Mar 20, 2016 - Let R a the unitary. arXiv:1509.02703v2 [quant-ph] 20 Mar 2016 .... [20, 21], extremely large 2D crystals [22], and in partic- ular in the XY limit ...
Spin models and boson sampling Borja Peropadre,1 Alan Aspuru-Guzik,1 and Juan Jos´e Garc´ıa-Ripoll2 1 Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, United States 2 Instito de F´ısica Fundamental IFF-CSIC, Calle Serrano 113b, Madrid E-28006, Spain
arXiv:1509.02703v1 [quant-ph] 9 Sep 2015
In this work we proof that boson sampling with N particles in M modes is equivalent to short-time evolution with N excitations in an XY model of 2N spins. This mapping is efficient whenever the boson bunching probability is small, and errors can be efficiently postselected. This mapping opens the door to boson sampling with quantum simulators or general purpose quantum computers, and highlights the complexity of time-evolution with critical spin models, even for very short times.
Aaronson and Arkhipov recently showed that predicting the measurement statistics of a linear optics circuit is, in general, a classically hard problem if the input of the circuit is a highly non-classical state [1]. The conditions for hardness are (i) an optical circuit with M modes or paths, randomly sampled from the set of optical transformations with the Haar measure; (ii) an input state with N M photons, with at most one photon on each mode; (iii) photon counters at the output ports that post-select events with one photon per port. Under these conditions, the probability distribution for any configuration n ∈ ZM 2 , is proportional to the permanent of a complex matrix, p(n1 , n2 . . . nM ) = |γn |2 , an object whose computation is #-P hard to compute. This has deep implications: namely, linear optics and interferometers does have some computing power that exceeds those of classical computation. Moreover, this result brings a new algorithm to the list of problems that are easier to implement in a quantum system than to compute in a classical device. Since then, generalizations of this problem have been created to consider other input states [2, 3], extensions to Fourier sampling [4] or trapped ion implementations [5]. In addition, boson-sampling has be related to a useful problem that shows the connection with actual quantum chemistry computations [6]. A typical boson-sampling circuit requires N singlephoton emitters and M photodetectors, with N M , as sketched in Fig. 1a. A very natural idea is to rely on fewlevel systems for both photoemission and photodetection [cf. Fig. 1b]. Indeed, photons can be efficiently generated from an initial state of N excited qubits that are weakly coupled to the photonic circuit, and these photons could be efficiently mapped into other qubits with some impedance-match tricks [7, 8]. Under such circumstances, and for times short enough that photons cannot bounce back to their sources, we would have implemented a one-directional mapping of excitations from one ensembles of few-level systems (input set) to the other set of few-level systems (output set). The setup in Fig. 1b can also bee seen as a generalization of earlier proposals with qubits in photonic or plasmonic waveguides. An extension of earlier calculations [9–11] shows that there is a coherent, photon-mediated coherent interactions for the
a)
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b)
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FIG. 1. (a) A setup that consists of beam splitters and free propagation implements boson sampling if the input state has a fixed number of photons on each port. (b) We can regard those photons as arising from the spontaneous emission of two-level systems onto the circuit, which after propagation map onto other two-level systems at the end.
qubits in Fig. 1b, H=
X
+ − σout,j Rji σin,i + H.c.,
(1)
ij
defined by the linear transformation R that maps input modes into output photons. At this point we ask the main question in this work: irrespective of other dissipative contributions, can the XY Hamiltonian Eq. (1) reproduce the the boson-sampling states that are generated optically? More precisely, starting with an initial state of N excited input spins, does the XY model (1) recreate the boson-sampling distribution of excitations in the bottom M spins? We show that the answer to this question is positive, thus establishing a connection between the boson sampling problem and the short-time dynamics of long-range-interacting but diluted spin systems. This opens the door to the simulation of boson-sampling with general purpose quantum computers and quantum simulators, but also establishes connections between the complexity of simulating the dynamics of arbitrary spin models and classically intractable problems. The structure of this letter is as follows. We reinterpret the boson sampling problem as dynamics a bosonic Hamiltonian that after a finite time recovers the unitary
2 transformation R. We obtain the full time-dependent solution of the problem, showing that it is a linear superposition of different boson-sampling experiments, converging at a finite time to the exact distribution. We then study the dynamics with Eq. (1), interpreting it as a projection of the boson-sampling Hamiltonian into a subspace of bosons with a hard-core boson (HBC) condition (i.e. at most one photon per site). We prove that the error in this dynamics is proportional to the evolution time and to the boson-sampling postselection error, so that both computation models have similar efficiency. We finally discuss the implications of this result for quantum simulation, quantum computing and quantum optics. Bosonic Hamiltonian.- Before studying the spin model, we develop a Hamiltonian for the bosonic system. Let R a the unitary transformation implemented by the circuit in Fig. 1a. R is sampled from U (M ) according to the Haar measure, and is associated an equivalent Hamiltonian X † X HBS = (bj Rji ai + H.c.) + ω(b†j bj + a†j aj ). (2) i,j
j
for the input and output modes, a and b, of the problem. The frequencies, ω > 1, which ensure boundeness of the Hamiltonian, can be removed through an interaction picture. We build new orthogonal modes c†j = Rji a†i , that satisfy the appropriate bosonic commutation relaP ∗ tions, [cm , c†n ] = i Rmi Rni = (R† R)m,n = δn,m . These canonical modes almost diagonalize the previous Hamiltonian, which a sum of beam-splitter models P † becomes † HBS = j (bj cj + cj bj ). The dynamics of this Hamiltonian involves a swap of excitations from the normal modes ci into the output modes bi , so that after a time t = π/2 the initial state |φ(0)i =
a†1
· · · a†N
|vaci ,
(3)
is transformed into the N boson superposition |φ(π/2)i = (−i)N
N X Y i=1
Rji b†j |vaci ,
(4)
j
just as one would expect from the original photonic circuit. Measuring the photons in the b modes at this state would amount to computing the permanents that are associated to the boson sampling 2 †nM 1 |γn |2 = hvac|b†n · · · b |φ(π/2)i , ni ∈ {0, 1}. (5) 1 M Dilute limit.- The final state φ(π/2) in Eq. (4) contains a non-zero probability of two or more bosons accumulating in the same mode. In other words, we can split the bosonic state into two contributions [cf. Fig. 2] φ = QN φ + ε,
(6)
Bosons
HCB
FIG. 2. The distance between the full bosonic state, φ(t) and the state approximated with spins, ψ(t), is covered by two error vectors: one δ that lives in the hard-core boson space, and another one that covers the distance between the projected state Qφ and the full boson state, φ.
where QN φ is the projection onto states with zero or one boson per site and ε the remaing states with two or more coincidences of bosons. Following the original spirit of the boson-sampling problem, events in ε have to be post-selected. In order for these interference corrections to become negligible while maintaining the efficiency of the sampling, the number of modes must be larger than the number of excitations. Formally the limit seems to be M ' N 5 log2 (N ), while M ' N 2 is the suspected ratio [1] at which sampling becomes efficient, with bounds being tested theoretically and experimentally [12, 13]. To make our later statement more quantitaive, we rely on A. Arkhipov’s bound [13] for the total probability that a measurement of the boson number on φ returns a state in the HCB sector kQφk22 ' pHCB (N, M ) =
N −1 Y a=0
2 M −a ' e−N /M . M +a
(7)
The exponential bound is approximately true when the number of bosons is very small, N = o(M 3/4 ), and hints at an error probability that also decreases with increasing number of modes 2 N 2 −αN 2 /M kεk2 ∼ 1 − c e ≤O , (8) M In other words, decreasing the rate N 2 /M , we can bring the postselection errors down as much as we wish to, augmenting the efficiency of the boson sampling experiment at the cost of larger interferometric setups. Spin-boson sampling.- The assumption of ’diluteness’ of excitations, which is needed for the efficient sampling of bosons, not only ensures that we have a small probability of boson bunch at the end of the beam-splitter dynamics, but also at all times. More precisely, the probability of bunching of particles in the state φ(t), estimated by kQφ(t)k22 , roughly grows in time and is bounded by the final postselection success probability [cf. Ref. [14]].
3 In other words, boson sampling dynamics is efficient only when it samples states with at most one boson per mode, the so called hard-core-boson subspace or the spin space. In this situation one would expect that the models (1) and (2) become equivalent, with the soft-boson corrections becoming negligible. Continuing with this line of thought, we will now study how well the dynamics of the full bosonic system can be approximated by the hard-core boson model (1). We regard the spin Hamiltonian as the projection of the full boson sampling model onto the hard-core-boson subspace, H = QHBS Q. Using this idea, we will show that the boson-sampling dynamics is reproduced by this spin model at short times, with an error that decreases with decreasing boson density, and which is a fraction of the two-boson bunching probability kεk Let us assume that |ψi is a hard-core boson state that initially coincides with the starting distribution of the boson sampling problem, |ψ(0)i = |φ(0)i. This state evolves with the spin model as i∂t ψ = QHBS Qψ. We introduce the distance between this state and the full boson model, ψ = φ + δ + ε, adding to the error ε an additional contribution in the HCB space δ = Qφ − ψ.
(9)
Note that δ is the actual sampling error which we make by working with spins, since Qφ is the desired bosonsampling state. We find a differential equation for the error vector i∂t δ = QHBS Qδ + QHBS ε, with a formal solution Z t δ(t) = −i e−iQHBS Q(t−τ ) QHBS ε(τ )dτ.
(10)
(11)
0
This equation shows that the errors in approximating 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 vector into the hard-core boson sector. We now bound the maximum error probability as an integral of two norms Z t 1/2 = kδk2 ≤ kQN HBS P1bpair k2 kP1bpair φ(τ )k2 dτ, 0
(12) where QN is a projector onto HCB states with N particles and P1bpair is a projector onto the states with N − 1 isolated bosons and 1 pair of b bosons on the same site. As explained in the supplementary materials [14], the value kP1bpair φ(τ )k2 is the probability of finding a single bunched pair in the full bosonic state. Combining a similar bound by Arkhipov [13] with the actual structure of the evolved state, we find N √ , (13) kP1bpair φ(τ )k2 ≤ O M
FIG. 3. Numerical estimates of the norm kQN HBS P1bpair k2 as a function of the number of bosonic modes, M , for different number of excitations, N .
which works provided that N = o(M 3/4 ). We have also shown [14] that the operator norm kQN HBS P1bpair k2 is strictly smaller than the maximum kinetic energy of N bosons in the original model, HBS , so that kQN HBS P1bpair k2 ≤ N. Combining both bounds we finally end up with 2 N , kδ(t)k2 ≤ t × O √ M
(14)
(15)
so that the error probability at t = π/2 vanishes when N ∼ o(M 1/4 ). We have strong evidence that this bound can be significantly improved using the properties of random matrices Rij and the structure of QN HBS ε. In particular, we have numerical evidence that the average norm over the Haar measure is kQN HBS P1bpair k2 ∝ O(N 1/2 ), which improves the requirement for efficient spin sampling N ∼ o(M 1/3 ). Fig. 3 shows the average and standard deviation of the operator norm obtained by sampling random bosonic circuits with N = 2 − 6 particles in M = 7 − 60 modes, creating random unitaries according to the Haar measure and estimating the norm of the operator QN HBS P1bpair with a sparse singular value solver. Note how, despite the moderate sample size (200 random matrices for each size) the standard deviation are extremely small, indicating the low probability of large errors and the efficiency of the sampling. It is also important to remark that the errors in the spin wavefunction δ, arise from the attempted bunching of excitations, which is prevented by the hard-core condition. For times short enough, these errors amount to excitations being ”reflected” back to the ”in” spins. This means that sampling errors can be efficiently postselected in any given realization of these experiments, rejecting measurement outcomes where there are less than + N excitations in the σout,j spins. In this case, what we characterized as an error becomes a postselection success probability, Pok = 1 − kδk2 .
4 Quantum computing.- The core result in this work is the mapping of the boson sampling problem to a spin-boson sampling model where the sampling is implemented by means of an XY spin-wave dynamics. As we will discuss now, this map opens the door to studying the original problem in a wider variety of platforms, including, for instance, trapped ions and superconducting devices, but also in particular, in any general purpose quantum computer. The algorithm that matches our proposal consists on the following steps: (i) Prepare a quantum register with 2N qubits where the first one represent the input spins and the rest are the output spins. (ii) Initialize the quantum register to the sampling state |ψ(0)i = |11 , . . . , 1N , 0N +1 , . . . , 02M i. (iii) Implement the unitary U = exp(−iπHπ/2), where H is the Hamiltonian given by (1). This can be done efficiently in terms of elementary single- and two-qubit gates. (iv) Measure the quantum register. If the first N qubits are at zero, record the resulting state of the output qubits to estimate the sampling probability. Otherwise, drop the experimental outcome. Note that while full boson sampling can by mere definition be simulated in a general purpose computer, the mapping that we have established is much more efficient than encoding the free boson space into qubits, which carries an large overhead due to the necessity of encoding all possible bunching effects. Quantum simulation.- We can use a quantum simulator with spins to implement spin-boson sampling. As a concrete application, let us assume that we have a quantum simulator that implements the Ising model with arbitrary connectivity and coupling to a transverse magnetic field X X HIsing = Jij σix σjx + B σiz . (16) ij
Another suitable platform for this kind of simulations would be the D-Wave machine or equivalent superconducting processors with long-range tunable interactions. These devices can now randomly sample J from a set of unitaries over a graph that is a subset of the available connectivity graph. Since the number of spins is very large, with over 900 good-quality qubits available, we expect that the hardness of the sampling should be better than that of the best optical circuits that can be built nowadays, which would have less beam-splitter and phase shifters and lower connectivity than the Chimera graph itself. Complexity theory.- Our mapping of boson sampling to spin evolution shows that classically simulating the dynamics of long-range interacting spin models at short times is #-P hard. This idea connects to earlier results that relate the difficulty classically simulating timeevolution due to very fast entanglement growth [23, 24]. It also does not contradict the fact that free fermionic problems can be efficiently sampled because model (1) only maps to free fermions for a subclass of matrices, R, which are tridiagonal. Summing up, we have established that boson sampling can also be efficiently implemented using spins or qubits interacting through a rather straightforward XY Hamiltonian. This map opens the door to simulating this problem with quantum simulators of spin model, of which we have offered two examples: trapped ions and superconducting circuits. Moreover, the same map states that boson sampling can be efficiently simulated in a general purpose quantum computer. JJGR acknowledges support from Spanish Mineco Project FIS2012-33022, CAM Research Network QUITEMAD+ and EU FP7 FET-Open project PROMISCE.
i
In the limit of very large transverse magnetic field, |B| kJk, we can map this problem via a rotating wave approximation to the Hamiltonian (1) where the coupling matrix is Rij = Jij . The Ising interaction (16) is already present in trapped ions quantum simulators with phonon-mediated interactions [15], a setup which has been repeatedly demontrated in experiments [16–18], even for frustrated models [19, 20], extremely large 2D crystals [21], and in particular in the XY limit [22]. In this setup, Jij can be tuned in magnitude and sign, although already ferro- and antiferro interactions are available with long range decays, Jij ∼ |i − j|α . The problem with this platform is that J cannot be guaranteed to be unitary, though we expect that the hardness of the sampling problem should remain provided that J can take both positive and negative signs, and be randomly sampled from a large enough matrix space.
[1] S. Aaronson and A. Arkhipov, in Proceedings of the fortythird annual ACM symposium on Theory of computing (ACM, 2011) pp. 333–342. [2] A. P. Lund, A. Laing, S. Rahimi-Keshari, T. Rudolph, J. L. O’Brien, and T. C. Ralph, Phys. Rev. Lett. 113, 100502 (2014). [3] K. P. Seshadreesan, J. P. Olson, K. R. Motes, P. P. Rohde, and J. P. Dowling, Phys. Rev. A 91, 022334 (2015). [4] B. Fefferman and C. Umans, ArXiv e-prints (2015), arXiv:1507.05592 [cs.CC]. [5] C. Shen, Z. Zhang, and L.-M. Duan, Phys. Rev. Lett. 112, 050504 (2014). [6] J. Huh, G. G. Guerreschi, B. Peropadre, J. R. McClean, and A. Aspuru-Guzik, arXiv preprint arXiv:1412.8427 (2014). [7] K. Koshino, K. Inomata, T. Yamamoto, and Y. Nakamura, Phys. Rev. Lett. 111, 153601 (2013). [8] B. Peropadre, G. Romero, G. Johansson, C. M. Wilson, E. Solano, and J. J. Garc´ıa-Ripoll, Phys. Rev. A 84,
5 063834 (2011). [9] D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, Phys. Rev. Lett. 97, 053002 (2006). [10] A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, Phys. Rev. Lett. 106, 020501 (2011). [11] G. D´ıaz-Camacho, D. Porras, and J. J. Garc´ıa-Ripoll, Phys. Rev. A 91, 063828 (2015). [12] N. Spagnolo, C. Vitelli, L. Sansoni, E. Maiorino, P. Mataloni, F. Sciarrino, D. J. Brod, E. F. Galv˜ ao, A. Crespi, R. Ramponi, and R. Osellame, Phys. Rev. Lett. 111, 130503 (2013). [13] A. Arkhipov and G. Kuperberg, Geometry & Topology Monographs 18, 1 (2012). [14] “Supplementary material for “spin-boson sampling”,”. [15] D. Porras and J. I. Cirac, Phys. Rev. Lett. 92, 207901 (2004). [16] A. Friedenauer, H. Schmitz, J. T. Glueckert, D. Porras, and T. Sch¨ atz, Nature Physics 4, 757 (2008). [17] K. Kim, M.-S. Chang, R. Islam, S. Korenblit, L.-M. Duan, and C. Monroe, Phys. Rev. Lett. 103, 120502 (2009). [18] R. Islam, E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. J. Wang, J. Freericks, et al., Nature communications 2, 377 (2011). [19] K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. Edwards, J. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, Nature 465, 590 (2010). [20] R. Islam, C. Senko, W. Campbell, S. Korenblit, J. Smith, A. Lee, E. Edwards, C.-C. Wang, J. Freericks, and C. Monroe, Science 340, 583 (2013). [21] J. W. Britton, B. C. Sawyer, A. C. Keith, C.-C. J. Wang, J. K. Freericks, H. Uys, M. J. Biercuk, and J. J. Bollinger, Nature 484, 489 (2012). [22] P. Jurcevic, B. P. Lanyon, P. Hauke, C. Hempel, P. Zoller, R. Blatt, and C. F. Roos, Nature 511, 202 (2014). [23] J. H. Bardarson, F. Pollmann, and J. E. Moore, Phys. Rev. Lett. 109, 017202 (2012). [24] J. Eisert and T. J. Osborne, Phys. Rev. Lett. 97, 150404 (2006).
6
Supplementary material I. BOSON SAMPLING DYNAMICS
II. BUNCHING BOUNDS
Let us start with the beam-splitter Hamiltonian X † (17) HBS = (bj cj + c†j bj ). j
defined in terms of the transformed modes X Rji a†i . c†j =
(18)
i
We need to study how the initial state |ψi =
N Y
a†k |0i ,
(19)
II.A. One-bunching events
For the purposes of bounding the error from the spinsampling model, we need to bound the part of the error ε that contains a single pair of bosons on the same site, on top of a background of singly occupied and empty states. We have labeled that component kP1pair εk22 . However, as discussed in Ref. [13], bounding that probability is harder than bounding the probability pHCB (N, M ) of having no bunching event in a state with N bosons in M modes which is approximately uniformly distributed (as it is the case for random matrices Rji ). This probability is
k=1
evolves. For that we write down the Heisenberg equations for operators evolving as O(t) = e−iHt OeiHt d † b = −i[H, b†j ] = −ic†j , dt j d † c = −ib†j , dt j
(20) (21)
which has as solutions b†j (t) = cos(t)b†j (0) − i sin(t)cj (0),
(22)
c†j (t)
(23)
=
cos(t)c†j (0)
− i sin(t)bj (0).
pHCB (N, M ) =
N Y 2 M −a ' e−N /M M +a a=0
(27)
for dilute systems N = o(M 3/4 ). We now use (i) that the state ψ(t) in Eq. (25) is made of a superposition of states with n = 0, 1, . . . N bosons distributed through the M modes, (ii) that due to the randomness of R, each of these components is uniformly distributed [13], (iii) the probability distribution pHCB (N, M ) is monotonously decerasing with N . With this we bound kQN φ(t)k22 '
N X
cos(t)2(N −n) sin(t)2n pHCB (n, M )
n=0
Inverting the relation (18), we recover X ∗ † a†k (t) = Rjk cj (t)
≥ (24)
Dynamics with Hamiltonian (17) is coherently transferring population from the a to the b modes, as in ∗ cos(t)a†k − i sin(t)Rjk bj |0i .
(25)
At time t = π/2 all population is transferred N X Y k=1
∗ Rjk bj |0i ,
(28)
Using the fact that QN φ and ε are orthogonal and thus kφk22 = kQN φk22 + kεk22 , we can find a very loose bound for the error probability of single bunching events kP1pair εk22 ≤ kε(t)k22 ≤ 1 − pHCB (N, M ).
(29)
Note that this bound can be translated into an upper bound of O(N 2 /M ) using the fact that the exponential falls faster than 1 − N 2 /M .
k=1
|φ(pi/2)i = (−i)N
n=0
= pHCB (N, M ).
∗ † = cos(t)a†k (0) − i sin(t)Rjk bj (0).
N Y
cos(t)2(N −n) sin(t)2n pHCB (N, M )
= (cos(t)2 + sin(t)2 )N pHCB (N, M )
∗ ∗ † = cos(t)Rjk Rji a†i (0) − i sin(t)Rjk bj (0)
|φ(t)i =
N X
(26)
j
but in-between, φ(t) may be regarded as a coherent superposition of boson-sampling states with 0, 1 . . . , N bosons fed into the M modes. This is useful for proving bounds on the distance between φ(t) and the hard-coreboson subspace at all times.
II.B. HCB operator bound
In addition to bounding the error vector, we also need to bound the norm of an operator that brings back population from the error subspace into the hard-coreboson subspace. Because kP1bpair εk2 is already rather small, we can afford a loose bound for the operator
7 kQN HBS P1bpair k, which is the other part of the integral. The argument is basically as follows. First, we notice that all operators in the product, QN , HBS and P1bpair , commute with the total number of particles, which in our problem is exactly N . We can thus study the restrictions of these operators to this sector, which we denote as PN OPN for each operator. We then realize that kABk2 ≤ kAk2 kBk2 and since the projectors have norm 1, kQN HBS P1bpair k2 = kQN PN HBS PN P1bpair k2
(30)
≤ kQN k2 kPN HBS PN k2 kP1bpair k2
ize
HBS =
X † † (αk+ αk+ − αk− αk− ),
(32)
k
P † † where the constraint is the same k αk+ αk+ +αk− αk− = N . Since the largest eigenvalues (in modulus) are obtained by filling N of these normal modes with the same frequency sign, we have
= kPN HBS PN k2 Notice now that PN HBS PN is just the Hamiltonian of N free bosons, without hard-core restrictions of any kind. In other words, it is the restriction of X † HBS = (bk ck + H.c.) (31) k
P to a situation where k c†k ck + b†k bk =√N . We introduce superposition modes, αk± = (ck ± bk )/ 2, and diagonal-
kQN HBS P1bpair k2 ≤ kPN HBS PN k2 = N.
(33)
Note that this proof does not make use of any properties of H such as the fact that it is built from random matrices. As explained in the body of the letter, if we sample QN HBS P1bpair randomly with the Haar measure and average the resulting norms, the bound seems closer √ to O( N ).
