The Computational Power of Multi-Particle Quantum Walk

The computational power of multi-particle quantum walk

Andrew Childs David Gosset Zak Webb

arXiv:1205.3782, Science 339, 791-794 (2013) arXiv:1311.3297 Quantum walk Quantum analog of a random walk on a graph.

Idea: Replace probabilities by quantum amplitudes. Interference can produce radically different behavior!

classical

-60 -40 -20 0 20 40 60

quantum

-60 -40 -20 0 20 40 60 Quantum walk algorithms Quantum walk is a major tool for quantum algorithms (especially query algorithms with polynomial speedup).

• Exponential speedup for black-box graph traversal [CCDFGS 02] • Quantum walk search framework [Szegedy 05], [Magniez et al. 06] - Spatial search [Shenvi-Kempe-Whaley 02], [CG 03, 04], [Ambainis-Kempe-Rivosh 04] - Element distinctness [Ambainis 03] - Subgraph ﬁnding [Magniez, Santha, Szegedy 03], [CK 10] - Matrix/group problems [Buhrman, Špalek 04], [Magniez, Nayak 05] • Evaluating formulas/span programs - AND-OR formula evaluation [Farhi, Goldstone, Gutmann 07], [ACRŠZ 07] - Span programs for general query problems [Reichardt 09] - Learning graphs [Belovs 11] → new upper bounds (implicitly, quantum walk algorithms), new kinds of quantum walk search Continuous-time quantum walk Quantum analog of a random walk on a graph G =( V,E ) .

Idea: Replace probabilities by quantum amplitudes.

(t) = a (t) v | i v | i v V X2 amplitude for vertex v at time t

Deﬁne time-homogeneous, local dynamics on G . d i (t) = H (t) dt| |

Adjacency matrix: H = A(G)= u v | ih | (u,v) E(G) X2 Multi-particle quantum walk With m distinguishable particles: m states: v1,...,vm vi V (G) Hilbert space dimension: n | i m 2 Hamiltonian: H(m) = A(G) + G i U i=1 X Indistinguishable particles: bosons: symmetric subspace fermions: antisymmetric subspace

Many possible interactions: m on-site: = J nˆ (ˆn 1) nˆ = v v U v v v | ih |i v V (G) i=1 2 (with bosons,X this is the Bose-Hubbard model) X nearest-neighbor: = J nˆ nˆ U u v (u,v) E(G) X2 Dynamics are universal; ground states are hard Theorem: Any n -qubit, g -gate quantum circuit can be simulated by a multi-particle quantum walk of n +1 particles interacting for time poly( n, g ) on an unweighted graph with poly( n, g ) vertices. Consequences: • Architecture for a quantum computer with no time-dependent control • Simulating dynamics of interacting many-body systems is BQP-hard (e.g., Bose-Hubbard model on a sparse, unweighted, planar graph)

Theorem: Approximating the ground energy of the n -particle Bose- Hubbard model on a graph is QMA-complete. Consequences: • Computing the ground energy of the Bose-Hubbard model is (probably) intractable • New techniques for quantum Hamiltonian complexity … but not always

model dynamics ground energy

Local Hamiltonians BQP-complete QMA-complete

Bose-Hubbard model BQP-complete QMA-complete (positive hopping)

stoquastic Local BQP-complete AM Hamiltonians

Bose-Hubbard model BQP-complete AM (negative hopping)

ferromagnetic Heisenberg BQP-complete trivial model on a graph Universal computation by multi-particle quantum walk

arXiv:1205.3782, Science 339, 791-794 (2013) Universality of quantum walk Quantum walk can be efﬁciently simulated by a universal quantum computer. N-vertex graph circuit with max degree poly(log N) poly(log N) qubits ) efﬁciently computable neighbors poly(log N) gates

Conversely, quantum walk is a universal computational primitive: any quantum circuit can be simulated by a quantum walk. [Childs 09] graph with N dimensions poly(N,g) vertices g gates ) max degree 3 walk for time poly(g)

Note: The graph is necessarily exponentially large in the number of qubits! Vertices represent basis states. Single-particle quantum walk experiments

Vol. 22, No. 2/February 2005/J. Opt. Soc. Am. B 499

Do et al. PRL 100, 170506 (2008) PHYSICAL REVIEW LETTERS

Experimental realization of a quantum quincunx Realization of Quantum Walks with Negligible Decoherence in Waveguide Lattices week ending by use of linear optical elements 2 MAY 2008 ˚ . Nordlund, G. B. Scharmer, H. C. Spruit, 5. T. Heinemann, A 669, 1390 (2007). 691, Hagai B. Perets, 1, Binh Do Astrophys. J. Astrophys. J. * Yoav Lahini, 1 Francesca Pozzi, 2 REPORTS Although our simulation of large sunspots is 6. M. Rempel, M. Schüssler, M. Knölker, 2 1 2 659,812(2007). Department of ElectronicsFaculty & Electrical of Physics, Engineering, The Weizmann University Institute ofMarc Glasgow, of Science, Sorel, Glasgow 76100 G12Rehovot, 8QQ, Israel Scotland, United Kingdom realistic in terms of relevant physics, it does not 640 (2009). Astrophys. J. Roberto Morandotti, 3 lation between brightness and vertical flow direction 3 and Yaron Silberberg faithfully reproduce all aspects of the morphology 7. M. Kubo, T. Shimizu, S. Tsuneta, Institute National de la Recherche Scientifique, Universita 1 Department(12 of Physics,). This constitutes Purdue evidenceUniversity, for a West convective Lafayette, flow Indiana 47907-2036 8. More detailed information about the physical model, Michael L. Stohler of observed penumbral filaments. The penumbral the numerical code, and the simulation setupScience is Online. pattern that transports the energy flux emitted in , (Received 24 September 2007; published 2 May 2008) regions are considerably more extended than in available as supporting material on 316 ´ du Quebec, Varennes, Quebec J3X 1S2, Canada the penumbra. Other studies show a correlation Astron. Astrophys. Quantum random walks are the quantum counterpart of classical random walks, and were recently previous local simulations, but they are still some- 9. R. Keppens, V. Martinez Pillet, between intensity and line-of-sight velocities 229 (1996). 641, L73 (2006). studied in the context of quantum computation. Physical implementations of quantum walks have only Purdue University, West Lafayette, Indiana 47907-2036what subdued, probably owing to the proximity of Astrophys. J. Lett. (13), which for sunspots observed outside the 10. M. Schüssler, A. Vögler, Astron. Astrophys. been made in very small scale systems severely limited by decoherence. Here we show that the Department of Physics, Wabash College, Crawfordsville, Indiana 47933-0352 andthe Department periodic boundaries. of Physics, The filaments in the inner center of the solar diskSunder is dominated Balasubramanian by the hor- 11. D. Dialetis, P. Mein, C. E. Alissandrakis, propagation of photons in waveguide lattices, which have been studied extensively in recent years, are penumbrae appear to be too fragmented,17 and)formonly short, 147, 93 (1985). essentially an implementation of quantum walks. Since waveguide lattices are easily constructed at large izontal Evershed flow. This is consistent with our , 1357 (2007). dark lanes along bright filaments ( 12. J. Sánchez Almeida, I. Márquez,Astrophys. J. A. Bonet, J. 658 findings, because in the penumbra the horizontal I. Domínguez Cerdeña, scales and display negligible decoherence, they can serve as an ideal and versatile experimental occasionally, likely a consequence of the still- playground for the study of quantum walks and quantum algorithms. We experimentally observe quantum Departmentflow of velocity Physics, isPurdue correlated University, with the vertical West Lafayette, flow Indiana 47907-2036 13. L. R. Bellot Rubio, R. Schlichenmaier,453, 1117 (2006). A. Tritschler, Daniel S. Elliott limited spatial resolution of the simulation. Lastly, Astron. Astrophys. Astrophys. walks in large systems ( direction. 8)showsthatthespatial ˚. Nordlund, T. Heinemann, the initial condition of the magnetic field under- 14. G. B. Scharmer, A 100 Our detailed analysis ( , L149 (2008). including ballistic propagation, disorder,sites) and and confirm boundary quantum related walks effects. effects which were studied theoretically, lying the sunspot is quite arbitrary, owing to our J. Lett. 677 , 1597 (2007). scales of the flows providing the major part of the et al ., Science 318 ignorance of the subsurface structure of sunspots. 15. K. Ichimoto convective energy transport are similar for both DOI: 10.1103/PhysRevLett.100.170506 Notwithstanding these limitations, the present sim- 16. V. Zakharov, J. Hirzberger, T. Riethmüller,488, L17S. (2008). Solanki, School of ElectricalDepartment and Computerundisturbed of Physics, Engineering, granulation Purdue and University, Purdue penumbra. University, West The primaryLafayette, West Lafayette, Indiana Indiana47907-2036 47907-2005 and P. Kobel, Astron. Astrophys. Christopher Eash ulations are consistent with observations of global 420, 151 (2002). PACS numbers: 03.67.Lx, 05.40.Fb, 42.25.Dd, 42.50.Xa difference is that there is no preferred horizontal 17. G. B. Scharmer, B. V. Gudiksen, D. Kiselman,Nature M. G. Löfdahl, sunspot properties, penumbral structure, and system- L. H. M. Rouppe van der Voort, In classical random walks, a particle starting from an direction for granulation, whereas the energy- atic radial outflows. These and earlier simulations 18. High-performance computing resources were provided by initial site on a lattice randomly chooses a direction, and transporting flows in the penumbra are distinctly NCAR's Computational and Information Systems Laboratory. Department of Physics, Wabash College, Crawfordsville, Indiana(5, 6 47933-0352, 10)suggestaunifiedphysicalexplanationfor then moves to a neighboring site accordingly. This process to scale to much larger configurations. Moreover, even at asymmetric: Convective structuresEphraim are Fischbach elongated in NCAR is sponsored by NSF. umbral dots as well as inner and outer penumbrae in these very small scales, errors attributed to decoherence the radial direction of the sunspot. These properties 5, 6) is repeated until some chosen final time. This simple terms of magnetoconvection in a magnetic field were already indicated in earlier simulations ( Supporting Online Material random walk scheme is known to be described by a have been observed. with varying inclination. Furthermore, a consistent www.sciencemag.org/cgi/content/full/1173798/DC1 Departmentand suggested of Physics, as an Purdueexplanation University, for the Evershed West Lafayette, Indiana 47907-2036 2013 18, April on Gaussian probability distribution of the particle position, Here we suggest a very different implementation of 14). The simulation shown here con- physical picture of all observational characteristics Materials and Methods outflow in ( Michael A. Fischbach SOM Text where the average absolute distance of the particle from the CQWs using optical waveguide lattices. These systems of sunspots and their surroundings is now emerging. firms this suggestion and demonstrates the convec- Figs. S1 to S4 origin grows as the square root of time. First suggested by have been studied extensively in recent years [ tive nature of a fully developed penumbra. References Feynman [ not in the context of QWs and quantum algorithms. We References and Notes 11, 153 (2003). Movies S1 and S2 1] the term 11], but The horizontal asymmetry ofArthur the convective Mills Astron. Astrophys. Rev. to describe the random walkquantum behaviorrandom of a quantum walks was parti- defined show that these systems can serve as a unique and robust Department of Chemistry and Chemical Biology, Harvard University, Cambridge,1. S. K. Massachusetts Solanki, 02138-3800Annu. Rev. Astron. Astrophys. flows is also manifest in the correlation of 0.42 vx) 19 March 2009; accepted 8 June 2009 2. J. H. Thomas, N. O. Weiss, Published online 18 June 2009; cle. The coherent character of the quantum particle plays a tool for the study of CQWs. For this purpose we demon- between the corresponding flow component ( 42, 517 (2004). 69, 454 (1909). Mon. Not. R. Astron. Soc. 10.1126/science.1173798 major role in its dynamics, giving rise to markedly differ- strate three fundamental QW effects that have been theo- and the brightness. We find that the rms of the 3. J. Evershed, Sunspots and Starspots Include this information when citing this paper. vx)inthepenumbra ent behavior of quantum walks (QWs) compared with retically analyzed in the QW literature. These include outflowing velocity componentBenjamin ( Zwicklvy) 4. J. H. Thomas, N. O. Weiss, School of Electrical and Computer Engineering, Purdue University, West Lafayette,(Cambridge Indiana Univ. Press, 47907-2035 Cambridge, 2008). classical ones. For example, in periodic systems, the quan- ballistic propagation in the largest system reported to is much larger than the transverse component ( www.sciencemag.org date ( (perpendicular to the filament direction), showing tum particle propagates much faster than its classical 100 sites), the effects of disorder on QWs, and QWs with reflecting boundary conditions (related to Departmentan asymmetry of Physics, similar Purdueto that found University, by the scale West Lafayette, Indiana 47907-2036 counterpart, and its distance from the origin grows linearly analysis. The total rms velocity profile as a func- with time (ballistic propagation) rather then diffusively [ Berry’s ‘‘particle in a box’’ and quantum carpets tion of depth is very similar to its counterpart for Quantum Walk in Position Space with In disordered systems, the expansion of the quantum me- [12,13]). Waveguide lattices can be easily realized with Received Juneundisturbed 4, 2004; accepted granulation, August apart 23, from 2004; a slightly revised manuscript received August 30, 2004 chanical wave-function can be exponentially suppressed 2]. even larger scales than shown here ( higher peak value, confirming the physical sim- even for infinitesimal amount of disorder, while such sup- current fabrication technologies), with practically2 no4 de- We report the experimental realization of a quantum analog of the classicalSingle Galton quincunx Optically and discuss its Trapped Atoms 10 –10 sites with The primary

possible use as a deviceilarity for of quantum convection computation. in granulation Our and quantum penumbra. quincunx is implemented withLeonid linear Förster, optical Jai-Min2,5,13 Choi, Andreas Steffen, Wolfgang Alt, pression does not occur in classical random walks. coherence. The high level of engineering and control of * from Downloaded Michal Karski, elements that allow anThe incoming mass flux photon and energy to interfere flux show with similar itself while traversing all possible paths from the * In recent years QWs have been extensively studied these systems enable the study of a wide range of different source to the detector.properties We show with that respect the experimentally to the length scales determined and Here intensityDieter we describe distributions Meschede, the implementation Artur are in Widera excellent of the QW using ), indicating that most of the theoretically [ parameters and initial conditions. Specifically it allows the agreement with theory.200.0200,asymmetry © 2005 230.0230. ( Optical8 Society of America an optical quantum quincunx (QQ). 2] and have been used to devise new quan- OCIS codes: tum computation algorithms [ implementation and study of a large variety of CQWs and outflowing material emerges, turns over, and de-differenceThe betweenquantum walk theGalton is the quantum quincunx analog and ofa the QQ well-known is that random walk, which forms the basis tinuous time QWs (DQWs; CQWs)3 []. Both discrete and con- show experimental observations of their unique behavior. scends within the penumbra. In the deeper layers,an objectfor traversing models and applicationsthe Galtonquincunx in many realms travels of science. along Its a properties are markedly different The CQW model was first suggested by Farhi and Much recent theoretical workthere ishas some examined contribution the (of prop- the order of 10 to single pathfrom thefrom classical the source counterpart to thedetector, and might whereaslead to extensive each applications in quantum information studied. In DQWs the quantum particle hops between4–6] have been Gutmann [ erties of a quantum1–11 A walk two-dimensional (QW)20%) in to bothone energyand QW twois and the mass quantum fluxes by the large- incidentscience. object In in our a QQ experiment, can be viewed we implemented as simultaneously a quantum walk on the line with single neutral atoms lattice sites in discrete time steps, while in CQW the 6], where the intuition behind it comes from –10 traversingby deterministically all possible1 is paths delocalizingshown to the in Fig. detector. them 1, where over A the schematic in sites our of case a one-dimensional spin-dependent probability amplitude of the particle leaks continuously continuous time classical Markov chains. In the classical dimensions. scale flow cell surrounding the sunspots. polarization is split The analysis of our simulations indicates that versionoptical of a QW lattice. With the use of site-resolvedx or y fluorescence imaging, the final wave function is to neighboring sites. Experimentally, many methods have random walk on a graph, a step can be described by a analog of the classical random walk, a statistical concept yЈ) that are rotated that has been used to demonstrategranulation such and phenomena penumbral flows as are similar with the quantumcharacterized object by is local a photon. quantum Atx stateЈ eachand tomography, node (indicated and its spatial coherence is demonstrated. been suggested for the implementation of DQWs (see [ matrix M which transforms the probability distribution for Brownian motion and the transitionregard to energy from transport; a binomial the asymmetryto a be- by a darkOur systemcircle)allows a photon the observationwith of the quantum-to-classical transition and paves the way for but only a small scale system consisting of a few states was the particle position over the graph nodes (sites). The Gaussian distribution in thetween limit the of horizontal large statistics. directions The and the reduced into twoapplications, beams (denoted such as quantum by cellular automata. implemented, using linear optical elements [ 2]), entries of the matrix 12 which consists of a matrix of pins random walk can be implementedoverall energy with flux a device reflect known the constraints as im- counterpart, the random walk, is relevant in many Mj;k give the probability to go from a few suggestions have been made [ 7 site j to site k Galton’s quincunx, posed on the convective motions by the presence aspects of our lives, providing insight into diverse8), de- ]. For CQWs, in one step of the walk. The idea was to carry nterference phenomena with microscopic experimental method have been implemented8,9], by yet realizing only one this construction over to the quantum case, where the through which a ball travels,of a strong with probability and inclined magnetic 50:50 of go-field. The de- particles are a direct consequence of their1–5). The fields: It forms the basis for algorithms ( a small scale cyclic system (4 states) using a nuclear Hamiltonian velopment of systematic outflows is a direct con- scribes diffusion processes in physics or biology of the process is used as the generator matrix. ing right or left at each pin. quantum-mechanical wave nature ( magnetic resonance system [ The system is evolved using sequence0740-3224/2005/020499-06$15.00 of the anisotropy, and the similarities © 2005 OpticalI Society of America (8, 9), such as Brownian motion, or has been 10). prospect to fully control quantum properties of 10]. Such systems are difficult in some initial state U t between granulation and penumbral flows strong- used as a model for stock market prices ( exp iHt . If we start atomic systems has stimulated ideas to engineer in , evolve it under ÿ ly suggest that driving the Evershed flow does not Similarly, the quantum walk is expected to have and measure the positionsj of the resulting state, weU obtain a quantum states that would be useful for applica- 0031-9007 i for a time T require physical processes that go beyond the =08=100(17)= tions in quantum information processing,undamental for 170506(4) 170506-1 combination of convection and anisotropy intro- example, and also would elucidate f Institut für Angewandte Physik der Universität Bonn Wegeler- duced by the magnetic field. Weaker laterally © 2008 The American Physical Society questions, such as the quantum-to-classical straße 8, 53115 Bonn, Germany. overturning flows perpendicular to the main fila- transition ( 6). A prominent example of state engi- *To whom correspondence should be addressed. E-mail: ment direction explain the apparent twisting15, 16 mo-)and [email protected] (M.K.); [email protected] (A.W.) neering by controlled multipath interference7). Its classical is tions observed in some filaments ( the quantum walk of a particle ( lead to a weakening of the magnetic field in the6). flow channels through flux expulsion ( SCIENCE www.sciencemag.org 10 JULY 2009 VOL 325 174 Momentum states

Consider an inﬁnite path:

7 6 5 4 3 2 1 0 1 2 3 4 5 6 7

Hilbert space: span x : x Z {| ⌅ } Eigenstates of the adjacency matrix: k˜ with | x k˜ := eikx k [ , ) ⇤ | ⌅ ⇥ Eigenvalue: 2 cos k Wave packets A wave packet is a normalized state with momentum concentrated near a particular value k .

L 1 ikx Example: e x (large L ) p | i L x=1 X k !

1 0 1 2 3 4 5 6 7 8 9 10 11 12 13

dE Propagation speed: =2sin k dk | | Scattering on graphs Now consider adding semi-inﬁnite lines to two vertices of an arbitrary ﬁnite graph.

Before:

k ! Gˆ

After:

k k R(k) T (k) ! Gˆ The S-matrix This generalizes to any number N of semi-inﬁnite paths attached to any ﬁnite graph.

A (4, 1) B (3, 1) Incoming wave packets of (2, 1) momentum near k are mapped (4, 2) to outgoing wavek packets (of (1, 1) (3, 2) the same momentum)! with amplitudes corresponding to (1, 2) (2, 2) G entries of an N N unitary k (1,N) matrix S ( k ) , called⇥ the S-matrix. ! b (2,N) (3,N) (4,N) Encoding a qubit Encode quantum circuits into graphs.

Computational basis states correspond to paths (“quantum wires”). A A (4, 1) (4, 1) B B (3, 1) (3For, 1) one qubit, use two wires (“dual-rail encoding”): (2, 1) (2, 1) (4, 2) (4, 2) k k (1, 1) (1,(31), 2) (3, 2) ! ! (2, 2) (1, 2) (1, 2) (2, 2) G G k k (1,N) (1,N) ! ! b b(2,N) (2,N) (3,N) (3,N) (4,N) (4,N) encoded 0 encoded 1 | i | i Fix some value of the momentum (e.g., k = ⇡ / 4 ).

Quantum information propagates from left to right at constant speed. Implementing a single-qubit gate

To perform a gate, design a graph whose S-matrix implements the desired transformation U at the momentum used for the encoding.

0in 0out Gˆ 1in 1out

0 V S(k)= U 0 ✓ ◆ Universal set of single-qubit gates

0in 0out 0in 0out

1in 1out 1in 1out

10 1 i1 0 i p2 1i ⇥ ✓ ◆

momentum for logical states: k = ⇡/4 Two-particle scattering In general, multi-particle scattering is complicated. But scattering of indistinguishable particles on an inﬁnite path is simple.

Before:

k p !

After:

i✓ p k e ⇥ !

Phase ✓ depends on momenta and interaction details. Momentum switch To selectively induce the two-particle scattering phase, we route particles depending on their momentum. A B 0c,in 0c,out

1c,in 1c,out 1 2 1 2 = 3 3 0m,in 0m,out

1m,in 1m,out

Particles with momentum ⇡ / 4 follow the single line. Particles with momentum ⇡ / 2 follow the double line. Controlled phase gate Computational qubits have momentum ⇡ / 4 . Introduce a “mediator qubit” with momentum ⇡ / 2 . We can perform an entangling gate with the mediator qubit.

A B 0c,in 0c,out 1c,in 1c,out 100 0 010 0 1 2 1 2 C✓ = = 0001 01 3 0 0 B000ei✓C 3 m,in m,out B C @ A 1m,in 1m,out Hadamard on mediator qubit

A B C 0in 0out 0in 0out 0in 0out

1in 1out 1in 1out 1in 1out

[Blumer-Underwood-Feder 11] Proving an error bound Proof ideas: • With long enough incoming square wave packets (length L ), the outgoing wave packets are well-approximated by the effect of the 1/4 S-matrix (error O ( L ) ). We prove this for single-particle scattering (for any graph) and for two-particle scattering on an inﬁnite path. • Truncation Lemma: If the state is well-approximated by one that is well-localized to some region, then changing how the Hamiltonian acts outside that region has little effect.

Theorem: The error can be made arbitrarily small with L = poly( n, g ) . Example: For Bose-Hubbard model, L = O ( n 12 g 4 ) sufﬁces. Open questions • Improved error bounds • Simpliﬁed initial state • Are generic interactions universal for distinguishable particles? • New quantum algorithms • Experiments • Fault tolerance The Bose-Hubbard model is QMA-compete

arXiv:1311.3297 Quantum Merlin-Arthur QMA: the quantum analog of NP Merlin wants to prove to Arthur that some statement is true. Merlin Arthur

quantum proof efficient quantum | i verification circuit • If the statement is true, there exists a that Arthur will accept with probability at least 2/3. | i • If the statement is false, any will be rejected by Arthur with probability at least 2/3. | i Complexity of ground energy problems • k-Local Hamiltonian problem: QMA-complete for k≥2 [Kempe, Kitaev, Regev 2006] • Quantum k-SAT (is there a frustration-free ground state?): in P for k=2; QMA1-complete for k≥3 [Bravyi 2006; Gosset, Nagaj 2013] • Stoquastic k-local Hamiltonian problem: in AM [Bravyi et al. 2006] • Fermion/boson problems: QMA-complete [Liu, Christandl, Verstraete 2007; Wei, Mosca, Nayak 2010] • 2-local Hamiltonian on a grid: QMA-complete [Oliveira, Terhal 2008] • 2-local Hamiltonian on a line of qudits: QMA-complete [Aharonov, Gottesman, Irani, Kempe 2009] • Hubbard model on a 2d grid with a site-dependent magnetic field: QMA-complete [Schuch, Verstraete 2009] • Heisenberg and XY models with site-dependent couplings: QMA- complete [Cubitt, Montanaro 2013] Bose-Hubbard Hamiltonian is QMA-complete Bose-Hubbard model on G :

H = t A(G) a† a + J nˆ (ˆn 1) G hop uv u v int v v u,v V (G) v V (G) X2 2X Theorem: Determining whether the ground energy for n particles on the graph G is less than ne 1 + ✏ or more than ne 1 +2 ✏ is QMA- complete, where e 1 is the 1-particle ground energy.

• Fixed movement and interaction terms ( A ( G ) is a 0-1 matrix) • Applies for any ﬁxed thop,Jint > 0 • It is QMA-hard even to determine whether the instance is approximately frustration free • Analysis does not use perturbation theory Dependence on signs of coefﬁcients

thop

QMA- QMA 2 complete

Jint

AM QMA 2 \

stoquastic (no sign problem) XY model We encode computations in the subspace with at most one boson per site (“hard-core bosons”)

Thus we can translate our results to spin systems, giving a generalization of the XY model on a graph:

i j + i j 1 i x x y y + z 2 2 A(G)ij =1 A(G)ii=1 Xi=j X 6

Theorem: Approximating the ground energy in the sector with 1 i z magnetization i 2 = n is QMA-complete. P Containment in QMA Ground energy problems are usually in QMA Strategy: • Merlin provides the ground state • Arthur measures the energy using phase estimation and Hamiltonian simulation Only one small twist for boson problems: project onto the symmetric subspace The quantum Cook-Levin Theorem Theorem: Local Hamiltonian is QMA-complete [Kitaev 99]

Consider a QMA veriﬁcation circuit U ...U U with witness t 2 1 | i The Feynman Hamiltonian t H = (I j j + I j 1 j 1 U j j 1 U † j 1 j ) ⌦ | ih | ⌦ | ih | j ⌦ | ih | j ⌦ | ih | j=1 X t 1 has ground states hist = U ...U j | i p j 1| i⌦| i t +1 j=0 X • Implement the “clock” using local terms • Add a term penalizing states with low acceptance probability Single-qubit gates

t =1 Construct a graph encoding a H H universal set of single-qubit gates in t =8 t =2 the single-particle sector:

• Start from Feynman-Kitaev Hamiltonian H H for a particular sequence of gates

• Obtain matrix elements {-1,0,+1} by t =7 t =3 careful choice of gate set and scaling • Remove negative entries using an ancilla (HT)† HT

Ground state subspace is spanned by t =6 t =4

1 HT (HT)† z,0 = z ( 1 + 3 + 5 + 7 ) t =5 | i p8 | i | i | i | i | i g0

(0, 1) (0, 2) (0, 1) (0, 4) (0, 1) (0, 5) (0, 3) (0, 8) (0, 3) (0, 6) (0, 3) (0, 7) + H z ( 2 + 8 )+HT z ( 4 + H6 ) ! HT (1, 1) (1, 2) (1, 1) (1, 4) (1, 1) I (1, 5) | i | i | i | i | i (1, 3) | i(1, 8) | (1,i3) (1, 6) (1, 3) (1, 7) z,1 = z,0 ⇤ some ancilla state | i | i g0 for z 0, 1 G (q, z, t, j): q =1,...R,z 0, 1 ,t 1,...,8 ,j 0,...,7 2 { } { 2 { } 2 { } 2 { }} Two-qubit gates

t =1 H H

t =8 t =2

Two-qubit gate gadgets: 4096-vertex H H

graphs built from 32 copies of the t =7 t =3

single-qubit graph, joined by edges (HT)† HT

t =6 t =4

HT (HT)† and with some added self-loops t =5 x32

(0, 1) (0, 2) (0, 1) (0, 4) (0, 1) (0, 5) (0, 3) (0, 8) (0, 3) (0, 6) (0, 3) (0, 7)

(1, 1) H (1, 2) (1, 1) HT (1, 4) (1, 1) I (1, 5) (1, 3) (1, 8) (1, 3) (1, 6) (1, 3) (1, 7)

Single-particle ground states are associated with one of two inputG (q, z, t, j): q =1,...R,z 0, 1 ,t 1,...,8 ,j 0,...,7 regions or one of two output regions: { 2 { } 2 { } 2 { }}

(States also carry labels associated with the logical state & complex conjugation.)

Two-particle ground states encode two-qubit computations: 1 + U p2 + ⌦ | i + ⌦ | i! Constructing a veriﬁcation circuit Connect two-qubit gate gadgets to implement the whole veriﬁcation circuit, e.g.:

Some multi-particle ground states encode computations:

+ U + U | i 1| i 1| i + U + U + U U 1| i 1| i 2 1| i But there are also ground states that do not encode computations (two particles for the same qubit; particles not synchronized). To avoid this, we introduce a way of enforcing occupancy constraints, forbidding certain kinds of configurations. We establish a promise gap using nonperturbative spectral analysis (no large coefficients). Open questions • Remove self-loops? • Complexity of other models of multi-particle quantum walk - Attractive interactions - Negative hopping strength (stoquastic; is it AM-hard?) - Bosons or fermions with nearest-neighbor interactions - Unrestricted particle number • Complexity of other quantum spin models defined on graphs - XY model - Antiferromagnetic Heisenberg model