QQubit Construction in 6D SCFTs
Jonathan J. Heckman ∗ Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA
Abstract
We consider a class of 6D superconformal field theories (SCFTs) which have a large N limit and a semi-classical gravity dual description. Using the quiver-like structure of 6DSCFTs we study a subsector of operators protected from large operator mixing. Theseoperators are characterized by degrees of freedom in a one-dimensional spin chain, and theassociated states are generically highly entangled. This provides a concrete realization ofqubit-like states in a strongly coupled quantum field theory. Renormalization group flowstriggered by deformations of 6D UV fixed points translate to specific deformations of theseone-dimensional spin chains. We also present a conjectural spin chain Hamiltonian whichtracks the evolution of these states as a function of renormalization group flow, and studyqubit manipulation in this setting. Similar considerations hold for theories without AdS duals, such as 6D little string theories and 4D SCFTs obtained from compactification of thepartial tensor branch theory on a T .July 2020 ∗ e-mail: [email protected] a r X i v : . [ h e p - t h ] J u l Introduction
There are intriguing connections between between information theoretic structures and theemergence of spacetime. An undergirding assumption in this setup is that entangled “qubits”in a conformal field theory (CFT) serve to encode the structure of a bulk gravitational dual.There are by now a number of suggestive holographic structures in this setting as found inreferences [1–5]. There are also proposed connections to phenomena from condensed matterand quantum information systems such as references [6–17].This also raises some additional questions. For example, to make further tests of theseproposals it would be nice to have some explicit examples of qubits in interacting conformalfield theories with genuine stringy holographic duals. Additionally, there is a suggestivegeometric “lattice structure” built into many discussions connecting holography and tensornetworks, but this also poses some puzzles for the role of the lattice suggested by a tensornetwork and its connection to spacetime locality.Our aim in the present note will be to construct some explicit examples of qubit systems insuperconformal field theories (SCFTs). Perhaps surprisingly, we find it simplest to constructthese states in 6D SCFTs, as well as in their dimensional reduction on a T . The class oftheories we consider resemble generalized quiver gauge theories which take the form:[ G ] − G − ... − G N − − [ G N ] . (1.1)Here, each G i for i = 1 , ..., N denotes a gauge symmetry factor, with G and G N flavorsymmetry factors. The links between these gauge groups correspond to “conformal matter,”as in references [18, 19].The main class of 6D SCFTs we consider arise from the worldvolume of N M5-branesfilling R , and probing the transverse geometry R ⊥ × C / Γ ADE where Γ
ADE ⊂ SU (2) is afinite subgroup. The subscript here serves to remind us that there is an ADE classificationof such finite subgroups, and these are in one to one correspondence with the ADE series ofLie algebras. Indeed, M-theory on R , × C / Γ ADE engineers 7D Super Yang-Mills theorywith the corresponding gauge group. Placing M5-branes on top of an ADE singularity butseparated from each other in the transverse R ⊥ direction leads to a quiver-like structureas in line (1.1) with gauge group factors G i = G ADE , with the strength of each gaugecoupling depending inversely on the separation length between neighboring M5-branes. The“matter” of the generalized quiver corresponds to degrees of freedom localized on each M5-brane. We reach a non-trivial conformal fixed point by making all M5-branes coincide,which corresponds to a point of strong coupling in the moduli space of the 6D field theory.In the large N limit, we also achieve a holographic dual description in M-theory given bythe spacetime AdS × S / Γ ADE , with orbifold fixed points at the north and south poles ofthe S . These fixed points correspond to the left and right flavor symmetry factors G and G N . There are similar long quivers without a semi-classical gravity dual [20, 21]. One can1lso produce 6D little string theories (LSTs) by gauging a diagonal subgroup of G × G N ,as in reference [22]. This last class of examples produces a non- AdS holographic dual witha linear dilaton background [23].The main message of recent work in classifying 6D SCFTs (see [20, 21] and [24] for areview) is that the vast majority of such theories resemble quiver gauge theories, and infact all known theories can be obtained through a process of fission and fusion [21, 25, 26]built from such quiver-like structures. So, lessons learned here apply to a broad class of 6Dtheories and their lower-dimensional descendants obtained from further compactification.To construct some explicit qubits, we make use of the recent work of reference [27] whichfound that for operators with large R-charge, there are nearly protected operator subsectorswhich only mix with themselves. That this is possible at strong coupling follows from thefact that at large R-charge, there is a further suppression in operator mixing which goesinversely in the R-charge, much as in references [28, 29].Using this fact, reference [27] identified a class of gauge invariant local operators of theform: O m ...m N = X ( m )1 ...X ( m i ) i ...X ( m N ) N , (1.2)where we view the X ( m i ) i as bifundamental operators between neighboring pairs of groups: G i − X ( mi ) i −→ G i . (1.3)We note that our indexing convention here is slightly different from [27]. These operators areconstructed on the partial tensor branch of the 6D SCFT, and to reach the conformal fixedpoint one must take a further decoupling limit in which momentum transverse to the stackof M5-branes is set to zero. This imposes a mild condition on the spectrum of quasi-particleexcitations in the 1D lattice of spins, and for the most part we will keep this point implicitin what follows. See reference [27] for details.Treating the G i − and G i as flavor symmetries, the X i define “conformal matter” opera-tors which have have non-trivial scaling dimension ∆ X and transform in a spin s X represen-tation of the SU (2) R R-symmetry. In the special case where all the G i = SU ( K ), s X = 1 / A K D K E E E s X / / . (1.4)The key feature found in [27] is that for the composite gauge invariant operators O m ...m N ,the action of the one-loop Dilatation operator on the O m ...m N is simply that of a 1D spinchain Hamiltonian. In the case where we have A-type gauge groups with G i = SU ( K ), the2ne-loop Dilatation operator takes the form:∆ = E (0) − λ A N (cid:88) i =1 −→ S i · −→ S i +1 . (1.5)Here, the −→ S i denote the usual angular momentum operators in the spin 1 / −→ S N +1 = 0. For a spin chain with periodic boundary conditions we wouldinstead set −→ S N +1 = −→ S . The constants E (0) and λ A > XXX s =1 / Heisenberg spin chain [30]. It also shows upprominently in the context of N = 4 Super Yang-Mills theory (see e.g. [31–33]) which servedas a motivation for reference [27].In the case of the D- and E-type spin chains, similar considerations hold, but we insteadget a Hamiltonian constructed from a polynomial in the −→ S i · −→ S i +1 . The precise form of thispolynomial can be fully fixed by assuming that the integrable structure present in the A-typetheories persists in this broader setting. The spectrum of excitations can now be studiedusing methods such as the algebraic Bethe ansatz (see [34,35] and [36] for a review). One canin principle also extend this to more general excitations of the full superconformal algebra osp (8 ∗ | N is quite large. The case of periodic boundary conditionsoccurs anyway in the study of 6D LSTs [27]. 3 Qubits in 6D SCFTs
We now proceed to build a system of qubits in a 6D SCFT. As already mentioned, we areinterested in the class of operators O m ...m N where m i = ± /
2. Working with the radiallyquantized SCFT, we see that each local operator specifies a state in the Hilbert space: O m ...m N (0) | GND (cid:105) = |O m ...m N (cid:105) ∈ H D . (2.1)On the other hand, the m i also specify a state in a 1D spin chain Hilbert space: | m ...m N (cid:105) ∈ H D . (2.2)From all that we have said, H D defines a protected subsector of states in the 6D SCFT.This is the qubit system we wish to study.The spatial direction of the spin chain is clearly related to the R ⊥ direction of the M5-brane probe theory. Note that in the holographic dual, the spin chain direction correspondsto a great arc passing from the north pole to the south pole of S / Γ ADE .Returning to our spin chain Hamiltonian:∆ = E (0) − λ A N (cid:88) i =1 −→ S i · −→ S i +1 , (2.3)we observe that the lowest energy states actually have a large degeneracy. To see this,observe that the total angular momentum operator: −→ S = N (cid:88) i =1 −→ S i (2.4)commutes with ∆, namely (cid:104) −→ S , ∆ (cid:105) = 0. So, we can organize our energy eigenstates into rep-resentations of −→ S . The system is also gapless in the sense that it costs very little energy toproduce an excitation above the lowest energy states. Note also that there is a non-relativisticdispersion relation with energy (cid:15) ( p ) ∼ p , so we get a scale invariant but non-Lorentz invari-ant system. Some examples of entanglement entropy calculations for Heisenberg spin chainsand deformations thereof have been carried out in references [37–39].Let us now discuss in more detail the lowest energy states of the system. To begin, takethe state with all m i = 1 /
2. This is the highest weight state of a spin N/ S ± = S x ± iS y , we reach the other states with the same scaling dimension bysuccessive applications of the lowering operator S − . The resulting form of the states obtained4rom M such spin flips are discussed in [39] and are given by: | N, M (cid:105) = 1 (cid:112) C M,N (cid:88) σ (cid:12)(cid:12) m σ (1) ...m σ ( N ) (cid:11) , (2.5)where we sum over all permutations of M down spins on N sites. Here, we have also intro-duced the combinatorial factor: C M,N = N ! M !( N − M )! . (2.6)We claim that the resulting qubits of this ground state are highly entangled states,and similar considerations hold for quasi-particle excitations of the spin chain. Indeed,introducing the pure state: ρ M,N = | N, M (cid:105) (cid:104)
N, M | , (2.7)we get a mixed state by performing a partial trace of N − n spins, not necessarily in a singlecontiguous block. We then get the reduced density matrix: ρ ( n ) M,N = Tr ( N − n ) ρ M,N . (2.8)The entanglement entropy for this was computed in [39] for periodic boundary conditions.In the thermodynamic limit where N → ∞ with M/N = p held fixed and n/N ≤ / S ( n ) = − Tr ρ ( n ) M,N log ρ ( n ) M,N ≈
12 log n + 12 log(2 πepq ) , (2.9)with p + q = 1.What is the interpretation of this in the original M-theory picture? We can considerstarting with our stack of N M5-branes, and can perform a partial trace over all but n ofthem. Doing so, we get a highly entangled stated. Indeed, the proof of this is that in our1D system we have an entanglement entropy proportional to log n .The pure states ρ M,N realize explicit examples of N qubit W-states as well as generaliza-tions thereof, as opposed to GHZ states. For example, in a system with N qubits, we canintroduce the pure states: | GHZ (cid:105) = 1 √ |↑ ... ↑(cid:105) + |↓ ... ↓(cid:105) ) (2.10) | W (cid:105) = 1 √ N ( |↓↑ ... ↑(cid:105) + ... + |↑↑↑ ... ↓(cid:105) ) , (2.11) As mentioned before, we are ignoring the zero momentum constraint on quasi-particle excitations of thespin chain. We expect this to be a subleading effect so we neglect it in the discussion which follows. The original terminology applies to the three qubit case [40], but it clearly extends, with the caveat thatthere are many ways to entangle four or more qubits [41].
Starting from a conformal field theory, we can consider deformations which trigger a flow toanother conformal fixed point in the infrared (IR). In the case of 6D SCFTs, the availableoptions for supersymmetry preserving renormalization group flows are quite limited. Theseare always specified by background operator vacuum expectation values (vevs) and are re-ferred to as tensor branch flows and Higgs branch flows [42, 43] (see also [44]). We considerboth sorts of flows.
Consider first tensor branch flows. Some examples of tensor branch flows arise from justseparating the M5-branes from one another. In the case of M5-branes probing an A-typesingularity, this turns out to be the only possibility. For more general 6D SCFTs other tensorbranch deformations are possible and they are all classified by suitable K¨ahler deformationsof the associated F-theory model [20, 21].Sticking to the simplest case where we separate our M5-branes into two stacks N and N such that N + N = N , we can clearly see that in the deep infrared, our single spin chainhas broken up into two independent spin chains:[ G ] − G − ... − G N − − [ G N ] ⊕ [ G N ] − G N +1 − ... − G N − − [ G N ] (3.1)where we have indicated how the gauge group G N has become a flavor symmetry. In thedeep infrared, this flavor symmetry acts independently on the two decoupled SCFTs (seefigure 1).Now, we can ask about the structure of the Hilbert space of the 6D SCFT in this limit.Clearly, we expect a split into two “decoupled” SCFTs in the infrared, so we can write: H IR = H N ⊗ H mix ⊗ H N , (3.2)where here, H mix denotes the Hilbert space of a TQFT coupled to some free fields (seee.g. [45]).In the spin chain, we can visualize this process by working with a slightly more general6igure 1: Depiction of a tensor branch flow from the UV to the IR. In the M5-brane picture(left) this involves separating a stack of N = N + N M5-branes into two stacks. In theassociated spin chain (right), the resulting spins separate into two decoupled sectors. Therecan still be significant entanglement between the two sectors.Dilatation operator / spin chain Hamiltonian:∆ IR = E (0)IR − N (cid:88) i =1 λ i −→ S i · −→ S i +1 , (3.3)where the couplings can now be position dependent. The limit we are discussing amountsto setting: λ = λ = ... = λ N − , (3.4) λ N = 0 , (3.5) λ N +1 = λ N +2 = ... = λ N − . (3.6)We note that the computation of the spin chain couplings performed in reference [27] displaysa non-trivial dependence on N , so in particular when N (cid:54) = N , we do not expect the couplingson the left and righthand sides of the decoupled spin chains to be the same in the deep IR.We can also see that there is a great deal of entanglement between the two separatedM5-brane sectors of line (3.2). Indeed, from our discussion in the previous section, we knowthat even in our spin chain subsector this scales as log N (in the case where N < N ). Thisprovides evidence for the existence of a non-trivial TQFT which couples these two sectors,in accord with the general considerations presented in [46] where 6D SCFTs were visualized7s “edge modes” of a bulk 7D theory (see also [47–49].We can generalize this to multiple boundaries by considering other partitions of N : N = N + ... + N k . (3.7)In this way, we can build multi-party entangled qubits. See figure 2 for a depiction. It wouldbe interesting to see whether this provides a higher-dimensional analog of the situationconsidered in references [12, 13, 16, 17]. Consider next the case of Higgs branch flows. The distinguishing feature here is that the SU (2) R R-symmetry is broken along the flow, but a new R-symmetry emerges in the deepinfrared. The resulting class of theories which can be achieved in these cases again resemblequiver gauge theories, but in which there can now be different ranks of gauge groups inthe generalized quiver, as well as possible decorations by conformal matter on the left andrighthand sides [21, 25, 26]. Importantly, in the vast majority of Higgs branch flows, theactual number of gauge group factors again remains of order N . This means that even inthe associated spin chain generated in the deep IR, we again have the same number of spins,but can now have more general position dependence:∆ IR = E (0)IR − N (cid:88) i =1 λ i −→ S i · −→ S i +1 . (3.8)Indeed, in such situations, we can ask about the structure of the “ground states” found forthe ultraviolet (UV) Hamiltonian. Note, however, that −→ S = −→ S + ... + −→ S N still commuteswith H IR . So, we again have a large degeneracy in the ground state of the spin chain.Moreover, even though the spectrum of excitations has moved around, the impact on thestructure of the spin chain Hilbert space is relatively mild. At this point, it is interesting to ask about how to interpret the structure of the spin chainand its Hamiltonian as we proceed from a perturbation of the ultraviolet fixed point to anew one in the deep infrared. Here we discuss some speculative comments in this direction.First of all, we note that in the 6D SCFT, the spin chain Hamiltonian has the inter-pretation as the one-loop Dilatation operator. Once we break conformal symmetry, ourinterpretation must also be suitably loosened. That being said, it is also clear that we canstill take a state and ask how it evolves as a function of scale. In the holographic dual setup,this corresponds to motion from the “UV brane” to the “IR brane”. Observe that at least for8igure 2: Depiction of a multi-throat spacetime generated by pulling N = N + ... + N k M5-branes apart into separate stacks. Starting from a configuration of spins in the parenttheory, we get a multi-party entangled state in the IR theory.tensor branch flows, this is immediately realized in terms of conventional branes: We simplytake some number of M5-branes and pass them down the throat of the AdS geometry (seefigure 2).As a first generalization, then, we can consider a family of spin chain Hamiltonians, onefor each step in the RG direction. Labelling this family as ∆( z ) such that z UV correspondsto the UV and z IR corresponds to motion into the IR, we now allow our nearest neighborinteractions to depend on RG time, writing λ i ( z ). In this context, it is appropriate to alsopermit our spin operators to be z dependent as well, so we write −→ S i ( z ) to reflect this fact.At a given RG time slice, we now can write:∆( z ) = E (0) ( z ) − N (cid:88) i =1 λ i ( z ) −→ S i ( z ) · −→ S i +1 ( z ) + .... (4.1)We can thus consider a slightly broader class of “time dependent” spin chains in which weevolve from ∆( z UV ) to ∆( z IR ).There are good reasons to generalize this slightly further. For one thing, we note thatas written, this Hamiltonian still preserves SU (2) R R-symmetry. On the other hand, wealso know that at least in Higgs branch flows, we expect the R-symmetry to be broken, onlyto reemerge deep in the IR. As a further generalization, we therefore allow various sorts ofR-symmetry breaking as generated by vevs of operators in the parent UV theory.9n the 1D spin chain, this suggests a further generalization where we allow SU (2) R R-symmetry breaking terms. One option is to just include some background magnetic fieldterms. We can also allow various SU (2) R braking terms of the sort appearing in integrable XY Z models. Including both sorts of terms, we get, in the obvious notation:∆( z ) = E ( z ) − N (cid:88) i =1 3 (cid:88) a =1 (cid:16) λ ( a ) i ( z ) S ( a ) i ( z ) · S ( a ) i +1 ( z ) + h ( a ) i ( z ) · S ( a ) i ( z ) (cid:17) . (4.2)The main condition we need to impose is that once we reach the IR where we recover a 6DSCFT, all SU (2) R breaking terms go to zero as z → z IR . Given the suggestive form of our spin chain system, it is of course interesting to ask whetherwe can directly manipulate the associated qubits, and using this, probe additional structurein these systems. From the perspective of the 6D SCFT, the natural operations on statesinclude acting with the symmetry generators of the conformal field theory, including theDilatation and R-symmetry operators. In particular, the Dilatation operator corresponds tothe Hamiltonian of the spin chain, governing time evolution as a function of renormalizationgroup scale in the 6D SCFT. At first pass, the use of the R-symmetry generators provides away for us to rotate qubits, but in the interacting SCFT, we really have only the operator: −→ S = N (cid:88) i =1 −→ S i , (5.1)which would simultaneously manipulate many qubits all at once.Using the M5-brane picture, however, we can see how to build a general protocol forqubit manipulation. As a warmup, consider the 4D SCFT obtained by compactifying thetensor branch theory on a T . This system has basically the same qubit structure as the6D theory, but there is no zero momentum constraint [27]. To manipulate an individualqubit, we consider a deformation which involves separating a single M5-brane from all of itsneighbors in the quiver. In this limit, a link between G i − and G i is isolated from the restof the system, and it has its own emergent R-symmetry in the infrared. So, we can applya unitary SU (2) R transformation along with an overall phase factor exp( iγ ) exp( − i −→ θ · −→ S ).Doing so, we can generate the standard single qubit operations, visualized as rotations onthe Bloch sphere. Some simple examples include the bit flip operation, or Pauli X -gate: σ X = (cid:20) (cid:21) = exp( iπ/
2) exp( − iπS ( x ) ). (5.2) We thank A. Kar for some comments which prompted us to consider this possibility. z . The starting point is to separate M5-branes fromone another. This is followed by a general Bloch sphere / SU (2) R rotation. After this, theM5-branes are recombined. In the case of 6D SCFTs this is followed by a projection ontothe zero momentum sector of the 1D spin chain Hilbert space.Another example is the Hadamard gate: H = 1 √ (cid:20) − (cid:21) = exp( iπ/
2) exp (cid:18) − i π √ (cid:0) S ( x ) + S ( z ) (cid:1)(cid:19) . (5.3)Now we can see a general way to start manipulating individual qubits: We begin bypulling all the M5-branes away from each other. In the spin chain Hamiltonian this corre-sponds to specific z -dependent behavior for the nearest neighbor interactions. After theyare well separated, they each have their own emergent SU (2) R R-symmetry and we canmanipulate their qubits individually. After this, we can bring the M5-branes back together,and evolve further with the Dilatation operator. Note that this is a flow in moduli spacewith the local coordinate of the flow playing the role of time evolution in the qubit system.Similar considerations clearly hold for the 6D SCFT system, with the mild caveat thatwe need to impose the zero momentum constraint on possible excitations. This means, forexample, that after manipulating individual qubits and bringing the stack of M5-branes backtogether that we need to perform a further projection onto the zero momentum sector of the1D spin chain Hilbert space. So, in an actual quantum computation we would perform thisoperation at the very end.As a final generalization, we can now see how to build far more involved qubit operations.11n this case we consider separating a stack of M M5-branes from the rest of the system viaa tensor branch flow, manipulate that individual set of qubits, and then bring it back tothe rest of the configuration. An example of this sort of qubit manipulation is depicted infigure 3. It would be interesting to study further the class of qubit operations which can beengineered in this way.
Acknowledgements
We thank F. Baume and C. Lawrie for helpful discussions and an inspiring collaboration onrelated work. We also thank M. Dierigl for helpful discussions. We thank A. Kar and O.Parrikar for several insightful comments on an earlier draft. The work of JJH is supportedby a University Research Foundation grant at the University of Pennsylvania.
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