Revealing symmetries in quantum computing for many-body systems

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Revealing symmetries in quantum computing for many-body systems © 2024 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft Published version Leeuwen, Robert van Leeuwen, R. V. (2024). Revealing symmetries in quantum computing for many-body systems. New Journal of Physics, 26, Article 103023. https://doi.org/10.1088/1367-2630/ad8677 2024 PAPER • OPEN ACCESS Revealing symmetries in quantum computing for many-body systems To cite this article: Robert van Leeuwen 2024 New J. Phys. 26 103023 View the article online for updates and enhancements. 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PAPER Revealing symmetries in quantum computing for many-body systems Robert van Leeuwen Department of Physics, Nanoscience Center, University of Jyväskylä, Jyväskylä, Finland E-mail: roleeuw[email protected] Keywords: symmetries, quantum computing, many-body systems Abstract We develop a method to deduce the symmetry properties of many-body Hamiltonians when they are prepared in Jordan–Wigner form in which they can act on multi-qubit states. Symmetries, such as point-group symmetries in molecules, are apparent in the standard second quantized form of the Hamiltonian. They are, however, masked when the Hamiltonian is translated into a Pauli matrix representation required for its operation on qubits. To reveal these symmetries we prove a general theorem that provides a straightforward method to calculate the transformation of Pauli tensor strings under symmetry operations. They are a subgroup of the Clifford group transformations and induce a corresponding group representation inside the symplectic matrices. We finally give a simplified derivation of an affine qubit encoding scheme which allows for the removal of qubits due to Boolean symmetries and thus reduces effort in quantum computations for many-body systems. 1. Introduction Recent years have seen steady progress in the design of quantum computing devices with an increasing number of qubits [1]. These developments are especially interesting for applications in many-body physics [2] and quantum chemistry [3] as quantum computers are intrinsically advantageous to solve quantum many-body problems. Moreover instead of the exponential computational cost required to perform such calculations on classical computers, on quantum computers the cost is envisioned to grow only polynomially. This has spurred many efforts in the quantum chemistry [3] and condensed matter physics [2] community to develop schemes to solve interesting many-body problems on a quantum computer. For the simulation of such quantum systems we need to represent the many-body Hamiltonian as a linear operator on a system of qubits. This is usually done by first writing the Hamiltonian in a second quantized form, i.e. as an operator in Fock space, and then use the Jordan–Wigner transformation [4–6] to translate the creation and annihilation operators into appropriate tensor products of Pauli matrices that can act on systems of qubits. The anti-symmetrized kets or Slater determinants in Fock space are then mapped to tensor products of qubits, while the anti-symmetry properties of the Fock space states are now encoded into the Pauli product form of the Hamiltonian. Although this representation is exceedingly useful for practical calculations, the Jordan–Wigner transformed representation of the Hamiltonian generally conceals the symmetry properties of the original second quantized form. One of the main results of this work is the derivation of a practical way to recover these symmetries directly from the Jordan–Wigner form of the Hamiltonian. The preservation of symmetries in quantum computing is often essential in attaining the correct computational output. As was pointed out in [7] there is often no convenient basis in the space of multiple qubits that is also an eigenbasis of a relevant symmetry operator and when the symmetry is not imposed there is a risk of mixing in the wrong symmetry states leading to a meaningless result. For this and other reasons the related question of enforcing the right symmetries in a calculation has been studied in several works [7–12]. Apart from being of intrinsic scientific interest, the elucidation of symmetries also has important practical applications in the reduction of the number of qubits needed and therefore several © 2024 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft New J. Phys. 26 (2024) 103023 R v Leeuwen techniques have been developed to this aim. A recent scheme was developed by Picozzi and Tennyson [12] which uses a clever affine encoding technique to address the issue. In this work we give a considerably shorter proof of their main result and illustrate all our results with the elucidating example of the Hubbard dimer, a system that was studied before in the context of qubit reduction using projection and shift operators [13]. The main result of this work, however, addresses a different issue and is concerned with the question how unitary operations that represent permutation symmetries in the fermionic representation of the Hamiltonian are represented in the Jordan–Wigner form of the Hamiltonian. We prove a general theorem on how these operators are constructed and act on Pauli strings and show how they can be used to construct symmetry invariant Hamiltonians directly in the Jordan–Wigner representation. We finally like to point out that recently an iterative scheme based on conserved charges has been implemented for qubit reduction [14]. However, unlike the work in [12], it does not allow for an easy identification of the symmetry labels that characterize the many-body systems at hand. The paper is structured as follows: we first give a discussion of the permutational symmetries of Hamiltonians defined in a local atomic basis, such as Hubbard lattices, and review their symmetry properties. Next we illustrate, by means of an example, the masking of these symmetries after Jordan–Wigner transformation and subsequently prove a general theorem to resolve this problem and which constitutes the main result of this work. We then address the issue of qubit reduction by symmetries as was discussed in [12] in which we make a small correction to their recent proof and provide a much shorter proof instead using the proof technique developed in the first part of our work. We finally present an application and our conclusions. 2. Revealing symmetries in Jordan–Wigner transformed Hamiltonians 2.1. Permutation symmetries in many-body Hamiltonians In quantum chemical and condensed matter applications one commonly studies second quantized molecular Hamiltonians of the general form ˆ H= M ∑ ij hija† iaj+ M ∑ ijkl vijkl a† ia† jakal(1) where a† iand aiare creation and annihilation operators for electrons in one-particle state i[15,16]. Let us to be concrete assume that the Hamiltonian represents a molecule in a localized basis, such as Löwdin orthogonalized atomic orbitals [15,17], and that the coefficients hij and vijkl represent oneand two-electron integrals over such orbitals. If the molecule has a non-trivial point group, then every point group operation will permute a given local orbital from one atomic site to another. Since the Hamiltonian of the molecule is invariant under point group symmetries it then follows that for such a permutation we have the following identities: hP(ij)=hij vP(ijkl)=vijkl (2) where Pis a permutation of the labels of the localized orbitals and and we used the notation P(i1,...,iM) = P(i1)...P(iM). Equivalently we can write that the Hamiltonian of equation (1) has the symmetry ˆ H= M ∑ ij hija† P(i)aP(j)+ M ∑ ijkl vijkl a† P(i)a† P(j)aP(k)aP(l)(3) for any permutation Pof the localized orbitals compatible with the point group symmetry. We note that according to Cayley’s theorem any finite group is isomorphic to a subgroup of a permutation group. Therefore any symmetry operation for any finite group, and in particular any point group, can thus be represented by a permutation P. In this work we study the general class of Hamiltonians that have symmetries of the kind displayed in equation (3), whether they are derived from a molecular model or not, as may be the case for certain model Hamiltonians in condensed matter physics. By making various simplifying assumptions about the one-electron matrix elements hij and the two-electron integrals vijkl in equation (1) one can derive various classes of popular model Hamiltonians that are widely studied in chemistry and physics. In chemistry this encompasses, for example, the Parr–Pariser–Pople Hamiltonians [16,18,19] which are used in the study of complex unsaturated molecules while in condensed matter physics an important class consists of the Hubbard Hamiltonians [20] which are widely studied as they exhibit a metal–insulator transition and may offer insights in the currently still 2 New J. Phys. 26 (2024) 103023 R v Leeuwen marginally understood phenomenon of high-temperature superconductivity. The general form of the Hubbard Hamiltonian is ˆ H=−t∑ ⟨i,j⟩,σ (a† iσajσ+a† jσaiσ)+U∑ i a† i↑ai↑a† i↓ai↓(4) where <i,j>is a notation for nearest neighbor pairs in a lattice of a certain symmetry, and σ∈{↑,↓} denotes the electron spin. The Hubbard systems are studied as standard models for strongly correlated electrons [20] and consequently it has been a long standing goal to solve these models for sufficiently large number of lattice sites to exhibit the behavior of strongly correlated extended systems. It is therefore not surprising that in the quantum computing community there has been a large interest in these systems [21–24] since a quantum computer may be a novel way to reach that long standing goal and indeed there have been important recent advances [23,24] in that direction. To solve a Hubbard cluster, or generally a Hamiltonian of the form of equation (1), on a quantum computer one needs to translate the terms in the Hamiltonian from creation and annihilation operators to tensor products of Pauli matrices as is routinely done with the Jordan–Wigner transformation [4–6]. We can then ask what happens to the permutation symmetries of equation (3) for the new representation of the Hamiltonian; in other words, how do the various Pauli operators transform among one another under the permutation symmetries? Answering this question is one of the main topics of this work, in which we will derive a general law for the transformation of Pauli operators under symmetries. However, before addressing the general case it is helpful to clarify the issue using an example, which we will present in the subsequent section. 2.2. An illustrative example To elucidate the general problem we start by giving an example that is as simple as possible but still sufficiently interesting to illustrate the key aspects and to which we can refer back later when discussing the general case. The example is that of spin-less fermions on a triangular cluster [25] of three atoms described by the Hamiltonian ˆ H=a† 1a2+a† 2a1+a† 1a3+a† 3a1+a† 2a3+a† 3a2(5) where a† iand aiare the creation and annihilation operators for a particle on site i. For this system any permutation (1,2,3)→(P(1),P(2),P(3)) of the labels of the creation and annihilation operators leaves the Hamiltonian invariant, which constitutes the symmetry group S3, or C3vin point group language. Let us study this Hamiltonian in the 2-particle subspace of Fock space [16] spanned by the basis vectors |12⟩=a† 2a† 1|0⟩ |13⟩=a† 3a† 1|0⟩ |23⟩=a† 3a† 2|0⟩(6) and consider the cyclic permutation P(123)=(231). This permutation of labels induces a permutation U(P)|ij⟩=|P(ij)⟩on the 2-particle states given by U(P)|12⟩=|23⟩U(P)|13⟩=|21⟩=−|12⟩U(P)|23⟩=|31⟩=−|13⟩(7) where the minus signs in the equations arise from the anti-commutation properties of the creation operators in equation (6). The matrix representations Hof the Hamiltonian ˆ Hand of U(P) of equation (7) in the basis {|12⟩,|13⟩,|23⟩}are readily calculated to be H=  0 1 −1 101 −1 1 0  U(P) =   0−1 0 0 0 −1 1 0 0  (8) where U(P) is a unitary matrix which we can check to be commuting with H, i.e. [H,U(P)] = HU(P)−U(P)H=0⇒H=U(P)HU†(P)(9) and therefore His invariant under a unitary conjugation induced by P, which precisely is what defines a symmetry operation for H. Similar considerations can be done in the one-particle subspace of dimension 3 and the zero-and three particle subspaces of dimension 1. This gives commuting blocked matrices H=H0⊕H1⊕H2⊕H3and U(P) = U(P)0⊕U(P)1⊕U(P)2⊕U(P)3(where the subindex of the blocks refers to the particle number) acting in the full 1 +3+3+1=8-dimensional Fock space. In general symmetries are represented by signed permutation matrices U(P)nthat commute with the Hamiltonian in each sector of fixed particle number n. 3 New J. Phys. 26 (2024) 103023 R v Leeuwen Let us now translate the problem to qubit space where we directly consider the 8-dimensional space spanned by the 3-qubit tensor products |ν⟩=|ν1,ν2,ν3⟩=|ν1⟩⊗|ν2⟩⊗|ν3⟩with νi∈{0,1}. Let now Xi,Yi,Zibe the standard Pauli matrices acting on qubit iand further denote σ± i= (Xi±iYi)/2. Then the creation operators in Jordan–Wigner form [4–6] are given by A† 1=σ− 1⊗Z2⊗Z3 A† 2=11⊗σ− 2⊗Z3 A† 3=11⊗12⊗σ− 3 and the corresponding annihilation operators follow from the replacement σ−→σ+in these expressions. From this we can then construct the corresponding form of the Hamiltonian and obtain the expression ˆ H=1 2[X1X2(1−Z1Z2) + X1Z2X3(1−Z1Z3) + X2X3(1−Z2Z3)] (10) where we used Yj=iXjZjto write the Hamiltonian solely in terms of Pauli Xand Zmatrices. This Hamiltonian does not have an obvious symmetry under relabeling of the indices. To find the symmetries we first need to construct the equivalent of the transformation U(P) but acting on tensor products |ν⟩. We denote the corresponding operator by CPand discuss its general construction in detail in the next section. If we use CP(vide infra) we find that the Xiand Zioperators transform according to CPX1C† P=X2Z1 CPX2C† P=X3Z1 CPX3C† P=X1Z2Z3 CPZ1C† P=Z2 CPZ2C† P=Z3 CPZ3C† P=Z1 (11) We thus see that the Zioperators transform cyclically according to the permutation Pbut that the Xi operators instead transform to a product of Xand Zoperators. We can then check that under these transformations we indeed find an invariance of the Hamiltonian, i.e. ˆ H=CPˆ HC† P(12) and therefore the transformations in equation (11) reveal a symmetry that was hidden in the explicit form of the Hamiltonian of equation (10). The actual structure behind the transformations in equation (11) remains mysterious at this point, but some insight is attained by writing them in the form of a so-called Clifford matrix or tableau [26–28] which we will define now. If CPXr1 1...XrM MZs1 1...ZsM MC† P=Xr′ 1 1...Xr′ M MZs′ 1 1...Zs′ M M(13) where r,sand r′,s′are M-dimensional vectors with entries 0 and 1 then the matrix MPthat maps vector (r,s) to vector (r′,s′)is called the Clifford matrix or tableau [26–28]; in general it is defined in a similar way for more general unitary transformations than CPthat we use here, but the definition above suffices at this point. In our example MPhas the explicit form MP=         001 100 010 110001 001100 001010         =(ΠP0 QPΠP)(14) where the empty block is just filled with zero entries. The upper left and lower right blocks are readily identified as the permutation matrix ΠPfor our symmetry, i.e. ΠP,ij has an entry equal to 1 when i=P(j) and has zero entries otherwise. It remains to explain the structure of the lower left block QPof the matrix MP; we will show that it is given by QP=LΠP+ ΠPL(15) where Lis the triangular matrix L=  000 100 110 (16) 4 New J. Phys. 26 (2024) 103023 R v Leeuwen in which all entries below the diagonal are filled with ones and the remaining entries with zeroes. Equation (15) is a special case of a general theorem valid for arbitrary M-qubit Hamiltonians. The derivation of that result is the content of the next sections. 2.3. Mapping symmetries from Fock space to multi-qubit space The mapping from Fock space states to occupation number or multi qubit states was already studied by Jordan and Wigner [4] and has been found to be great practical use in quantum computing applications. Here, however, we focus on an aspect of this mapping that has not received much attention. Since symmetries correspond to permutations of the single particle states, they lead to permutations of anti-symmetrized kets of Slater determinants in Fock space where the permutations typically introduce sign factors. On the other hand, multi-qubit space consists of tensor products of qubits that have no particular symmetry under permutations. This means that in the translation from a Fock space to a multi-qubit representation, those sign factors have to be introduced explicitly and in the following we will describe how to do this. As it is easy to loose track of factors and meanings of the quantum states we have tried to make the notation and formulation as clear and precise as possible. The final result of this section is a precise general definition of the operator CPthat we used in equation (11) for our motivating example. Let J= (j1,...,jN)be a multi-index and consider the many-particle states [16] |J⟩=|j1,...,jN⟩(17) representing a fermionic state with one particle in state j1, another in j2, etc. Such states are anti-symmetric, i.e. for any permutation P∈SNof Nlabels we have |P(J)⟩=|jP(1),...,jP(N)⟩= (−1)|P||j1,...,jN⟩(18) where |P|is the parity of the permutation, and therefore to choose a linearly independent set of many-body kets it is useful to define kets |J⟩with an ordering, like j1<j2< ... < jN. There are (M N)such ordered kets which form an orthonormal basis. We then define the creation operators a† ito create such many-body kets. If we start with the empty state |0⟩we have [16] |i1⟩=a† i1|0⟩ |i1,i2⟩=a† i2|i1⟩ |i1,...,iN⟩=a† iN|i1,...,iN−1⟩=a† iN...a† i1|0⟩.(19) The action of the corresponding annihilation operators aifollows from the definition of the adjoint of an operator and is found to be [16] ai|j1,...,jN⟩= N ∑ l=1 (−1)N+lδi,jl|j1...jl−1,jl+1,...jN⟩(20) i.e. removing jNhas a plus sign, removing jN−1a minus etc and this continues in an alternating way. The Hilbert space set of anti-symmetric N-particle kets on Mis often denoted HN= ΛNH1[25], i.e. the N-fold wedge product of one-dimensional tensors. The full Fock space is F=H0⊕H1⊕...⊕HMdimF= M ∑ k=0(M k)=2M.(21) The space relevant for quantum computing consists of all linear combinations of the M-fold tensor products of qubits, defined more precisely as CM 2={span(|ν1⟩⊗...⊗|νM⟩)|νj∈{0,1}}dimCM 2=2M.(22) As opposed to the anti-symmetric states |J⟩in Fock space the states in CM 2have no particular permutation symmetry and therefore this feature needs to be build in the operators acting on them. The basis states in F and in CM 2can related in a one-to-one manner by using an occupation number representation of the one-particle states representing the anti-symmetric kets in F. With |J⟩,j1< ... < jNwe associate a vector ν= (ν1,...,νM)for which νj1=νj2=...=νjN=1 while all other elements of νare zero. We then define the one-to-one mapping J:F →CM 2on basis states by J(|j1,...,jN⟩) = |ν1⟩⊗|ν2⟩⊗...⊗|νM⟩=|ν1...νM⟩(23) 5 New J. Phys. 26 (2024) 103023 R v Leeuwen and use linear extension to define the mapping on all of F, such that superpositions of anti-symmetric kets are mapped to superpositions of multi-qubit states. For example if M=3,N=2 then J(|12⟩+|13⟩) = |1⟩⊗|1⟩⊗|0⟩+|1⟩⊗|0⟩⊗|1⟩=|110⟩+|101⟩.(24) The translation of the creation operators a† k:F →F to operators A† k:CM 2→CM 2, and similarly for the annihilation operators, is furnished by the Jordan–Wigner transformation in which the creation and annihilation operators are represented as 2M×2Mmatrices acting on CM 2vectors [4–6]: Ak=11⊗...⊗1k−1⊗σ+ k⊗Zk+1⊗...⊗ZM(25) A† k=11⊗...⊗1k−1⊗σ− k⊗Zk+1⊗...⊗ZM(26) which allows for the translation of any second quantized Hamiltonian as in (1) to a qubit Hamiltonian of the form ˆ H= M ∑ ij hijA† iAj+ M ∑ ijkl vijkl A† iA† jAkAl(27) which, when written out in terms of Pauli matrix tensor products, represents a 2M×2M-matrix acting on multi-qubit states (note that our definition corresponds to the ordering of the labels as in equation (19), reversing the order gives another prescription). Our discussion so far aimed to set the stage for the question how symmetry operations are mapped from Fock space to multi-qubit space. In equation (18) we considered permutations that only reordered the contents of the ket. Now we extend this to a permutation P∈SMof all M>Nlabels and again we denote |P(J)⟩=|jP(1)...jP(N)⟩(28) which is of the same form except that now there are in general labels in P(J) that do not occur in J. For example for M=3, N=2 if |J⟩=|13⟩and P(123)=(231)then |P(13)⟩=|21⟩=−|12⟩. As this example shows, the labels |P(J)⟩may not be in ascending order, but we can reorder them using a permutation of N elements. The actual form of this permutation is not relevant, but its sign is important. Let the labels P(J) be reordered in ascending order to¯ J, i.e.¯ jk∈P(J)with¯ j1< ... <¯ jN. Then |P(J)⟩= (−1)πP(J)|¯ J⟩(29) where πP(J)is the parity (even or odd) of the permutation of Nelements that reorders P(J) to¯ J. Its value can be calculated as follows: we first construct the M×Mpermutation matrix ΠP,ij with entries equal to one if i=P(j)and zero entries otherwise. Then we consider the N×N-submatrix of ΠP, which we denote by (¯ J|J), having rows¯ Jand columns J. Then πP(J)is equal to the number or row swaps needed to transform (¯ J|J)to the N×Nidentity matrix. Equivalently we have (−1)πP(j)=det(¯ J|J)(30) i.e. the determinant of the submatrix (¯ J|J). For example, for the cyclic permutation P(123)=(231)we have ΠP=  001 100 010 (13|13)= (12|13) = (0 1 1 0).(31) Clearly we need only one row swap to convert (12|13)to the identity matrix and its determinant is equal to −1 which agrees with the sign in our example where |P(13)⟩=−|12⟩. With this preparation we are ready to generalize the derivation of our example in the previous section. Our starting point is the relation for N-electron multi-indices Kand L ⟨K|ˆ H|L⟩=⟨P(K)|ˆ H|P(L)⟩(32) where Pis a permutation of the Mlabels of the creation and annihilation operators in the second quantized form of ˆ Hthat keeps ˆ Hinvariant, as in equation (3). The relation above is a simple consequence of the fact that, written out in terms of creation and annihilation operators as in equation (19), the left and the right 6 New J. Phys. 26 (2024) 103023 R v Leeuwen hand side only differ in a renaming of all operators which can not affect the value of the expectation value. We thus have ⟨K|ˆ H|L⟩=⟨P(K)|ˆ H|P(L)⟩=∑ I,J⟨P(K)|I⟩⟨I|ˆ H|J⟩⟨J|P(L)⟩ =∑ I,J UKI (P)⟨I|ˆ H|J⟩U† JL (P)(33) where we sum over ordered multi-indices I,Jand we defined the unitary matrix UKI (P) = ⟨P(K)|I⟩= (−1)πP(K)⟨¯ K|I⟩= (−1)πP(K)δ¯ KI (34) where δ¯ KI is a multi-index Kronecker delta. The mapping P→U(P)is an (in general reducible) unitary representation of the symmetry group of the Hamiltonian and we have from equation (33) that ˆ H=U(P)ˆ HU†(P)⇒[ˆ H,U(P)]=0 (35) for all symmetries described by permutations P. If we evaluate equation (34) for the example in the previous section we exactly recover the matrix U(P) of equation (8). We now again consider the effect of a symmetry permutation in multi qubit basis. Our goal is to derive an operator CPthat leads to similar equation to (35) for the Hamiltonian in Jordan–Wigner form acting on multi-qubit states, i.e. we search for an operator CPdefined on CM 2such that ˆ H=CPˆ HC† P⇒[ˆ H,CP]=0 (36) for every permutation Pcorresponding to a symmetry of the Hamiltonian. The starting point of our definition of this operator is equation (29). If a permutation transforms anti-symmetric kets |J⟩to |P(J)⟩, then in ket |J⟩the states j1,...,jNare occupied, i.e. νj1=...=νjN=1 and all other occupation numbers zero, while in |¯ J⟩all states P(j1),...,P(jN)are occupied, i.e. νP(j1)=...=νP(jN)=1 and all other occupation numbers zero. The mapping |J⟩→|P(J)⟩can therefore be represented in occupation number language as a mapping CP|ν⟩= (−1)τP(ν)|ΠPν⟩(37) where ΠPis a permutation matrix with F2entries defined by ΠP,ij =1 if i=P(j)and zero otherwise (this is a slight abuse of notation as we defined ΠPbefore but with integer entries rather than integers mod 2, but this does generally not lead to confusion as in practice the matrix forms are identical). It remains to specify the value of the parity τP(ν)for a given νand P. This must correspond to the number of row swaps in (¯ J|J)given by πP(J)in the equivalent Fock space expression (29), which we now must translate to an occupation number expression that we can calculate given νand P. If positions Jin ascending order in vector νhave occupation 1, then we define [ν] = J. For example, if |ν⟩=|101⟩then [101] = 13. We then define τP(ν)to be the number of row swaps in submatrix ([ΠPν]|[ν]) of ΠPrequired to transform it to the identity matrix, or equivalently (−1)τP(ν)=det([ΠPν]|[ν]) (38) where [ν] = Jis a set of labels for the occupied states in νand [ΠPν] =¯ Jis a set of labels for the occupied states in the image vector ΠPν. So finally, with definition (38), we can then represent CPand its adjoint C† Pas operators on CM 2as CP=∑ ν∈FM 2 (−1)τP(ν)|ΠPν⟩⟨ν|(39) C† P=∑ ν∈FM 2 (−1)τP(ν)|ν⟩⟨ΠPν|.(40) These equations are the main result of this section and an overview of the general structure of our mappings is summarized in figure 1. We thus see again a general feature of the mapping from Fto CM 2; the anti-symmetry properties are not incorporated in the basis states |ν⟩but appear as explicit signs (−1)τP(ν)in the transformation operators. We will derive how these operators transform strings of X’s and Z’s in the next section. 7 New J. Phys. 26 (2024) 103023 R v Leeuwen The commutator therefore vanishes if we can find r′and s′such that modulo 2 0=ri·s′+si·r′(83) for all terms qi. This means that the vector (s′,r′)is in the kernel or nullspace of a matrix Kwhich has all the vectors (ri,si)as rows. The construction of this kernel is precisely the procedure outlined in [8], who then proceed to design a transformation that maps Pauli strings deduced from the kernel vectors to single Pauli X-matrices. We will now deviate from that reference as their additional computational problem was very much simplified by the alternative method of Picozzi and Tennyson [12] for which we gave the much shortened derivation above. Instead of mapping to single X-matrices the map is to single Z-matrices while a numerical construction as in [8] is avoided. To elucidate the general theory we give an example of the procedure in the next section, in which we furthermore also illustrate the theory of the earlier sections. 3.3. Example: removing three qubits for the Hubbard dimer We can now illustrate all the theory from the preceding two sections with a simple interacting quantum system, namely the Hubbard dimer. We label the creation and annihilation operators with the labels (1,2,3,4)=(↑1,↑2,↓1,↓2)where each entry σiinside the last brackets stands for spin σ∈{↑,↓}on Hubbard site i∈{1,2}. Then the Hubbard Hamiltonian takes the form ˆ H=−t(a† 1a2+a† 2a1+a† 3a4+a† 4a3)+U(a† 1a1a† 3a3+a† 2a2a† 4a4)(84) and which transformed Jordan–Wigner notation has the form ˆ H=−t 2(X1X2(1−Z1Z2) + X3X4(1−Z3Z4)) +U 4(2−Z1−Z2−Z3−Z4+Z1Z3+Z2Z4).(85) The conservation of number of particles with a given spin quantum number makes the Hamiltonian commute with the operators C↑=Z1Z2C↓=Z3Z4(86) corresponding to n↑= (1,1,0,0)and n↓= (0,0,1,1)in equation (73). However, it is readily seen that there are additional order two operators related to permutation of the Hubbard sites and in particular we consider the Boolean subgroup given by the unit permutation and P(1234)=(2143). The latter permutation is represented by a permutation matrix ΠPand a QPmatrix (calculated from equation (49)) with the explicit form ΠP=    0100 1000 0001 0010     QP=    1 0 0 0 0 1 0 0 0 0 1 0 0001.     (87) From the corresponding Clifford tableau MPof equation (62) it then follows that under the symmetry operation the Xitransform as CPX1C† P=X2Z1CPX2C† P=X1Z2 CPX3C† P=X4Z3CPX4C† P=X3Z4 while CPZiC† P=ZP(i), which indeed is readily seen to preserve the symmetry of the Hamiltonian. If we diagonalize ΠPwe find using equation (67) that the new symmetry adapted creation and annihilation operators are given by c1,2=1 √2(a1±a2)c3,4=1 √2(a3±a4) where c1,c3are the +combinations and c2,c4are the—combinations of the aion the right hand side of the equations. The corresponding equations for the c† iare obtained by taking the adjoint of these expressions. By exposing the symmetry character of the one-particle states we have the following correspondence between symmetry labels and the operators ci: (c1,c2,c3,c4)↔(g↑,u↑,g↓,u↓)(88) 14 New J. Phys. 26 (2024) 103023 R v Leeuwen where gand uare gerade (even) and ungerade (odd) symmetries, i.e. under the symmetry Pthe g→gand u→−u. So, for example, the application of c1removes a spin up particle from a gerade orbital, etc. The second quantized Hamiltonian in terms of the new creation and annihilation operators attains the form ˆ H=−t(c† 1c1−c† 2c2+c† 3c3−c† 4c4) +U 2(c† 1c1+c† 2c2)(c† 3c3+c† 4c4)+U 2(c† 1c2+c† 2c1)(c† 3c4+c† 4c3)(89) and further translated to Jordan–Wigner form this becomes ˆ H=−t 2(−Z1+Z2−Z3+Z4) + U 8(2−Z1−Z2)(2−Z3−Z4) +U 8X1X2(1−Z1Z2)X3X4(1−Z3Z4).(90) To calculate the operators that commute with this Hamiltonian we can construct the matrix Kcontaining the row vectors (ri,si)for each term XriZsiin the Hamiltonian and find its kernel (see the discussion following equation (83)). In our case we find kerK=span((1,1,0,0,0,0,0,0),(0,1,0,1,0,0,0,0),(0,0,1,1,0,0,0,0)) (91) corresponding to the Z-strings Z1Z2,Z2Z4and Z3Z4that commute with the Hamiltonian. The first three rows of Afor our symmetry encoding must therefore contain the Z-part of these kernel vectors, i.e. we have the rows (1,1,0,0),(0,1,0,1),(0,0,1,1)for the first three rows of A. We take the last row to be linearly independent of these, for which a simple choice is (0,0,0,1). In [12] the vector bof the affine encoding is chosen in such a way that the eigenvalues of the Ziin the new representation can be taken to be equal to 1 for the symmetry sector they are interested in. We simply take the translation vector b=0 for the affine mapping as we do not want to select particular eigenvalues yet for the group generators transformed to Zi. With this we find for the Clifford tableau of equation (79) that MA=(A0 0(A−1)T)=             1100 0101 0011 0001 1000 1100 0010 1111             .(92) From this we can read off the transformations of the Xand Zoperators, and we indeed find that Z1Z2,Z2Z4,Z3Z4→Z1,Z2,Z3so that the new Hamiltonian ˆ H′=Cˆ HC†becomes ˆ H′=−t 2(−Z1Z2Z4+Z2Z4−Z3Z4+Z4) + U 8(2−Z1Z2Z4−Z2Z4)(2−Z3Z4−Z4) +U 8X4(1−Z1−Z3+Z1Z3)(93) which now contains a single X4-operator and otherwise only Zi. We readily see that the Hamiltonian indeed commutes with Z1,Z2and Z3so that all these operators can be replaced by their eigenvalues ±1 such that the Hamiltonian reduces to a simple single qubit Hamiltonian. The 2-particle Sz=0 multi-qubit states of gerade symmetry |x⟩=|1010⟩and |x⟩=|0101⟩(see also equation (88)) are mapped by Ato |Ax⟩=|1010⟩and |Ax⟩=|1011⟩and therefore from the action of the Z1,Z2and Z3matrices on the first three entries of these vectors we see that we have Z1=Z3=−1 and Z2=1 as eigenvalues. The 2-particle Sz=0 multi-qubit states of ungerade symmetry |x⟩=|0110⟩and |x⟩=|1001⟩are mapped to |Ax⟩=|1110⟩and |Ax⟩=|1111⟩ corresponding to Z1=Z3=Z2=−1 as eigenvalues. So in the Sz=0 sector, where Z1=Z3=−1 we have the form ˆ H′=−t(1+Z2)Z4+U 2(1+X4).(94) For the gerade states we have Z2=1 and then: ˆ H′=(−2t+U 2 U 2 U 22t+U 2.)(95) 15 New J. Phys. 26 (2024) 103023 R v Leeuwen We have for the eigenvalues λ=1 2(U±√16t2+U2)containing the well-known singlet gerade ground state energy of the Hubbard dimer which is a combination of a Slater determinant with two gerade (bonding) orbitals and a Slater determinant with two ungerade (anti-bonding) orbitals. On the other hand for the ungerade states we have Z2=−1 and ˆ H′=U 2(1 1 1 1)(96) which has eigenvectors (1, 1) with eigenvalue Uand (1,−1)with eigenvalue 0. A similar procedure as discussed above applies to other choices of the eigenvalues of Z1,Z2and Z3; there are six other choices that each lead to a single qubit problem and therefore our original 16 ×16-problem has blocked into eight 2 ×2 single qubit problems by using symmetry. In this example we found a large reduction from four qubits to a single one. In general applications one studies larger systems in which the relative reduction is less spectacular, but the universal procedure for qubit reduction as outlined here nevertheless applies the same way. 4. Conclusions and outlook In this work we considered the symmetry properties of many-body Hamiltonians and the way they are encoded in a Jordan–Wigner transformed manner for implementations on quantum computers. Whereas these symmetries are explicit in the original second quantized Hamiltonian defined on Fock space, they are concealed in its Pauli tensor form when the Hamiltonian is translated to multi-qubit space. We proved a general theorem that allows us to straightforwardly derive the symmetry transformation properties of the Pauli operators and thereby making explicit again the symmetries of the Hamiltonian. We further gave a short and simplified proof of a recently introduced qubit reduction scheme based on affine mappings of qubits, and illustrated all our theory with an example. The results presented here may be extended via several avenues of research. Instead of regarding the symmetry properties of Jordan–Wigner transformed encodings we could study various other forms of encoding such as the Bravyi–Kitaev transformation [30] or perhaps even more general ones [6], and ask for a generalization of our main theorem to that case; at first glance there does not seem to be any major obstruction in our proofs that would prevent such a generalization of the main theorem. Such a transformation may, for example, be optimized to reduce circuit depth for noise reduction on noisy quantum devices. Regarding the affine encoding scheme we could wonder if our results could be generalized to non-Boolean but still Abelian groups with group elements of higher order than two. Some of our equations do remain valid and an extension to general Abelian groups would widen the range of applications for qubit reduction. At least if we are willing to replace qubits by qudits [29] in Fq, where qis the largest order of a group element in the Abelian group that we consider, a symmetry reduction of qudits seems an achievable goal. It would require the replacement of Pauli Z-matrices by more general phase shift matrices and some other technicalities would arise due to the fact that not all group elements in general have the same order. Finally there may be several connections to error correction schemes in quantum computing. The symmetry operators CPthat we discussed were elements of the Clifford group used to develop stabilizer codes [26–28] and in our case we find subspaces of CM 2stabilized by symmetry operations, so that perhaps a useful connection can be made to ways to correct symmetry errors in quantum computing schemes. Data availability statement No new data were created or analyzed in this study. Acknowledgments I wish to acknowledge support from the Finnish Academy under Project Number 356906. 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