Adiabatic Topological Quantum Computing

CESARE, LANDAHL, BACON, FLAMMIA, AND NEELSPHYSICAL REVIEW A 92, 012336 (2015)system [17]. Information encoded into the ground space shouldtherefore remain well protected from the detrimental effectsof decoherence. Further, if one immerses the system in abath with a temperature lower than that of the energy gapin the system, then one should expect a suppression of thermalexcitations out of the ground space. The decay rate of thequantum information encoded into the ground space is not setby a length scale in the system, but instead the lifetime scales asexp(cβ ) where β is the inverse temperature, is the energygap of the Hamiltonian, and c is a constant [18]. Crucially, thisimplies that the lifetime of the information is exponentiallylengthened as a function of the inverse temperature. Whileone does not obtain, using Kitaev’s original idea, a methodfor protecting quantum information with a lifetime that growswith the size of the system, a hallmark of “self-correcting”quantum memories [8,19], for a suitably low temperature,the information lifetime will be long enough for all practicalpurposes. Thus, via the use of a static many-body Hamiltonian, Kitaev proposed that quantum information could beprotected without resorting to active quantum error-correctingalgorithms.Following Kitaev’s introduction of this idea, numerousauthors put forward similar approaches. Many of these ideasstayed within the realm of topological protection [15,20–24],but others explored energetic protection without referenceto topological ideas [25–28]. Here, we will focus on thetopological models, but many of our results apply in the moregeneral setting.Kitaev noted in his original proposal that the excited statesof his Hamiltonian act as particles with exotic statistics. Inparticular, he showed that the excitations were quasiparticles called anyons [29], particles that exist in two spatialdimensions that exhibit statistics different from fermions andbosons and which interact by braiding around one another inspace-time, an interaction that only depends on the topologyof the anyon worldlines. These excitations not only describeerrors in the code space, but can also be thought of as quantuminformation carriers in their own right. Indeed, for somemany-body Hamiltonians, it is possible to have non-Abeliananyons (anyons whose braidings do not commute) that performuniversal quantum computation in the label space of theanyons. This is known as topological quantum computing[4,30–32], the principal model of quantum computing we willconsider here.In a topological quantum computation, one creates anyonsfrom the vacuum, braids them around one another in spacetime, fuses them together, then records their label types.Although the topological nature of the anyonic interactionprovides a degree of control robustness, it is not immediatelyclear why the processes of anyon creation and fusion couldnot create new unwanted anyons. Such anyons could inturn wander and disrupt the desired braid. The initializationprocess in particular is quite subtle [33]. Moreover, therewill likely be a background of thermal anyons and anyonsarising from material defects which could also disorder thequantum computation. On top of all of this, even if a space-timebraid is topologically correct, the mere act of moving anyonsaround at any nonzero speed has the potential to generate newexcitations because the adiabatic approximation is not exact.Measurement-based topological quantum computation [34,35]has the potential to overcome this last problem, but the otherproblems remain. In summary, the great merit of topologicalquantum computation is that the “only” thing that can corruptit is uncontrolled anyons, the problem is that there are manyways that uncontrolled anyons can arise. Even somethingas seemingly innocuous as a lack of complete knowledgeof the system’s Hamiltonian could do this because it couldlead to anyons being trapped or leaking out of the systemunbeknownst to the computer operator [15]. We do not claimto address every possible adversarial scenario for topologicalquantum computation here; our focus is on constructing anarchitecture that limits the chances for uncontrolled anyons toappear.The second line of research relevant to our proposal isthe recent use of code deformations to perform quantumcomputation on topological quantum error-correcting codes[8–10,12,32]. In this approach, one works directly with thequantum error-correcting code used in topological quantumcomputing without introducing a Hamiltonian to provideenergetic protection of the quantum information. Instead, onefocuses on active error correction, but performed with thetopological quantum codes. Consideration of such codes forquantum error correction was first examined in detail byDennis et al. [8]. In this approach, qubits are arranged ona two-dimensional surface with a boundary, resulting in asingle encoded qubit for each such surface. In order to build aquantum computer with more than one qubit, such surfacesare stacked on top of each other so that transversal gatescan be achieved between the neighboring surfaces. Since theoriginal analysis, modifications [10,36] of this architecturehave been introduced that have considerable advantages overthe three-dimensional stacking of Dennis et al. In these models,one takes a surface code and “punctures” it by removing thequantum check operators (stabilizer generators) from a region,creating a defect [37]. For each defect, one obtains an encodedqubit with a code distance that is the minimum of the perimeterof the defect and the distance from the defect to the nearestappropriate boundary (which may lie on another defect). Onecan show that, via a sequence of adaptive measurements,one can deform the boundary of the defect, and, by usingsuitable deformations, braid defects in such a way that logicaloperations are performed between the logical qubits associatedwith the defects.The third line of research relevant to our proposal is therecent discovery of methods to perform holonomic [38] andopen-loop holonomic [39] universal quantum computationin a stabilizer code setting [6,7,40]. In holonomic quantumcomputing, adiabatic changes of a Hamiltonian with degenerate energy levels around a loop in parameter space induceunitary gates on each energy eigenspace. The enacted gatedepends on geometric properties of the Hamiltonian pathand not on the exact timing used to traverse it (to withinthe limits of the adiabatic approximation), thus offering amethod to avoid some timing errors. Universal quantumcomputation using holonomic methods was originally studiedin Ref. [38]. Recently, Oreshkov et al. demonstrated a novelmanner for achieving universality within the context of faulttolerant quantum computing [6]. In particular, this resultshowed how to perform gates on information encoded intoa quantum stabilizer code. Building along these lines, two012336-2

ADIABATIC TOPOLOGICAL QUANTUM COMPUTINGPHYSICAL REVIEW A 92, 012336 (2015)of the present authors (D.B. and S.T.F.) have shown how toachieve similar constructions within the context of open-loopholonomic quantum computation [7,41]. In this setting, insteadof using cyclic evolutions, one can quantum compute usingnoncyclic evolutions. A consequence of this is a schemeknown as adiabatic gate teleportation where one mimics gateteleportation via a very simple interpolation between two-qubitinteractions [7]. Another consequence is that it is possible toperform measurement-based quantum computing [42] usingonly adiabatic deformations of a Hamiltonian [41]. Further,and more suggestive for this work, it is possible to performholonomic quantum computation on symmetry-protected spinchains [43]. Holonomic quantum computation, whether performed cyclically or noncyclically, should be distinguishedfrom (universal) adiabatic quantum computation, in which theground state is always nondegenerate throughout the noncyclicadiabatic evolution [44–48].In this work, we combine many of the above insights into amethod for computing on information encoded into the energylevels of a Hamiltonian. We consider a situation where, asin the first line of research, quantum information is encodedinto the ground state of a topologically ordered many-bodysystem. Rather than storing information in the label space ofanyons themselves, we consider information stored in defects,which act somewhat like anyons, as in the second line ofresearch. We then examine explicit adiabatic interpolationsbetween Hamiltonians that simulate code deformation, as inthe third line of research. This is all done while keeping theenergy gap in the system constant, a necessary requirement touse these techniques to maintain the topological protectionoffered by these systems. Further, we demonstrate howto prepare quantum information into fiducial states usingadiabatic evolutions. The gap only closes when we changethe ground-state degeneracy, and in this case the degeneratestates are connected only by operators with a weight that growswith the system size, making this part of the evolution robustto low-order error processes. Some of these state-preparationprocedures are robust to error, but some (e.g., the preparationof certain “magic states” [49]) are not robust and thus requiredistillation protocols. Finally, we discuss how one can usecode deformations to facilitate measurements of certain logicaloperators. We discuss all of these procedures first within thecontext of Kitaev’s surface codes with defects, and then wediscuss how these results can be extended to the topologicalcolor codes [50].The systems and protocols we use are not strictly faulttolerant. Without active error correction, the lifetime of thecodes studied is a constant independent of the system size[18]. As mentioned above, here we rely on a couplingto a cold (with respect to the gap) thermal bath, whichsuppresses the creation of errors exponentially in the size ofthe gap. We retain robustness to things like control errors byvirtue of the holonomic nature of the logical operations weimplement, and robustness to correlated fluctuations inducedby the environment by keeping defects well separated duringbraiding. Once the environment creates an excitation, it isfree to wander and corrupt the computation. We prevent theenvironment from doing this by ensuring that it is cold, and weprevent ourselves from introducing excitations accidentally bycarefully designing our procedures.ZZp ZZXXsXXFIG. 1. (Color online) Stabilizer generators (checks) for the surface code. An example of a plaquette check Sp and a vertexcheck Sv .II. SURFACE CODES WITH DEFECTSWe begin by working with a simple class of surface codeswith defects to establish the main ideas behind our procedures.In Sec. VII, we extend these ideas to the topological colorcodes. We assume that the reader is familiar with the theoryof stabilizer codes [51], toric codes [4], and surface codes[37], which are specializations of toric codes to boundedplanar surfaces. However, we review these subjects to set ournotation.Let L be a two-dimensional square lattice (see Fig. 1) thatis l edges (or links) wide and l edges tall, with the leftmostl vertical edges and bottommost l horizontal edges removed.(Other lattices are possible; we make this restriction only tobe concrete.) We call the sides of the lattice with the edgesremoved the rough or X-type boundaries and the other sides thesmooth or Z-type boundaries (see Fig. 2). A qubit is associatedwith each edge of the lattice so that there are 2l 2 qubits intotal. For each plaquette (or face) p of the lattice, define theplaquette operator Sp e p Ze where p denotes the edgesbounding the plaquette and Ze is the Pauli Z operator actingon the qubit at edge e. In other words, Sp acts as the tensorproduct of Z operators on the qubits touching the plaquettep and acts trivially everywhere else in the lattice (see Fig. 1).Similarly, for each vertex (or site) in the lattice, define a vertexoperator Sv e δv Xe , where δv denotes the edges incidentat vertex v and Xe is the Pauli X operator acting on the qubitat edge e. In other words, Sv acts as a tensor product of PauliX operators on all the edges surrounding a vertex and actstrivially on all the other qubits in the lattice, as shown inFig. 1.It is important to note that the rough and smooth boundariesstill have plaquette and vertex operators defined on them; theseoperators simply act nontrivially on fewer qubits than theoperators in the bulk of the lattice. Since the lattice L has l 2plaquettes and l 2 vertices, there are also l 2 plaquette operators012336-3

CESARE, LANDAHL, BACON, FLAMMIA, AND NEELSPHYSICAL REVIEW A 92, 012336 (2015)ZZXXFIG. 2. (Color online) A smooth (Z-type) defect. A logical Zoperator is defined by a closed loop of Z’s on the lattice that surroundsthe defect and a logical X operator is defined by a connected path ofX’s on the dual lattice from the defect to a smooth (Z-type) boundary.By removed, we mean that stabilizers in the gray region are removedfrom the stabilizer group. Also, the star operators on the boundary ofthe lattice or the defect have lower weight in general. Here, we depictthe removed region by removing that part of the lattice; this simplyindicates that the code of the system factors into a code in the drawnregion and a code inside of the defect.and l 2 vertex operators. These operators are all independentin the sense that no strict subset can generate the rest, and,moreover, they all commute since they are incident on eachother an even number of times.The collection of all the Sp and Sv operators comprises theset of stabilizer generators for a quantum surface code, thecode space being defined by the simultaneous 1 eigenspaceof all the stabilizer generators. This set generates the stabilizergroup for the code, which is simply the set of all the productsof generators. The above description actually specifies a singlestate rather than a code space since it has 2l 2 checks on 2l 2qubits. This is a consequence of the particular way in whichwe chose the boundary of the lattice, which disallows theexistence of any additional operators that commute with allof the generators but which are not elements of the stabilizergroup. Encoding quantum information in the lattice requiresthe constructions described next.Consider a closed simple curve c on L that does notcross itself and that does not touch the boundary of L. Callthe interior of this loop, excluding c itself, Ic . Consider“removing” all of the qubits in Ic . Here, by “removing” wedo not mean physically removing the qubits, but rather thatwe consider a new code in which the stabilizer generatorsexterior to the region Ic are consistent with the descriptionabove, while the region Ic has a different set of stabilizergenerators (not necessarily of the plaquette and vertex type).We call this process puncturing (not to be confused with thenotion of puncturing associated with classical coding theory[52]), and the resulting region of removed qubits is called adefect. Given such a defect, we can study the properties of theFIG. 3. (Color online) A rough (X-type) defect. A logical Xoperator is defined by a closed loop of X’s on the dual lattice thatsurrounds the defect and a logical Z operator is defined by a connectedpath of Z’s on the lattice from the defect to a rough (X-type) boundary.new code induced on the exterior of Ic . Careful counting ofthe stabilizer generators and qubits in this new code revealsthat the puncturing procedure has created a logical qubit [37].The logical operators for the new logical qubit can be chosenas follows: an encoded Z is a closed loop of Z operators onthe lattice L that encircles the defect and an encoded X is aconnected path of X operators on the dual lattice L that startson the smooth (Z-type) boundary of the defect and ends on asmooth (Z-type) boundary of the lattice L other than the loopc (see Fig. 2). The distance of this code is the minimum ofthe length of curves on L bounding the defect and the lengthof paths connecting the defect to a smooth (Z-type) boundaryof L. We note that the curve c itself is the minimum-weightchoice for the encircling logical Z operator. Similarly, insteadof starting with a simple closed curve on the lattice, we canconsider a simple closed curve on the dual lattice and removethe interior of this curve. To be consistent with the definitiongiven for the former kind of defect, we must define the encodedX to be a closed loop c of X operators on the dual lattice L that encircles the defect and the encoded Z to be a connectedpath of Z operators on the lattice L that starts on the rough(X-type) boundary of the defect and ends on a rough (X-type)boundary of the lattice L other than the loop c (see Fig. 3).Puncturing the surface code creates a single encoded qubit.By puncturing multiple times we can create a code with morethan one encoded qubit, one for each additional puncture. Theboundary curves of these defects can be on the lattice, in whichcase we call the defect smooth (Z type), or on the dual lattice,in which case we call the defect rough (X type). The distanceof such a code is the minimum of the distance between defects,the distance between a defect and the boundary of the lattice,and the circumference of a defect. This is most easily seen byappealing to the homological nature of the code [53].Surface codes with defects were first explored withinthe framework of active quantum error correction. Here,012336-4

ADIABATIC TOPOLOGICAL QUANTUM COMPUTINGPHYSICAL REVIEW A 92, 012336 (2015)we consider an alternative situation in which we constructa Hamiltonian with a ground space that is degenerate andidentical to the code space of a quantum error-correctingcode. The construction of such a Hamiltonian is easy froma theoretical point of view; it is simply the negative sum of thestabilizer generators G:H S.2 S G(1)The constant in front is chosen so that all errors will have anenergy penalty of a least (errors adjacent to a boundary willhave this penalty, while errors away from boundaries will havea penalty of 2 ). Since the set of generators is commutative,the eigenspaces of H can be labeled by their eigenvalues withrespect to the operators S. Because the eigenvalues of all the Sare 1, the ground state of this Hamiltonian is equivalent to thecode space of the quantum code generated by G: S ψ ψ for all S G.Hamiltonians such as those in Eq. (1), which we call stabilizer Hamiltonians, have interesting properties for protectingquantum information. The first property is that operators thatact nontrivially on the code space (the degenerate groundspace) must be nonlocal, having a Pauli weight at least aslarge as the code’s distance. This allows the system to retainits information even when perturbed by a local Hamiltonian[17,54]. For toric codes, surface codes, color codes, and, moregenerally, codes formed from quantum double models [55],this is a partial indication of a topological order in the system.(A more robust indicator would be a nontrivial topologicalentanglement entropy [56–59].)While a stabilizer Hamiltonian is robust to local perturbations, if the system is immersed in a thermal bath, the lifetimeof information encoded into the ground state does not necessarily scale with the size of the system (or the size of the defect fora surface code with defects). For example, for the toric code,the lifetime of this information is proportional to exp(2β )[18], where β (kB T ) 1 is the inverse temperature of thebath. It is widely believed that all stabilizer Hamiltonians withlocal terms embedded in two spatial dimensions have a similarlifetime [60]. The more challenging issue is how to computewith them without increasing the rate at which information isdestroyed. As mentioned in Sec. I, if a stabilizer Hamiltoniandescribes a topologically ordered system possessing anyonswith a sufficiently rich non-Abelian structure, then quantumcomputation can be carried out by creating, braiding, andfusing the anyons. However, it is not entirely clear that onecan controllably create single excitations without also creatingother uncontrolled excitations that could then disorder thesystem, nor how one can move the anyons without causingother anyons to be produced. This has led to the search forself-correcting quantum systems where the excitations arenot pointlike particles like anyons but structures that haveboundaries with dimension [8,19,60]. The energetic cost of anexcitation in such a system is proportional to the size of itsboundary and thus would be robust to errors during creationand movement processes; such a system would energeticallyfavor shrinking the boundaries of the errors to zero, causingthem to vanish. In particular, it has been argued that suchsystems would have a lifetime proportional to their size,indicating that the system and the environment to which itis coupled participate in a form of “self-correction” in whichthe environment that creates the errors can also fix the errors;at a low enough temperature, the rate of the latter processdominates the rate of the former. In this paper, we do notdirectly address the question of self-correction; instead, weattempt to better understand how computation can be doneadiabatically within existing models.III. ADIABATIC CODE DEFORMATIONSBefore showing how to perform the adiabatic deformationsand creation of fiducial states, we briefly review a schemefor performing adiabatic gate teleportation [7] (AGT), as thisgives an idea of how the protocols we introduce below operate.AGT is a procedure for transferring information in one qubitto information in another qubit (with a possible gate applied tothis information) via the use of an adiabatic evolution andan ancillary qubit. The following example is on a systemcomposed of three qubits in which the first and third qubitsare swapped (without a gate applied during the swapping).Initially, the system evolves under a Hamiltonian given byHi (I1 X2 X3 I1 Z2 Z3 ),(2)where Pi represents the operator P acting on the ith qubitand where we soon omit the identity operators I . A finalHamiltonian is defined asHf (X1 X2 I3 Z1 Z2 I3 ).(3)The AGT protocol begins with the information encoded in thefirst qubit and Hi turned on. Then, Hi is adiabatically turnedoff while simultaneously turning on Hf . In other words, theevolution is described byH (t) f (t)Hi g(t)Hf ,(4)where f (0) 1, f (T ) 0, g(0) 0, and g(T ) 1 and T isthe time taken to perform the evolution. If f (t) and g(t) arechosen to be slowly varying and the time T is long enoughsuch that the evolution is adiabatic (meaning here that theprobability of exciting the system out of its ground space ismade small), then the above evolution will take informationin the first qubit and send it to information in the third qubit.For example, one may choose f (t) 1 g(t) and g(t) Ttso that the evolution is made adiabatic for sufficiently large T .A constant error can be achieved for a fixed constant T .To see that a constant energy gap is maintained during theabove evolution and that the information is transported fromthe first to third qubit, it is convenient to use the formalismof stabilizer codes to describe this evolution. Indeed, it isactually useful to define three codes. The first code, call itS1 , is defined by the stabilizer generators X2 X3 and Z2 Z3 andthe logical Pauli operators Z Z1 Z2 Z3 and X X1 X2 X3 . Asecond code, call it S2 , is defined by the stabilizer generatorsX1 X2 and Z1 Z2 and the logical Pauli operators Z Z1 Z2 Z3and X X1 X2 X3 . Suppose information is encoded into thestabilizer code S1 so that it is in the 1 eigenstate of bothX2 X3 and Z2 Z3 . Notice then that because X1 X(X2 X3 )and Z1 Z(Z2 Z3 ), information encoded into this code canbe accessed by making a measurement on the first qubit.Similarly, information encoded into the second code, S2 , is012336-5

CESARE, LANDAHL, BACON, FLAMMIA, AND NEELSPHYSICAL REVIEW A 92, 012336 (2015)localized in the third qubit. The adiabatic evolution in Eq. (4)can now be seen as adiabatically dragging a Hamiltonian thatis a sum over stabilizer generators in S1 to a sum over stabilizergenerators in S2 such that the information in the encoded qu

universal quantum computation in the label space of the anyons. This is known as topological quantum computing [4,30-32], the principal model of quantum computing we will consider here. In a topological quantum computation, one creates anyons from the vacuum, braids them around one another in space-