CCustom fermionic codes for quantum simulation
Riley W. Chien ∗ and James D. Whitfield † Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755
Simulating a fermionic system on a quantum computer requires encoding the anti-commutingfermionic variables into the operators acting on the qubit Hilbert space. The most familiar ofwhich, the Jordan-Wigner transformation, encodes fermionic operators into non-local qubit oper-ators. As non-local operators lead to a slower quantum simulation, recent works have proposedways of encoding fermionic systems locally. In this work, we show that locality may in fact be toostrict of a condition and the size of operators can be reduced by encoding the system quasi-locally.We give examples relevant to lattice models of condensed matter and systems relevant to quantumgravity such as SYK models. Further, we provide a general construction for designing codes to suitthe problem and resources at hand and show how one particular class of quasi-local encodings canbe thought of as arising from truncating the state preparation circuit of a local encoding. We endwith a discussion of designing codes in the presence of device connectivity constraints.
I. INTRODUCTION
A well-known duality between spins and fermions dueto Jordan and Wigner [1] has been famously employedin the solutions of spin chains [2]. Recent years haveseen a new applications of spin-fermion dualities as a wayof encoding systems of indistinguishable fermions into asystem of distinguishable qubits. Such transformationsare employed in the simulation of fermionic systems onquantum computers. The idea of a quantum simulatorwas conceived in [3] and further expanded upon in [4].It is now expected that quantum computers will becomean invaluable tool in the study of physical properties ofstrongly correlated systems which are out of reach of clas-sical computers such as for example quantum chemistry[5]. Target problems include chemical reaction mecha-nisms [6] and the Hubbard model [7].Working in second quantization, it is necessary to en-code the anti-commuting nature of the fermionic opera-tors into the local qubit degrees of freedom. The solu-tion used in the Jordan-Wigner transformation (JW), isto map local operators on one side of the duality to non-local operators on the other side. When mapping fromfermions to spins, the Pauli Z strings are non-local a j → (cid:89) k In this section we review a certain construction intro-duced by Setia et al. in [17] for representing fermionicsystems in terms of qubits in a local fashion. This methodis reminiscent of the construction of [29] for generatinghighly entangled states using Majorana modes at ends ofnanowires.Typically, our problem setting will be that we are in-terested in some property of a system of N fermionicmodes with dynamics governed by a Hamiltonian H = (cid:88) jk h jk a † j a k + (cid:88) jklm h jklm a † j a † k a l a m + . . . (3)where the fermionic operators satisfy the usual anti-commutation relations { a j , a † k } = a j a † k + a † k a j = δ jk and { a j , a k } = { a † j , a † k } = 0 . As explored in [30], the Hamil-tonian could also couple the fermions to a gauge field, forexample in the context of a high energy physics simula-tion.It will be useful to work in the Majorana basis offermionic operators a j = 12 ( γ j − + iγ j ) (4) a † j = 12 ( γ j − − iγ j ) (5) { γ j , γ k } = 2 δ jk . (6)From these, we can form two types of operators quadraticin the Majoranas which we will from here on refer to asedge and vertex operators, A jk = − iγ j − γ k − (edge) (7) B j = − iγ j − γ j (vertex) . (8)These operators suffice to generate the full algebra ofparity preserving fermionic operators. The vertex opera-tor B j is the parity operator for the mode j and the edgeoperator A jk is involved in all hopping, pairing, and scat-tering processes. Note that edge operators involve onlyodd-indexed Majoranas. Appropriately multiplying anedge operator by vertex operators allows for coupling twofermionic modes by their even indexed Majoranas. Forexample, a hopping term can be expressed as a † j a k + a † k a j = − i ( A jk B k + B j A jk ) / . (9)We here explicitly place the fermionic system on agraph which we refer to as the interaction graph withfermionic modes associated to vertices j ∈ V . Whenone writes the fermionic Hamiltonian in terms of edgeand vertex operators, if an edge operator, A jk , couplingmodes j, k is required to write the Hamiltonian in thisform, then an edge is placed on the graph connectingvertices j and k , ( j, k ) ∈ E . The graph is the given asthe pair Γ = ( V, E ) . Further, we can identify the set ofplaquettes on the graph P and the boundary of a givenplaquette p is the set of edges which form it, ∂p .For each vertex j on the graph, we place a number ofqubits equal to half the degree of the vertex, d ( j ) / (ac-tually (cid:100) d ( j ) / (cid:101) but we will assume graphs of even degreefor simplicity). We next define d ( j ) Pauli operators act-ing on the d ( j ) / qubits on site j which we will refer toas local Majoranas c j , . . . , c d ( j ) j ∈ P d ( j ) / (10)We use the letter c for the local Majoranas to distin-guish these Pauli operators from the physical Majoranafermion operators γ . These local Majoranas may be any-thing except that they must satisfy the following prop-erties: (1) they must satisfy Majorana anti-commutationrelations with the other operators defined on that ver-tex (they commute with operators defined on other ver-tices) and (2) they must generate the full Pauli group (a)(b) FIG. 1: (a) Local Majorana modes for a degree d = 6 vertex. (b) Two types of plaquette stabilizers for thetriangular lattice with the above encoded Majoranas.(Left) Orange lines connecting Majoranas indicate thecorresponding edge operators. (Right) Thecorresponding qubit operators.on d ( j ) / qubits. Any choice of definition for the localMajoranas is related to any other choice by a Cliffordcircuit acting only on qubits at that vertex. In the fig-ures throughout this paper, we will always use Jordan-Wigner to encode the local Majoranas on a given ver-tex starting from the top clockwise, { c j , c j , c j , c j , . . . } →{ X j , Y j , Z j X j , Z j Y j . . . } (See Fig. 1). A choiceof definition for the Majoranas based on Fenwick treescan give a decrease in the Pauli weight from O ( d ) to O (log( d )) . As such, the Fenwick tree choice will bepreferable for graphs of large degree. We refer readersto [14, 17] for a discussion of Fenwick tree encodings.The recently proposed ternary tree construction of [11]would provide a further reduction in Pauli weight.Each local Majorana is associated to one of the edgesincident to the vertex. Each edge then has two localMajoranas associated to it, one from each vertex at itsendpoints and we will define our encoded edge operatorsto be ˜ A jk = (cid:15) jk c pj c qk (11)where (cid:15) jk = +1 if j < k and = − if j > k and k isthe p -th neighbor of j and j is the q -th neighbor of k .Encoded vertex operators are a product of all Majoranasat a given vertex, ˜ B j = i d ( j ) / c j c j . . . c d ( j ) j . (12) If Jordan-Wigner is used to encode the local Majoranason a given site, then the vertex operator will be Pauli Zacting on all the qubits on that vertex so the occupancyof the mode is stored in the collective parity. Anotherchoice could be made such that the occupancy is storedin a single qubit. One could achieve this an appropriateapplication of CNOTs.Finally, for each plaquette p on the graph, we havea stabilizer which is given by a product of all the edgeoperators around the boundary of the plaquette S ( ∂p ) = i | ∂p | (cid:89) ( j,k ) ∈ ∂p ˜ A jk . (13)Whereas in the toric code, the plaquette stabilizers de-tected the presence of a flux, here the plaquette stabiliz-ers detect the presence of a flux without an accompany-ing charge and vice-versa. In dimensions higher than 1,the logical subspace is that in which charges have fluxesattached. Recall that charge-flux pairs are fermionic innature which provides the basis for this construction. Anexample on a triangular lattice is shown in Fig. 1. Forgeometries with non-contractible loops, e.g. a torus, wefix boundary conditions to be periodic with a stabilizerconsisting of a product of edge operators around the non-contractible loop.As a final consideration, we consider the odd fermionicparity sector. On a graph with even degree it is notpossible to directly encode an odd number of particles.One must introduce an additional non-physical auxiliarymode and create a pair of particles with one occupy-ing the auxiliary mode. The total parity of the physi-cal modes is then odd and the odd-parity simulation canproceed. If however there is a vertex on the graph withan odd degree, then there will be a single unpaired Majo-rana operator on that vertex. Acting with the unpairedMajorana operator can create or destroy a single particleat that vertex without violating any of the stabilizers. Asa simple example without any stabilizers, we consider a1D chain with open boundaries. At the two ends of thechain, there are two unpaired local Majorana operators.If we for example act with the unpaired Majorana on thefirst site in the chain, we create a single particle. In thisway, acting with unpaired Majorana operators changesour parity sector. We can then proceed with our odd-parity sector simulation. In the next section we showthat if we pair the local Majorana operators at the endsof the chain, we impose periodic boundary conditions andlose the ability to enter the odd parity sector. A. 1D chain recovers Jordan-Wigner To further illustrate the construction in a familiar set-ting that we hope will give some intuition for the latersections, we will encode a 1D chain of fermions with anonsite potential and periodic boundary conditions. TheHamiltonian of this system is H = t (cid:88) j ( a † j +1 a j + a † j a j +1 ) + U (cid:88) j a † j a j (14) = − it (cid:88) j ( A j,j +1 B j +1 + B j A j,j +1 ) + U (cid:88) j B j . (15)Each vertex in this 1D chain obviously has d = 2 so asingle qubit is placed at each vertex. We choose c j = Y j , c j = X j for the Majoranas at each site. All theedge operators are then ˜ A j,j +1 = X j Y j +1 and the vertexoperators are ˜ B j = Z j . The transformed Hamiltonianthen takes the familiar form of the XY chain ˜ H = t (cid:88) j ( X j X j +1 + Y j Y j +1 ) + U (cid:88) j Z j . (16)Notice that given the basis chosen for the local Majoranasat each vertex, the Jordan-Wigner transformation is re-covered. Indeed, an edge operator between two modesnot nearest-neighbor connected is necessarily a productof edge operators in a path connecting the two targetedvertices A j,j + n = A j,j +1 . . . A j + n − ,j + n (17) ˜ A j,j + n = ( − i ) n − X j Z j +1 . . . Z j + n − Y j + n (18)giving back the Jordan-Wigner strings of Pauli Zs thatall local fermionic encodings are attempting the alleviate.Finally, as we have periodic boundary conditions andthus a (large) loop in our interaction graph, we have a sta-bilizer which is the product of all edge operators aroundthe loop. This operator, which is given by S = (cid:81) j Z j ,corresponds to the fact that global fermionic parity ispreserved and constrained to be even. We could as wellchoose not to restrict to the subspace stabilized by theloop as previous described in our discussion of the oddparity sector. In that case, edge operators coupling sites , N would have a Pauli weight extensive in the systemsize. The total fermionic parity will always be a symme-try of the Hamiltonian by virtue of only even parity op-erators being physical. The above shows a consequenceof imposing the total parity to be a symmetry of thestates as well. This clear physical interpretation doesnot however generalize to plaquette stabilizers in higherdimensions. III. MAIN RESULT: CUSTOM FERMIONICCODES Our main result is centered on the idea that regardlessof the interaction graph determined by the Hamiltonianterms, we may choose to encode the system into whatevergeometry we wish. As such, we will begin discussing twoseparate graphs for the remainder of the paper, the inter-action graph as determined by the Hamiltonian and thesystem graph that we will encode with our qubit system. (a)(b) FIG. 2: (a) Two nearest-diagonal-neighbor couplings(left) fermion picture (right) qubit picture (b) The sametwo nearest-diagonal-neighbor couplings with diagonaledges omitted from the system graph.The system graph must have at least as many verticesas the interaction graph in order to accommodate thefermionic modes. Also, if an edge operator coupling twomodes is present in the Hamiltonian is expressed usingthe operators of (7 - 8), then a path must connect thosetwo vertices on the system graph. Otherwise the systemgraph may be arbitrary.We will discuss a number of useful modifications ofthe interaction graph. We will briefly discuss sparsifica-tions of the interaction graph - simply omitting edges inthe case of lattice models. We then discuss using a vir-tual geometry including virtual modes that provide pathsacross which interactions may take place. We there givean example of a system featuring all-to-all interactions.Next, we give a construction that allows for balanc-ing qubit resource requirements and code locality in thecase of lattice models through a blocking construction.We will find it convenient at that time to discuss statepreparation. Finally, we will discuss encodings in thepresence of constraints on connectivity between qubitsusing the recently proposed heavy-hexagon lattice as anexample. A. Omitting edges As shown above in (18), if two modes are meant to becoupled with an edge operator but the two modes do notshare an edge on the system graph, the two modes arestill able to be coupled together, but the interaction willbe not strictly local. This will lead to a generalization ofthe JW Z string where the intermediary vertices will allbe acted on by a product of two local Majoranas. Thesegeneralized Jordan-Wigner strings are similar to stringoperators in quantum error correcting codes.For concreteness, consider a L × L square lattice offermionic modes interacting with nearest-neighbors andnearest-diagonal-neighbors, those across the diagonal ofa square. An example of such a problem is the nearest-diagonal-neighbor Hubbard model. The vertices of theinteraction graph are then of degree d ( j ) = 8 . If wechoose the system graph to match the interaction graph,then we require qubits at every vertex giving L qubitsin total. As shown in Fig. 2b, we omit the diagonal edgesfrom our system graph such that each vertex has onlydegree d ( j ) = 4 . The diagonal edge operators are thena product of two nearest-neighbor edge operators. Thepath taken around the square plaquette does not matteras the upper and lower path in each case are equal up tomultiplication by a stabilizer. We see that in this case,the Pauli Weight of the qubit operators is smaller forthe two paths without the diagonal edges in the systemgraph. This can be seen in Fig. 2. Also the presenceof the additional qubits also increases the Pauli weightof the nearest-neighbor edge operators and the vertexoperators. So, we have shown that strict locality is notalways optimal and relaxing to quasi-locality is in somecases beneficial.The system graph can be sparsified arbitrarily relativeto the interaction graph to save qubits as long as pathsconnect modes that must be coupled with edge operators.In this construction, the qubit requirement is determinedby the vertex degrees and not the number of edges as inthe local mappings of [10, 21]. Thus, if a reduction ofqubit requirement is sought, one should delete edges withthe aim of reducing the degrees of vertices with targetdegrees being even numbers.We would like to mention here that if our system has asquare lattice interaction graph and we sparsify the graphin certain ways, we recover the auxiliary qubit mappingsof Steudtner and Wehner [19]. Thus, this constructioncontains the auxiliary qubit mappings as special case. B. Adding virtual modes Additional virtual modes can be added to the systemgraph which can in some cases lead to a reduction inthe Pauli weight of the transformed Hamiltonians. Asalways we require closed loops in the system graph tobe stabilized by the corresponding plaquette operators.Virtual modes are stabilized by their vertex operators asthey will always be unoccupied and so have parity +1 .Using virtual modes to reduce Pauli weights could beespecially useful in cases where one has a complete ornearly complete interaction graph. Nearly complete in-teraction graphs (within spin sectors) are known to arisein small molecular Hamiltonians using atomic orbital ba-sis sets leading to large simulation costs with strictly FIG. 3: (left) Schematic of an operator coupling twomodes at the boundary of the MERA geometry. Theorange arc connecting the two endpoints indicates thepath of the generalized Jordan-Wigner string. The paththrough the virtual space depicted in gray is shorterthan the path along the boundary. (right) The qubitoperator corresponding to a coupling with thegeneralized Jordan-Wigner string following the discretegeodesic.local encodings [16]. A number of all-to-all interact-ing fermionic models have also become popular in recentyears in the study of scrambling of quantum informationand of AdS/CFT, the most notable of which is the SYKmodel [31, 32]. 1. All-to-all coupled fermions We now give an example of a system for which thequantum simulation cost is decreased by using virtualmodes. The SYK model consists of N Majoranafermions with random strength q -body interactions cou-pling all Majoranas. Proposals regarding quantum simu-lation of the q = 4 SYK model have previously been putforward [33–35].We will consider the q = 2 case, H = − i N (cid:88) j We will now consider an approach to finding a balancebetween locality and qubit requirement for a system on a L × L square lattice. A linear (JW) geometry will require L qubits but will suffer from long JW strings whereasa strictly local encoding will require roughly L qubits(minus a few on the boundaries). Again, we can reducethe required number of qubits by relaxing the necessityfor strictly local interactions. We will divide the sys-tem into a number of blocks. Interactions within blockswill be non-local incurring Jordan-Wigner strings thatincrease in length with the size of the blocks. Interac-tions between blocks, however, remain local in that thePauli weight of operators are independent of the size ofthe full lattice.The blocking goes as follows. Partition the latticeinto the desired number of blocks, b , determined by theavailable resources. Treat the modes within the block asthough they were on a 1D chain. The lattice is then acoarser lattice with each vertex connected to a 1D chain.The first mode in the chain remains connected to the lat-tice such that those vertices are of degree d = 5 in thebulk of the system and so require qubits. The totalnumber of qubits required is then L + 2 b (up to bound-aries). With this construction, we are free to interpolatebetween a strictly local encoding with L blocks of size and block of size L by choosing blocks to be of thedesired size.The idea of using a segmented encoding was explored in[14] where segmented versions of the Bravyi-Kitaev andFenwick tree encodings were explored for the 2D Hubbardmodel. 1. As truncated state preparation We now hope to provide an intuitive picture for theblocking construction. This will also be a convenienttime to address state preparation. For simplicity, we willagain consider a square lattice of dimension L × L . Aspreviously mentioned, this encoding manages to encodethe fermions in a local way by representing them as ex-citations of the toric code which are odd under particleexchange. Thus, to use this encoding, one must preparea toric code state which is well known to be topologicallyordered and therefore long-range entangled [40].Utilizing a MERA quantum circuit, one can preparea toric code state by introducing entanglement scale-by-scale beginning with long-range entanglement and endingwith entanglement at the final lattice scale [41]. The cir-cuit is comprised of a number of levels U = U . . . U log L .Each level k takes as input a state on a lattice of linearsize l and a number of ancilla qubits and outputs a stateof linear size l , U k | ψ l (cid:105) | . . . (cid:105) = | ψ l (cid:105) . (22)The state at each level has four times as many plaque-ttes as the previous level and so, with corrections at theboundaries, has about four times as many qubits in addi-tion to the L qubits associated to the fermionic modes.Again, at the final level, the total number of qubits is L with correction at the boundaries. Each layer con-sists of acting with Hadamards and CNOTs locally withrespect to the lattice at the given level. The exact formof the circuit can be found in [41]. Upon application ofthe log L layers of the circuit, the toric state is prepared.At this point, a constant depth unitary is performed tosatisfy the stabilizers which differ slightly from the truetoric code model. The exact form of this circuit dependson the basis chosen for the local Majoranas. If a Jordan-Wigner basis is chosen, the circuit consists merely of asingle layer of Pauli X and Z gates.The blocking construction can be thought of as a trun-cation of the state preparation circuit. Truncating thecircuit results in a coarse-grained toric code state relativeto the lattice of fermionic modes. As such, the fermionicoperators which are local due to the topological order ofthe toric code state, are now only local with respect tothe toric code lattice. Operators may be non-local up tothe scale of the toric code lattice spacing. The benefit,however, is a savings in qubit resources as each MERA FIG. 6: Here we show the trade-off between qubitrequirement and operator locality. The circles representthe fermionic modes while the lines represent the latticeof the topologically ordered state underpinning the localencoding. Each level represents a case in which oneadditional level of the state preparation circuit isapplied, creating a finer lattice for the underlying toriccode. Operators are local only with respect to thelattice spacing of the toric code state. At the top levelwith no topological order, the fermionic operators maybe fully non-local e.g. long JW strings. At the bottomlevel, the operators are fully local with respect to thelattice spacing of the fermions but twice as many qubitsare required.layer requires additional qubits. Thus by utilizing topo-logical order on a coarse-grained lattice, one can realizethe trade-off between operator locality and qubit require-ment as depicted in Fig. 6.Although each level of the MERA unitary is local withrespect to the lattice at each level, it is not local withrespect to the final lattice of qubits. With strictly localoperations, preparing the topologically ordered toric codestate takes a time proportional to the linear size of thelattice [42]. 2. Further generalizations We now discuss a number of ways this above construc-tion can be generalized. Going beyond a square lattice ofblocks, one could perform the partitioning of the systemof N modes into a set of general sites S = { s , . . . , s | S | } each containing a number of modes n ( s i ) and where eachsite is connected to d ( s i ) others on the lattice of sites.Then given the construction above, the total number ofqubits required would be of qubits = N + | S | (cid:88) i (cid:24) d ( s i )2 (cid:25) . (23)Further, one is free to choose any encoding for themodes within each site. For example, one could choose toencode the modes on a 1D chain as described or chooseto use a Fenwick tree [14] or Jiang et al.’s ternary treeencoding [11].To emphasize the generality of the construction weare proposing here and to illustrate how the geometryof the interactions should inform the geometry of thequbit system, we would like to sketch how one mightencode a system of interest lately, that being a latticeof SYK islands [43, 44]. The system is a lattice of is-lands S = { s , . . . , s | S | } each containing n ( s i ) Majoranafermions with two types of interactions, quartic interac-tions between Majoranas within each island with randomstrength and quadratic interactions between Majoranason adjacent islands on the lattice. We propose that sucha system would be best simulated as a lattice of sites,where the modes on each site, are placed at vertices onthe “ N branches” geometry or a hierarchical geometry asdescribed above. The chosen geometry is then attachedto the lattice at the central vertex. We highlight sucha system as it frustrates many of the existing encodingschemes, featuring both highly connected regions as wellas a notion of locality.Finally, we reiterate that one is free to use any ba-sis for the local Majorana operators at each vertex. AJordan-Wigner encoding was chosen for simplicity butan improvement in Pauli weight can be achieved by us-ing a different local basis. For example, a Fenwick treebasis for local Majorana operators as proposed in [17] orJiang et al.’s ternary tree basis [11] would give a Pauliweight for Majorana operators logarithmic in the numberof qubits. D. Device connectivity constraints Many of the quantum computing platforms under de-velopment are subject to connectivity constraints. No-tably, these include superconducting qubits which haverecently been used to achieve a “quantum supremacy”result [46]. The processor used in the supremacy ex-periment features qubits laid out on a square grid withnearest-neighbor connectivity. Recent work on increas-ing the capabilities of quantum computers as measuredin so-called quantum volume [47] has led to progress ondevices with qubits laid out on lower-degree graphs. Inparticular, the heavy-hexagon lattice has been identifiedas a candidate system geometry for realizing quantum er-ror correction while mitigating hardware challenges pre-sented by cross-talk and frequency collisions [45]. Thislattice features qubits placed on vertices of a hexagonallattice as well as on edges. FIG. 7: (top left) The heavy-hexagon geometry aspresented in [45]. 65 qubits are shown. (top right) Weshow a choice of grouping pairs of qubits together toencode the degree vertices. (bottom) We show thesystem graph where each larger circle corresponds toone of the 49 encoded fermionic modes. Smaller circlesrepresent the qubits associated to each mode.On such devices subject to connectivity constraints, itmay be preferable to encode a fermionic system into agraph informed by the connectivity of the device. Tothat end, we present here a candidate geometry for en-coding a fermionic system into a heavy-hexagon lattice.In Fig. 7, we show a qubit heavy-hexagon lattice. Toeach degree vertex, we associate an additional qubit.Thus, with degree vertices, we are able to encode49 fermionic modes into the 65 qubit heavy-hexagon lat-tice. As above, coupling modes which do not share anedge in the system geometry will require edge operatorscontaining generalized Jordan-Wigner strings.Designing device-specific encodings for other platformswould proceed similarly, identifying qubits to group to-gether to encode vertices of appropriate degree and em-bedding a problem withing the device-informed geome-try. We leave an investigation of optimal device-specificgeometries to future work. IV. CONCLUSION In this paper, we have presented a very generalconstruction for designing quantum codes to simulatefermions on quantum computers. The construction re-alizes the trade-off between qubit resources and operatorlocality in such a way that one can tailor the encoding tobest fit the resources at hand. We have also shown thatin some cases, locality is too strict of a constraint and oneis better off seeking a quasi-local encoding. We showedthis occurs in systems such as square lattice models with0nearest- and nearest-diagonal-neighbor coupling where itis best to simply encode the square lattice and realize thediagonal couplings quasi-locally. We also presented thecase of a fermionic system with all-to-all connectivity anddemonstrated that one should encode this system witha virtual geometry so that generalized Jordan-Wignerstrings traverse paths through the virtual geometry. Wediscussed how quasi-local codes can be interpreted as lo-cal codes with truncated state preparation circuits. Fi-nally, we considered the design of custom codes to suit de-vice connectivity constraints. We expect that the encod-ing construction presented here will find use in quantumsimulation experiments ranging from quantum chemistry, to quantum gravity, to condensed matter physics. ACKNOWLEDGEMENTS RWC and JDW were funded by the NSF (PHYS-1820747) and the Department of Energy (Grant DE-SC0019374). 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