SU(2) hadrons on a quantum computer via a variational approach

Quantum computers have the potential to create important new opportunities for ongoing essential research on gauge theories. They can provide simulations that are unattainable on classical computers such as sign-problem afflicted models or time evolutions. In this work, we variationally prepare the low-lying eigenstates of a non-Abelian gauge theory with dynamically coupled matter on a quantum computer. This enables the observation of hadrons and the calculation of their associated masses. The SU(2) gauge group considered here represents an important first step towards ultimately studying quantum chromodynamics, the theory that describes the properties of protons, neutrons and other hadrons. Our calculations on an IBM superconducting platform utilize a variational quantum eigensolver to study both meson and baryon states, hadrons which have never been seen in a non-Abelian simulation on a quantum computer. We develop a hybrid resource-efficient approach by combining classical and quantum computing, that not only allows the study of an SU(2) gauge theory with dynamical matter fields on present-day quantum hardware, but further lays out the premises for future quantum simulations that will address currently unanswered questions in particle and nuclear physics.

In the following, we consider the continuum version of the SU(2) Yang-Mills Hamiltonian and show how its discretisation leads to the Kogut-Susskind Hamiltonian given in the main text. We regard the non-Abelian SU(2) Yang-Mills model describing fermions and antifermions interacting via color electric fields in one spatial dimension. In this model, the fermions and antifermions carry a color charge which we will refer to as red and green. The interactions between the fermions are mediated by the gauge field or "color electric field". We denote the gauge field vector potential at position z with temporal and spatial componentsÂ a 0 pzqT a andÂ a 1 pzqT a respectively, where T a " σ a {2 are the three generators of the SUp2q Lie algebra, and σ a the a-th Pauli matrix (a " x, y, z). From now on, we adopt the temporal or Weyl gaugeÂ a 0 pzq " 0 and Einstein's summation convention will apply on repeating indices in color space, but not on lattice indices. The color electric field also carries a group index a and is given byL a pzq "´B tÂ a 1 pzq. As the canonical conjugate momentum ofÂ a 1 pzq, the color electric field satisfies rÂ a 1 pzq,L b pyqs " iδ ab δpz´yq. In the continuum, the Yang-Mills Hamiltonian is given by [ whereψpzq " pψ 1 pzq,ψ 2 pzqq T is a two-component spinor representing the matter fields. The fermion mass is denoted by m, and g quantifies the matter-field coupling constant, while γ µ are the Dirac matrices satisfying the anticommutation relations tγ µ , γ ν u " 2η µν with η " diagp1,´1q the metric tensor. We further use the : yyatas@uwaterloo.ca ; jingleizl@gmail.com § jan.frhaase@gmail.com shorthand notationψpzq "ψ : pzqγ 0 . In one dimension, a convenient representation for the Dirac matrices is given by γ 0 " σ z and γ 1 " iσ y . Note that the first term in Supplementary Eq. (1) represents the gauge-invariant kinetic energy, the second contribution corresponds to the mass term, and the last part gives the color electric energy. For quantum (and classical) simulation purposes, it is more convenient to work with a discretised version of the continuum Hamiltonian, which is defined on a spatial lattice whose points are separated by a distance a l . In this work, we adopt the staggered formulation of Kogut and Susskind, where fermions and antifermions occupy separate lattice sites, and are arranged in an alternating pattern along the lattice. This yields the lattice Hamil-tonianĤ (2) The matter field at each lattice site n is described by a two-component fermionic fieldφ n " pφ 1 n ,φ 2 n q T , where the upper index labels the two possible colors. These are related to the components of the continuum spinor in Supplementary Eq. (1) in the limit of vanishing lattice spacing a l Ñ 0 as i p p´1q nφ 2n`p { ? a l Ñψ p`1 pzq with p " 0, 1. In the first (kinetic) term, the parallel transporter (or connection)Û n " exp´iΩ a n T a¯a cts on the link between sites n and n`1 and mediates the interaction between the internal color degree of freedom of the fermions on neighbouring sites. Its presence ensures the invariance of the Hamiltonian under local gauge transformations. The angular variablesΩ a n are related to the continuum gauge field on the link n asΩ a n {pa l gq ÑÂ a 1 pzq when the lattice spacing goes to zero.
The last term in the Hamiltonian corresponds to the invariant Casimir operator of the theory and represents the color electric field energy stored in the gauge links. More precisely,L 2 n "L a nL a n "R a nR a n whereL a n andR a n (with a " x, y, z) are respectively the left and right color electric field components on link n. They are conjugate momenta of the vector potential [2] and are related to the continuum variable viaL a n ÑL a pzq{g. The operatorŝ L a n andR a n satisfy the algebra rR a n ,R b m s " i abcR c n δ mn , rL a n ,L b m s "´i abcL c n δ mn , and rL a n ,R b m s " 0, where abc is the Levi-Civita symbol. For a non-Abelian gauge group, the right and left color electric field are related via the adjoint representationR a n " pÛ adj n q abL b n , with pÛ adj n q ab " 2Tr Concluding, if one follows the mapping between the lattice and continuum variables prescribed above, and employs the correspondence a l ř N n"1 f pa l nq Ñ ş dz f pzq as a l Ñ 0, one can recover the continuum Hamiltonian in Supplementary Eq. (1) from the lattice version given in Supplementary Eq. (2).

Supplementary Note 2. Elimination of the gauge fields
For one spatial dimension and open boundary conditions the gauge fields can be eliminated and expressed in terms of the fermionic fields [3,4]. The decoupling is achieved by a unitary transformationΘ that acts on the fermionic fields and eliminates the gauge connectionŝ U n from the kinetic term and expresses the color electric energy in terms of the fermionic operators. This approach was recently introduced by the authors of [4], who performed tensor-network simulations of the SU(2) gauge theory to study real time dynamics of string breaking phenomena. In the following, we reproduce the main steps of their derivation.
We seek a unitary transformationΘ such that Θ´φ :

n´1Û
: n´2¨¨¨Û : 1φ n . One hence introduces the operatorsŴ whereQ m is the vector of the non-Abelian charges with componentsQ a m "φ i: m pT a q ijφ j m , a " x, y, z.
The three (operator) components vectorΩ k "´a l gÂ 1 is directly proportional to the spatial component of the gauge field at site k, and are related to the parallel transporters, sinceÛ k " exp´iΩ a k T a¯. In [4], it was shown that the desiredΘ-transformation is explicitly given byΘ "Ŵ 1Ŵ2¨¨¨ŴN . Under this transformation, the Kogut-Susskind Hamiltonian given by Supple-mentary Eq. (2) takes the form where the rotated color electric term is given bŷ andR 0 can now be interpreted as a background field. We now give more details on how Gauss's law has been used in order to arrive at Supplementary Eq. (7). In the initial frame (before theΘ transformation), Gauss's law readsĜ a n "L a n´R a n´1´Q a n , a " x, y, z, whereL a n andR a n´1 act on the links emanating from the site n, which itself carries the non-Abelian color chargê Q a n . Gauss's law operatorsĜ a n are the generators of the local gauge transformations, hence we have rĤ,Ĝ a n s " 0, @n, @a. Since we assume that there are no external charges in the system, the gauge invariant states must satisfy the identityL a n´R a n´1 "Q a n . In order to see how Gauss's law transforms underΘ, one needs to find the transformation rules forL a n ,Q a n , andR a n´1 . Using the following commutation relation [5] rL a n , pÛ n q pq s " pT aÛ n q pq , and recognising thatŴ n has the same matrix structure asÛ n , we find [4] rL a n ,Ŵ n s "˜ÿ mąnQ a m¸Ŵn , which leads toŴ As a consequence, under theΘ transformation the left electric field transforms aŝ To find how the non-Abelian charges transform under W k , we make use of the following identitŷ which can be easily derived from the commutation relation rQ a n ,φ i m s "´δ mn pT a q ijφ j n . By using the definition of the non-Abelian charges in terms of the fermionic field given in Supplementary Eq. (5), we find W kQ a mŴ : k "φ :i m pÛ k q ip pT a q pq pÛ : k q qjφ j m " pÛ adj k q abQ b m , (14) where the last equality was obtained from the definition of the adjoint representation given in Supplementary  Eq. (3), along with the following identity on the generators of SU(2) [6] pT a q ij pT a q kl " Finally, combining all the results above we can write the transformation of the left electric field aŝ The transformation rule of the right electric field can easily be obtained from the relationR a n " pÛ adj Gauss's law therefore transforms aŝ ΘĜ a nΘ : "L a n´R a n´1 " 0, @n ą 1.
In the rotated frame, Gauss's law can thus be solved recursively and giveŝ L a n " pÛ adj n´1Û adj n´2¨¨¨Û where we have used the relation between the right and left electric field via the adjoint representation. As previously pointed out, the transformation for the first site of the chain has to be studied on its own. For n " 1, Gauss's law transforms asΘL a

1Θ
: "R a 0`Q a 1 , where we have used the fact that the right hand side is invariant under theΘ transformation. In fact,R a 0 is invariant, sincê Θ is independent ofΩ a 0 by construction, furthermoreQ a which by virtue of the orthogonal nature of the adjoint representation, leads to the transformed electric field used in Supplementary Eq. (7).
By setting the background fieldR 0 " 0, and using the fact that the non-Abelian charges commute on different sites, we can rewrite the chromoelectric energy term appearing in the Hamiltonian aŝ which exhibits long-range interaction between the non-Abelian charges. Note that although the gauge fields do not appear explicitly, the non-Abelian physics is preserved in this formulation and reflected through the long range exotic interaction between non-Abelian charges. The formula given above is completely general and can be used for both Abelian and non-Abelian models as long as the appropriate expressions for the charges are used. For instance, in the staggered formulation of the Abelian Up1q theory, the electric charge at site n is given byQ n "φ : nφn´p 1´p´1q n q{2. Substituting this expression in Supplementary Eq. (24), we recover the electric energy of the Abelian Up1q Kogut-Susskind Hamiltonian which has been studied both theoretically and numerically [7,8] and has already been implemented on a quantum computer [9,10].
In the rotated frame, the Kogut-Susskind Hamiltonian is exclusively represented in terms of fermionic degrees of freedom. Although the gauge fields have been eliminated, their interaction with the matter field has been directly incorporated into the Hamiltonian by virtue of Gauss's law. Furthermore, we emphasize that gauge field observables are still accessible in this approach even though they do not appear explicitly in the transformed Hamiltonian.

Supplementary Note 3. Qubit encoding
In this section, we discuss the transformation from the fermionic Hamiltonian in Supplementary Eq. (2) to a for-Eq. (26), considering the symmetry constraints of the theory.
Since we consider the neutral charge sector, we can expand the eigenstates of the Hamiltonian in the basis of the zero mode eigenstates of the total non-Abelian charges, which in the qubit formulation read while the baryon number is given bŷ The z-component of the total non-Abelian charge is diagonal, therefore it is easy to find the eigenstates with eigenvalue zero. We note that the action ofQ z tot on a cell with spins pointing in the same direction gives zero, and the only non-zero contribution comes from cells with antiparallel spins. In particular,Q z tot |Öy k " 1 2 |Öy k and Q z tot |OEy k "´1 2 |OEy k , where |Öy k corresponds to the spin configuration at the spatial site (or cell index) k. The first qubit in the ket |Öy k is thus at position 2k´1 and the second one at position 2k of the encoded lattice. As a consequence, in order to be an eigenstate ofQ z tot with eigenvalue zero, a basis state must contain as many cells of type |Öy as |OEy. If we call n OE the number of such cells appearing in the basis state and n Ó the number of cells with both spins pointing down, then it is easy to see that the baryon quantum number of such state is B " N {2´n Ó´nOE with n OE " 0, 1, . . . , N {2 and n Ó " 0, 1, . . . , N´2n OE . This clearly shows that the baryon quantum number of a physical state is an integer Among all the eigenstates ofQ z tot with eigenvalue zero, some of them must be combined in order to be annihilated by the two other non-Abelian charges. This is obvious from the expression Supplementary Eq. (30) and Supplementary Eq. (31), where we see that these operators induce non-diagonal transitions between the computational basis states. As a simple illustration, let us consider the state |ÖOEÒÒy. It contains one cell of type |Öy and one cell |OEy and is thus an eigenstate ofQ z tot with eigenvalue zero as per our discussion above. It is however easy to see that this state alone is not an eigenstate of the two other non-Abelian chargesQ x,y tot . In order to be a simultaneous eigenstate of the three non-Abelian charges, it must be combined with its companion state in the table in the following way p|ÖOEÒÒy´|OEÖÒÒyq { ? 2. For an arbitrary number of sites N , the construction discussed above is a non-trivial task. This is why the design of a circuit incorporating the neutral charge constraint is generally harder and one can rely on the variational algorithm to restore the right color symmetry. This approach was for instance used to obtain our results in the N " 6 case (see Methods). For small lattice sizes, it is however possible to impose directly the color symmetry into the design of the circuit, as we did in Methods for the circuit generating the color symmetric ansatz for N " 4 spatial sites in the sector with baryon number B " 1.