement a boson sam- falls faster than 1 N 2/M. − pler one has to combine beam splitters that at different stages of the circuit cause the accumulation of the bosons. However, while those intermediate states in the III. HCB OPERATOR BOUND construct are essential to reach the final Boson Sampling states, ξ , none of those intermediate states belongs In addition to bounding the error vector, we also need | BS,ni to the family of states at the output of the circuit, ξBS,n , to bound the norm of an operator that brings back pop- which must have a low bunching probability (and| whichi ulation from the error subspace into the hard-core-boson do, as shown by Ref. [1]). subspace. Because P ε is already rather small, we k 1bpair k2 can afford a loose bound for the operator QHBSP1bpair , which is the other part of the integral. Thek argumentk is basically as follows. First, we notice that all operators in the product, Q, HBS and P1bpair, commute with the total number of particles, which in our problem is exactly A. One-bunching events N. We can thus study the restrictions of these operators to this sector, which we denote as PN OPN for each operator, where P is the projector onto the space with N For the purposes of bounding the error from the spin- N particles. We then realize that AB A B and sampling model, we need to bound the part of the error 2 2 2 since the projectors have normk 1, k ≤ k k k k ε that contains a single pair of bosons on the same site, | i on top of a background of singly occupied and empty QHBSP1bpair 2 = QPN HBSPN P1bpair 2 (32) 2 k k k k states. We have labeled that component P1pairε 2. k k Q 2 PN HBSPN 2 P1bpair 2 However, as discussed in Ref. [1], bounding that probabil- ≤ k k k k k k = P H P ity is harder than bounding the probability pHCB(N,M) k N BS N k2 of having no bunching event in a state with N bosons in Notice now that PN HBSPN is just the Hamiltonian of N M modes, distributed according to the random matrices free bosons, without hard-core restrictions of any kind. Rji. This probability is In other words, it is the restriction of X HBS = (b† ck + H.c.) (33) N k Y M a N 2/M k p (N,M) = − e− (29) HCB M + a ' a=0 P to a situation where k ck† ck + bk† bk = N. We introduce superposition modes, αk = (ck bk)/√2, and diagonalize ± ± for dilute systems N = (M 3/4). We now use (i) that the state φ(t) in Eq. (23)O is made of a superposition of X HBS = (αk†+αk+ αk† αk ), (34) | i − − − states with n = 0, 1,...N bosons distributed through the k M modes, (ii) that due to the randomness of R, each of P these components shares the same statistical properties where the constraint is the same k αk†+αk++αk† αk = of the boson-sampling states [1], (iii) the probability dis- N. Since the largest eigenvalues (in modulus)− are− ob- tribution pHCB(N,M) is monotonously decreasing with tained by filling N of these normal modes with the same 8

the variation distance between probability distributions (this is, 1-norm). However, our proof above can be written in a similar was as the one given by Aaronson and Arkhipov, that is X p p := p (n) p (n) , (36) | 1 − 2|1 | 1 − 2 | n which represents total diﬀerence between probabilities for all conﬁgurations n = (n1, . . . , nM ) of the occupations at the output ports. In our model, the probability distribution associated to boson sampling would be

p (n) = n φ(t) 2 =: φ(n) 2, (37) FIG. 3. Numerical estimates of the norm kQHBS P1bpairk2 as 1 |h | i| | | a function of the number of bosonic modes, M, for different number of excitations, N. where if we focus on events with ni 0, 1 , we can replace φ with Qφ. The corresponding∈ probability{ } for the spin model would be frequency sign, we have 2 p (n) = n ψ(t) =: ψ(n) 2, (38) 2 |h | i| | | QHBSP1bpair 2 PN HBSPN 2 = N. (35) Using the above expressions for the probability distri- k k ≤ k k butions, we can write down the following identities for Note that this proof does not make use of any prop- the total variation distance (36). erties of H such as the fact that it is built from random X X 2 2 matrices. As explained in the body of the letter, if we p1(n) p2(n) = ψ(n) φ(n) | − | || | − | | | sample QHBSP1bpair randomly with the Haar measure n n X and average the resulting norms, the bound seems closer = ψ∗(n)ψ(n) φ∗(n)φ(n) (39) to (√N). | − | O n We have strong evidence that this bound can be sig- 1 X = [ ψ(n) + φ(n) δ(n) + δ(n) ψ(n) + φ(n) ] , nificantly improved using the properties of random ma- 2 | h | i h | i | n trices Rij and the structure of QHBSε. In particular, we have numerical evidence that the average norm where δ(n) = ψ(n) φ(n) . Hence, it follows that 1/2 over the Haar measure is QHBSP1bpair 2 (N ), | i | − i k k ∝ O P 2 2 X which improves the requirement for efficient spin sam- n ψ(n) φ(n) = Re( ψ(n) + φ(n) δ(n) ) 1/3 || | − | | | | h | i | pling N (M ). Fig. 3 shows the average and stan- n dard deviation∼ O of the operator norm obtained by sam- X = Re(2φ∗(n)δ(n) + δ∗(n)δ(n) pling random bosonic circuits with N = 2 6 particles | | − n in M = 7 60 modes, creating random unitaries ac- X X − 2 φ(n) δ(n) + δ(n) 2 (40) cording to the Haar measure and estimating the norm of ≤ | || | | | n n the operator QHBSP1bpair with a sparse singular value !1/2 !1/2 solver. Note how, despite the moderate sample size (200 X X random matrices for each size) the standard deviation is 2 φ(n) 2 δ(n) 2 + δ 2 ≤ | | | | k k2 extremely small, indicating the low probability of large n n errors and the efficiency of the sampling. = 2 φ δ + δ 2, k k2k k2 k k2 Since the boson sampling wavefunction is normalized, φ 2 = 1, and δ 2 1 we finally get the next tight IV. VARIATION DISTANCE boundk k for the variationk k ≤ distance N 2 Throughout this manuscript, we have found bounds p1 p2 3 δ 2 = . (41) according to the 2-norm, in contrast to Aaronson and | − | ≤ k k O √M Arkhipov work, whose results are expressed in terms of

[1] A. Arkhipov and G. Kuperberg, Geometry & Topology Monographs 18, 1 (2012).