Approximate exchange-only entangling gates for the three-spin- 1/2 decoherence-free subsystem
aa r X i v : . [ qu a n t - ph ] M a r Approximate exchange-only entangling gates for thethree-spin- / decoherence-free subsystem James R. van Meter
1, 2 and Emanuel Knill
2, 31
Department of Mathematics, University of Colorado, Boulder, Colorado 80309, USA National Institute of Standards and Technology, Boulder, Colorado 80305, USA Center for Theory of Quantum Matter,University of Colorado, Boulder, Colorado 80309, USA
The three-spin-1 / σ y can beimplemented directly. Self-inverse gates that are constructible from these Hamil-tonians, such as the CNOT, can be implemented without the assumption on thephysical spins. We compare the control complexity of implementing CNOT to previ-ous methods and find that the complexity for fault-tolerant fidelities is competitive. I. INTRODUCTION
Three physical spin-1 / /
2, where collective noiseis induced by global fields acting with total spin operators along any axis. This motivatesthe use of the 3DFS as a logical qubit [1, 2]. The 3DFS’s observables are spanned by thethree Hermitian and unitary operators exchanging two of the physical spins, enabling imple-mentation of logical one-qubit gates. These operators are referred to as swaps when used asunitary gates, and as exchange interactions when used as generators of evolution. This def-inition of the exchange interaction agrees with the conventional one up to a multiple of theidentity. We also identify the swap operators with transpositions in the representation of thesymmetric group that acts by permuting the physical spins. The best-known application ofthe 3DFS is to quantum computing with quantum dots where evolution under the exchangeinteraction can be more accessible than universal one-quantum-dot evolutions [3]. Indeedthe requisite control over three quantum dots has been proven feasible by many experimen-tal demonstrations [4–10]. The application to quantum dots is further supported by theability to implement entangling logical gates between two blocks of physical spins carrying3DFSs with exchange interactions only [3], but these gates require many steps and are oftendifficult to construct and understand from first principles (with a possible exception being[11]).In theory, every logical two-qubit gate is generated by exchange Hamiltonians on thephysical spins comprising two 3DFS blocks [12]. In practice constructing such gates ischallenging, and constructing them to be independent of the total spin of the two blockseven more so. Since each block has spin 1/2, the total spin of the system is either 0 or1. While the action on each logical qubit of exchange interactions local to each block isindependent of the total spin, the action of cross-block exchange interactions is in generalspin-dependent. Thus constructing unitaries from cross-block exchange interactions that acton logical qubits independently of the total spin is nontrivial, but essential for entanglinggates that are robust under collective noise on the total system.Much of the work on exchange-only gates has focused on the CNOT gate. The firstexchange-only CNOT gate was found numerically by DiVincenzo et al [3]. It required atotal spin of 1 and was only approximate, albeit very accurate. Subsequently, exact CNOTgates from the same sequence of interactions were found in [13] and [14]. A simpler, exactspin-dependent CNOT was later found in [15]. A spin-independent, exchange-only CNOTgate was first published by Fong and Wandzura [16] (although a solution was also claimed,but not given, in [3]). Surprisingly this exact spin-independent solution was simpler, interms of the number of exchange gates and their analytic coefficients, than that of theoriginal spin-dependent solutions. Since then several other spin-independent, exchange-onlyCNOT gates have been found [17, 18].The question then arises of how to compare the efficiencies of these various implementa-tions. They each consist of a product of gates, each of which exponentiates a Hamiltonianthat depends linearly on exchange interactions. One measure of complexity, then, is to sumthe absolute values of the coefficients on the exchange interactions. Taking the absolutevalue assumes that the exchange interaction can be turned on with either sign (which is per-missible [19–24]). If the exchange interactions are all performed in series, then this measureis proportional to the total evolution time required for the physical operation. Alternatively,if some exchange interactions are performed in parallel, one could instead sum the largestcoefficients for each exponential in the product for the gate, and again this sum is propor-tional to the total evolution time when operating in parallel mode. By either measure, theFong and Wandzura construction [16] is one of the most efficient of the exact, exchange-onlyCNOT gates, and we use it as a benchmark.We give an alternative approach that constructs approximate entangling logical gates be-tween two 3DFSs. The strategy is to decouple cross-block exchange interactions by within-block operations that project the cross-block interactions into the computational two-qubitsubsystem determined by the two 3DFSs. With the orthogonal complement of the compu-tational subsystem thus decoupled, we proceed to construct two-qubit gates from exchangeinteractions and further show that a large subset of these gates are spin-independent. Thistask is facilitated by allowing non-commuting exchange interactions to be performed in par-allel, as made possible for example by semiconductor quantum dot technology [25]. Wethen find that for the spin-independent CNOT gate, an entanglement fidelity better than0 .
99 can be achieved with a total evolution time that is shorter than that of the exactimplementations, and slightly shorter still if spin-dependence is allowed.This paper is organized as follows. In Section II we review the symmetric group andunitary group representation theory needed to derive our results. In particular, we giveexplicit maps of the computational subsystems into the symmetric group representationsthat appear. In Section III we describe our procedure for decoupling the computationalsubsystem for two 3DFSs from its orthogonal, “leakage” space with exchange interactions.In Section IV we apply this method to construct a spin-independent exchange-only CNOTgate, and spin-dependent exchange-only evolutions optimized for total spin 1. We concludewith a discussion and open problems in Section V.
II. REPRESENTATION THEORY
Here we briefly review the relevant representation theory; for more comprehensive dis-cussion of symmetric group representations see [26], of unitary group representations see[27], and of both applied to quantum information see [28]. The special unitary group of d × d matrices is denoted by SU( d ), and the symmetric group of permutations of n elementsis denoted by S n , where we are interested in n = 3 or n = 6. Irreducible representations(irreps) of the groups SU( d ) and S n are conveniently labeled by partitions of n in a canonicalway that is made clear below. For this purpose it is convenient to introduce the notation ν ⊢ n signifying that ν is a partition of n . We also write | ν | for the size of ν , so | ν | = n ,and ν ) for the number of parts of ν . Now we have that for each ν ⊢ n there is an irrepof S n , which we denote by V ν , and for each ν ⊢ n such that ν ) ≤ d there is an irrep of SU ( d ), which we denote by U ν . By convention, each partition can be explicitly denoted bya Young diagram, consisting of a row of cells for each part, arranged in descending order oflength, with n cells total. So for n = 3 the partitions are , , (1)and therefore the irreps of S are V , V , and V and the irreps of SU (2) are U and U .A convenient choice of basis vectors for each irrep of S n is labeled by Young tableaux,defined as follows. A Young tableau is a Young diagram in which every cell is filled in witha number from 1 to n , with no repetitions, such that the numbers are increasing along everyrow from left to right and along every column from top to bottom. So for the Young diagramthe only Young tableau is , (2)and for the Young diagram the Young tableaux are: , . (3)Similarly, a convenient choice of basis vectors for each irrep of SU ( d ) is labeled by Weyltableaux (also known as semistandard tableaux), defined as follows. A Weyl tableau isa Young diagram of at most d rows with every cell filled in with a number from 1 to d ,with repetitions allowed, such that the numbers are nondecreasing along rows from left toright and increasing along columns from top to bottom. To avoid confusion with the Youngtableaux, and because we only consider d = 2, we substitute 1 and 2 in Weyl tableaux by anup arrow and a down arrow, respectively. So for the Young diagram the Weyl tableauxare: ↑ ↑ ↑ , ↑ ↑ ↓ , ↑ ↓ ↓ , ↓ ↓ ↓ , (4)corresponding to the four possible spins along the z axis of the spin 3 / ↑ ↑↓ , ↑ ↓↓ , (5)corresponding to the two states of the spin 1 / T be a Youngtableau and for 1 ≤ i, j ≤ n let d T ( i, j ) ≡ ( c T ( j ) − r T ( j )) − ( c T ( i ) − r T ( i )) where r T ( k )and c T ( k ) respectively denote the row and column containing number k . Then denoting thetransposition in S n that exchanges i and j by ( ij ), and its matrix representation on V λ by ρ λ (( ij )), we have ρ λ (( i i + 1)) T = 1 d T ( i, i + 1) T + s − d T ( i, i + 1) ( i i + 1) T, (6)where ( i i + 1) T denotes direct action on T via exchange of the positions of i and i + 1.Note that ( i i + 1) T results in an invalid tableau (only) when i and i + 1 are in the same rowor column, but then d T ( i, i + 1) = ± i i + 1) generate S n , the above definition is sufficient todetermine the representation of every permutation.Now consider the Hilbert space H of three physical spin-1 / H and ( V ⊗ U ) ⊕ ( V ⊗ U )that commutes with the action of S × SU (2). Such an isomorphism is called an intertwinerwith respect to the group S × SU (2). We denote the existence of such an intertwiner asfollows: H S × SU (2) ∼ = ( V ⊗ U ) ⊕ ( V ⊗ U ) . (7)The Hilbert space on the right-hand side can further be expressed in terms of tableau basisvectors: V ⊗ U ∼ = span { ⊗ ↑ ↑ ↑ , ⊗ ↑ ↑ ↓ , ⊗ ↑ ↓ ↓ , ⊗ ↓ ↓ ↓ } , (8) V ⊗ U ∼ = span (cid:26) ⊗ ↑ ↑↓ , ⊗ ↑ ↑↓ , ⊗ ↑ ↓↓ , ⊗ ↑ ↓↓ (cid:27) . (9). The above basis vectors can be related by linear transformation to the conventionalproduct states of the three physical spins, specified in terms of spin basis, by demandingthat each swap of the qubits should be consistent with the action of the representation ofeach transposition on the tableau basis. Here we identify the action of transpositions on theproduct basis with that of swaps in the obvious way:( ij ) | b · · · b i · · · b j · · · b n i = | b · · · b j · · · b i · · · b n i . (10)If we further assume the convention that the Weyl tableaux are eigenvectors of the z -component of the spin operator, then up to an overall phase factor we find the followingcorrespondence with the two-qubit 3DFS encoding of [3]: ⊗ ↑ ↑↓ √ ( | i − | i ) ≡ | L i| ↑i (11) ⊗ ↑ ↓↓ √ ( | i − | i ) ≡ | L i| ↓i (12) ⊗ ↑ ↑↓ √ (2 | i − | i − | i ) ≡ | L i| ↑i (13) ⊗ ↑ ↓↓ √ (2 | i − | i − | i ) ≡ | L i| ↓i . (14)Therefore the Young tableaux can be identified with logical 0 (0 L ) and logical 1 (1 L ) asindicated above. It is now clear in terms of representation theory that the independence ofthis qubit from collective noise on the spin is due to independence from the action of SU (2).We now consider entangling two such logical qubits. For the goal of constructing gates,we need to calculate the action of cross-block exchange interactions on these qubits. To thatend we make use of the following theorem, adapted from [28]: Theorem 1.
Given irreps V λ of S | λ | and V µ of S | µ | and irreps U λ and U µ of SU ( d ) , ( V λ ⊗ U λ ) ⊗ ( V µ ⊗ U µ ) S | λ | × S | µ | × SU ( d ) ∼ = M ν ⊢| λ | + | µ | ν ) ≤ d c νµλ V λ ⊗ V µ ⊗ U ν , (15) where the isomorphism is an intertwiner with respect to the indicated group action and SU ( d ) is assumed to act simultaneously on U λ and U µ on the left hand side. Further, the right hand side is isomorphic to a subspace of the Schur-Weyl sum, M ν ⊢| λ | + | µ | ν ) ≤ d c νµλ V λ ⊗ V µ ⊗ U ν S | λ | × S | µ | × SU ( d ) ֒ → M ν ⊢| λ | + | µ | ν ) ≤ d V ν ⊗ U ν , (16) where the inclusion map is a term-by-term intertwiner with respect to the indicated groupaction such that each Littlewood-Richardson coefficient c νλµ gives the multiplicity of the imageof V λ ⊗ V µ on the left hand side in each corresponding V ν on the right hand side. For the case of λ = µ = and d = 2, the Littlewood-Richardson coefficients are suchthat( V ⊗ U ) ⊗ ( V ⊗ U ) ∼ = ( V ⊗ V ⊗ U ) ⊕ ( V ⊗ V ⊗ U ) ֒ → ( V ⊗ U ) ⊕ ( V ⊗ U ) (17)where both maps are intertwiners with respect to S × S × SU (2), and term-by-term theinclusion map implies the following two intertwiners: V ⊗ V S × S ֒ → V (18) V ⊗ V S × S ֒ → V . (19)The left hand side encodes our two logical qubits, while its four-dimensional image in thefive-dimensional V and nine-dimensional V is the computational subspace of each. Thatthese intertwiners are with respect to S × S implies that the actions of transpositions localto each block, on the computational subspace, are consistent across irreps. These maps alsodetermine the actions of cross-block transpositions on the computational subspace, albeit inan irrep-dependent manner. This is one of our primary concerns for the remainder of thispaper.Before proceeding it is instructive to consider the SU (2) irreps in order to form a morecomplete physical interpretation. Observe that the single Weyl basis vector of U is ↑ ↑ ↑↓ ↓ ↓ , (20)and the Weyl basis vectors of U are ↑ ↑ ↓ ↓↓ ↓ , ↑ ↑ ↓ ↓↑ ↓ , ↑ ↑ ↓ ↓↑ ↑ . (21)By our convention, these basis vectors are eigenvectors of the z component of the totalspin operator, with eigenvalues 0 (Eq. (20)), and -1, 0, 1 (Eq. (21)) respectively. Thereforediagram is associated with states of total spin 0 and diagram is associated withstates of total spin 1.Returning to symmetric group irreps, and henceforth overloading Young diagrams todenote symmetric group irreps directly, notice that irrep has one extra dimension andirrep has five extra dimensions outside of the images of representation ⊗ underthe intertwiners of Eqs. (18-19). The extra dimension in irrep and one of the extradimensions in irrep each correspond to the image of representation ⊗ underthe respective intertwiner implied by Theorem 1 for λ = µ = , which is associated withstates for which each block has total spin 3 /
2. The four remaining dimensions of irrep ⊗ (cid:1) (cid:1) ☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ (cid:29) (cid:29) ❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁ ⊗ ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ ⊗ ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ ⊗ ] ] ❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁ @ @ ✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁ FIG. 1. The injective intertwiners with respect to S × S of the 4-dimensional representation ⊗ , the 2-dimensional representation ⊗ , the 2-dimensional representation ⊗ , andthe 1-dimensional representation ⊗ into the 5-dimesional irrep and the 9-dimensionalirrep . correspond to images of representations ⊗ and ⊗ , which are associated withproducts of logical qubit states with spin 3/2 states. The multiplicity of each image is onein every case, as determined by the Littlewood-Richardson coefficients, and further clarifiedby the following vector space isomorphisms: ∼ = (cid:18) ⊗ (cid:19) ⊕ ( ⊗ ) (22) ∼ = (cid:18) ⊗ (cid:19) ⊕ (cid:18) ⊗ (cid:19) ⊕ (cid:18) ⊗ (cid:19) ⊕ ( ⊗ ) . (23)These various maps are illustrated in Fig. 1.For the purpose of constructing gates we need to specify the intertwiners ⊗ ֒ → and ⊗ ֒ → explicitly. It suffices to map the Young tableaux basis (cid:26) ⊗ , ⊗ , ⊗ , ⊗ (cid:27) (24)of representation ⊗ into irreps and . The images of each vector in this basisare determined up to overall phase by demanding that the action of S { } × S { } on itbe consistent with the action of S { } × S { } on the original basis, where S { ijk } denotespermutations of { i, j, k } . We then find that for ⊗ ֒ → , ⊗ − √ , (25) ⊗
7→ − √ − , (26) ⊗ − √ , (27) ⊗
7→ − √ − , (28)and for ⊗ ֒ → , ⊗ − √ , (29) ⊗
7→ √ + 16 − √ , (30) ⊗ − √ , (31) ⊗
7→ √ + 16 − √ . (32)The resulting right hand sides give the computational basis in each irrep. Note the abovenumbering is consistent with the physical qubit “nearest neighbors” and corresponding prod-uct state ordering assumed by DiVincenzo et al. [3, 13]. III. DECOUPLING
Each computational basis defined above spans what we call the computational subspaceof each irrep, allowing us to further define the computational submatrix of each matrix rep-resentation of a permutation as its projection onto the computational subspace. We findthat these computational submatrices of representations of transpositions in irreps andare rich and varied enough that their linear combinations yield most of the computa-tional Pauli basis. Therefore if we were able to evolve according to projections onto thecomputational subspace using only exchange interactions, we would have a simple way toconstruct a wide variety of Hamiltonians and gates. In this section we present just such amethod, to good approximation. Our strategy is to remove transitions between the com-putational subspace and its complement with the general decoupling procedure introducedin [31, 32]. Decoupling involves interspersing pulses with evolution so that the resultingeffective Hamiltonian preserves the computational subspace while acting as the projectionof the original Hamiltonian on this subspace.For the purpose of evolving under operators projected onto the computational subspace,the symmetric sums of local transpositions prove to be useful:Σ a ≡
13 ((12) + (13) + (23)) (33)Σ b ≡
13 ((45) + (46) + (56)) . (34)It can be easily verified that Σ a and Σ b each act as zero on the computational subspace, byaccordingly exchanging the physical spins of the product states that constitute the compu-tational basis.For the sake of consistency with our representation theory approach we introduce thegroup algebra C S , which we have just used implicitly. This consists of all finite linearcombinations of S , C S ≡ { a σ + · · · + a m σ m | a i ∈ C , σ i ∈ S , m ∈ N } , (35)as for example Σ a and Σ b . The group algebra comes equipped with multiplication, addition,and scalar multiplication in the obvious ways. It is then straightforward to linearly extendYoung’s orthogonal representation from S to C S , which we assume henceforth. As expectedwe find ρ λ (Σ a ) and ρ λ (Σ b ) are each null on the computational subspace.The sums Σ a and Σ b have additional properties that make them well-suited to the taskof performing effective projection operations. As they consist of mutually nonoverlappingtranspositions, they commute, and therefore their representations can be simultaneouslydiagonalized. We then have for irrep , ρ (Σ a ) = diag(0 , , , , , (36) ρ (Σ b ) = diag(0 , , , , , (37)and for irrep , ρ (Σ a ) = diag(0 , , , , , , , , , (38) ρ (Σ b ) = diag(0 , , , , , , , , . (39)where here and henceforth the first four basis vectors of irrep are given by the right handsides of Eqs. (25-28), in that order, the first four basis vectors of irrep are given bythe right hand sides of Eqs. (29-32), in that order (thus corresponding in both cases to thecomputational basis {| i L , | i L , | i L , | i L } ), and the remainder of each basis is suchthat the following properties hold. The eigenspace of the final 1 in every case is the imageof representation ⊗ , the remaining two-dimensional 1-eigenspace in ρ (Σ a ) is theimage of representation ⊗ , and the remaining two-dimensional 1-eigenspace in ρ (Σ b )is the image of representation ⊗ . So Σ a and Σ b can be interpreted as projectors ontostates that are disallowed by the computational subspace. Furthermore Σ a + Σ b = 0 on thecomplement of the computational subspace in both irreps.To put these sums to use, we define the following unitaries, U a ≡ exp( iπρ λ (Σ a )) (40) U b ≡ exp( iπρ λ (Σ b )) , (41)where λ equals partition or . For λ = , and again in the diagonalizing basis0(which includes the computational basis), U a ≡ diag(1 , , , , −
1) (42) U b ≡ diag(1 , , , , −
1) (43)and for λ = , U a ≡ diag(1 , , , , − , − , , , −
1) (44) U b ≡ diag(1 , , , , , , − , − , − . (45)It follows that, for all H ∈ ρ λ ( C S ) and λ = or ,14 ( H + U a HU † a + U b HU † b + U b U a HU † a U † b ) = D ( H ) , (46)where D denotes a decoupling function which cancels cross terms with the computationalsubspace. Alternatively we can define U ≡ exp (cid:16) i π ρ λ (Σ a + Σ b ) (cid:17) , (47)and obtain in both irreps,14 ( H + U HU † + U † HU + U H ( U ) † ) = D ( H ) , (48)with the choice of unitaries depending on which is more convenient for the problem at hand.In terms of projection operators, D ( H ) ≡ (cid:26) Π H Π + Π ⊗ H Π ⊗ , H ∈ ρ ( C S ) , Π H Π + Π ⊗ H Π ⊗ + Π ⊗ H Π ⊗ + Π ⊗ H Π ⊗ , H ∈ ρ ( C S ) , (49)where Π projects onto the computational subspace while Π ⊗ , Π ⊗ , and Π ⊗ projectonto the subspaces comprising the complement of the computational subspace as indicatedin Fig. 1 and Eqs. (22-23). Noting that ρ ( C S ) ∼ = M ( C ), if we let H = ( h ij ) ∈ ρ ( C S )then we obtain D ( H ) = h h h h h h h h h h h h h h h h
00 0 0 0 h . (50)On the other hand noting that ρ ( C S ) ∼ = M ( C ), if we let H = ( h ij ) ∈ ρ ( C S ) then we1obtain D ( H ) = h h h h h h h h h h h h h h h h h h h h h h
00 0 0 0 0 0 h h
00 0 0 0 0 0 0 0 h . (51)We now recognize projections onto the four-dimensional computational subspace, the two-dimensional subspaces associated with ⊗ and ⊗ , and the one-dimensional sub-space associated with ⊗ , as discussed in Section II.Our task, then, is to express the above projection procedure in terms of physically-realizable, exchange-only operations. Therefore we “Trotterize”, that is approximateexp( iαD ( H )) = exp iα X j =1 U j HU † j !! (52)as a product of unitaries in the form of exponentials depending on linear combinations ofexchange interactions, where α is some angle and { U j } j =1 is one of { , U a , U b , U a U b } or { , U, U † , U } . Letting { A j } kj =1 be a set of non-commuting operators we can use for examplethe zeroth order exponential product expansion,exp k X j =1 A j ! = k Y j =1 exp (cid:18) n A j (cid:19)! n + O (cid:18) n (cid:19) , (53)or the first order Suzuki-Trotter expansion [33, 34],exp k X j =1 A j ! = k Y j =2 exp (cid:18) n A k +2 − j (cid:19) exp (cid:18) n A (cid:19) k Y j =2 exp (cid:18) n A j (cid:19)! n + O (cid:18) n (cid:19) , (54)or still higher order approximations. Therefore, noting thatexp (cid:16) iαU j HU † j (cid:17) = U j exp( iαH ) U † j , (55)and using { U j } j =1 = { U , U, , U † } , we have to zeroth order thatexp( iαD ( H )) = (cid:0) U exp( iδtH )( U ) † U exp( iδtH ) U † exp( iδtH ) U † exp( iδtH ) U (cid:1) n + O ( δt )(56)2and to first order thatexp( iαD ( H )) = (cid:18) U † exp (cid:18) i δt H (cid:19) U exp (cid:18) i δt H (cid:19) U exp (cid:18) i δt H (cid:19) U † U exp( iδtH )( U ) † × U exp (cid:18) i δt H (cid:19) U † exp (cid:18) i δt H (cid:19) U † exp (cid:18) i δt H (cid:19) U (cid:19) n + O ( δt ) (57)= U † (cid:18) exp (cid:18) i δt H (cid:19) U (cid:19) exp( iδtH ) (cid:18) U † exp (cid:18) i δt H (cid:19)(cid:19) ! n U + O ( δt ) . where δt = α/ n . IV. HAMILTONIAN AND GATE CONSTRUCTIONA. One qubit: Spin-independent Hamiltonians and gates
We now explore which combinations of exchange interactions yield useful computationalsubmatrices. First we review the relationship between local exchange interactions and Paulimatrices. The X and Z Pauli matrices on the first qubit can be constructed from any twolocal exchange interactions within the first block as follows: − √ − √ − Π ρ λ (12)(13) Π = √ √ − Π ρ λ (12)(23) Π= − √ √ Π ρ λ (13)(23) Π = XIZI , (58)where the representation ρ λ and projector Π are understood to operate on each componentof each column vector of transpositions. Thus for example the first row of the left-most termimplies − √ ρ λ (12)Π − √ ρ λ (13)Π = XI (59)where the projector appears in this equation only for consistency of matrix dimensions acrossthe equal sign (decoupling is not required for single-qubit gates), and by a component ofthe form AB , where A, B ∈ {
I, X, Z } , we mean a tensor product of the indicated Paulimatrix on the first qubit with the indicated Pauli matrix on the second qubit. Similarlylocal exchange interactions within the second block yield Eq. (58) but with (12), (13), and(23) replaced by (45), (46), and (56) respectively, and XI and ZI replaced by IX and IZ respectively.Now we focus on one of the blocks and so momentarily dispense with tensor productswith the identity. We can obtain any Hamiltonian of the form aX + bZ from local exchangeinteractions, up to phase. A Hamiltonian of the form aX + bY can then be obtained byconjugation with exp (cid:0) i π X (cid:1) , which transforms Z to Y , and a Hamiltonian of the form aY + bZ can be obtained by conjugation with exp (cid:0) i π Z (cid:1) , which transforms X to Y .More generally, for any single-qubit Hamiltonian H and time t it follows from a Cartan3decomposition of SU (2) (a generalized Euler angle decomposition of rotations) [35] thatexp( iHt ) = exp( iδI ) exp( iαX ) exp( iβZ ) exp( iγX ) , (60)for some α, β, γ, δ ∈ R , which can then be expressed in terms of local exchange interactionsby Eq. (58). Thus we can construct any single-qubit gate from local exchange interactions.Such constructions are independent of the choice of irrep ( or ) since the relevantintertwiners preserve the action of local exchange interactions. B. Two qubits: Spin-dependent Hamiltonians and spin-independent gates
Constructing exchange-only Hamiltonians for two-qubits is more challenging. Fromcalculations performed in Maple 2016 (see ancillary files spin0_exchange_gates and spin1_exchange_gates ) we find:
15 15 15 15 15 15 15 15 15 −√ √ −√ √ −√ √ − − − − − − −√ −√ −√ √ √ √ − − − − − − − −
32 32 √ √ −√ − √ − √ √ √ − √ √ − √ −√ √
12 12 −
12 12 − − − Π ρ λ (14)(15)(16)(24)(25)(26)(34)(35)(36) Π = a λ b λ IIIXIZXIZIXXXZZXZZ , (61)where a λ = b λ = 1 for λ = and a λ = − b λ = − / λ = . Thus withineach irrep we can construct any linear combination of Pauli Hamiltonians excluding Y bydecoupling sums of exchange interactions. As before, combinations of X and Y or Z and Y on one qubit can be obtained by appropriate conjugation with exp (cid:0) i π ( ZI ) (cid:1) , exp (cid:0) i π ( XI ) (cid:1) ,exp (cid:0) i π ( IZ ) (cid:1) , or exp (cid:0) i π ( IX ) (cid:1) .For any two-qubit gate G it follows from a Cartan decomposition of SU (4) that [35] G = K exp( iαXX + iβY Y + iγZZ ) K , (62)for some local unitaries K and K and real coefficients α, β, γ . The local unitaries can beexpressed in terms of X and Z as discussed previously. The remaining exponential can befurther decomposed since XX , Y Y , and ZZ all commute; for example: G = K exp( iαXX ) exp( iβY Y ) exp( iγZZ ) K (63)= K exp( iαXX ) exp (cid:16) i π XI + IX ) (cid:17) exp( iβZZ ) × exp (cid:16) − i π XI + IX ) (cid:17) exp( iγZZ ) K . (64)4The above Hamiltonian and gate constructions are irrep-dependent, in general. However,consider an exchange-only Hamiltonian H equal to a combination of the above Pauli terms inthe irrep that squares to the identity. Then using that H equals the same combination ofPauli terms multiplied by -3 in the irrep (neglecting a possible identity term contributionto overall phase) we have by Euler’s formula for matrices that exp (cid:0) i π H (cid:1) is irrep-independent.For example,exp (cid:18) i π ρ ((14) − (15) − (24) + (25))Π (cid:19) = exp (cid:18) − i π XX (cid:19) (65)= iXX = exp (cid:16) i π XX (cid:17) = exp (cid:18) i π ρ ((14) − (15) − (24) + (25))Π (cid:19) Similarly the general gate G in Eq. (64) is irrep-independent for coefficients α, β, γ ∈{− π/ , , π/ } . This procedure is used for the CNOT gate in the next section. C. Spin-independent CNOT
Observe that 12 ( IX − ZX ) = . (66)Further, letting N ≡ √
34 ((15) − (14) + (25) − (24)) , (67)we immediately obtain from subtraction of the 2nd and 8th rows of Eq. (61) thatΠ ρ ( N )Π = 12 ( IX − ZX ) , (68)and Π ρ ( N )Π = − ρ ( N )Π . (69)Clearly then exp (cid:16) i π ρ ( N )Π (cid:17) = exp (cid:16) i π ρ ( N )Π (cid:17) (70)by the same logic as in the example above (Eq. (66)), albeit with the Hamiltonian Π ρ ( N )Πsquaring to a two-dimensional rather than four-dimensional identity. Meanwhile I − (12),being local, acts the same on the computational subspace of both irreps:Π 12 ρ ( I − (12))Π = Π 12 ρ ( I − (12))Π = diag(1 , , , . (71)51 ✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕ ✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮ ✁✁✁✁✁✁✁ ❂❂❂❂❂❂❂ FIG. 2. Qubits arranged so that exchange interactions in the spin-independent CNOT are be-tween neighbors. Solid lines denote exchange interactions in the Hamiltonian yielding the CNOTsubmatrix while dotted lines denote local interactions used only for the decoupling procedure.
Thus by Eq. (58) the following expression maps to an approximate CNOT on the computa-tional subspace of both irreps,exp (cid:16) − i π I + (12)) (cid:17) U † (cid:18) exp (cid:18) i δt N (cid:19) U (cid:19) exp( iδtN ) (cid:18) U † exp (cid:18) i δt N (cid:19)(cid:19) ! n U = CNOT + O ( δt ) , (72)where δt = π/ n , and for ease of notation we have dropped the explicit ρ λ mapping as wellas the decoupled orthogonal projection.Conveniently, the above Hamiltonian and its Trotterization can be recast as neighboringexchange interactions in planar geometry (Fig. 2), where (12) has been removed from U because it commutes with N . (In general if any one local exchange interaction commuteswith the Hamiltonian, then since it also commutes with the sum of the remaining localinteractions it cancels during conjugation and can thus be removed.) That (12) commuteswith N is due to the fact that N is symmetric with respect to qubits 1 and 2 and thusinvariant under conjugation with (12). We add that the exact diagram in Fig. 2 may notbe physically practical due to the relative distances between qubits 1, 4 and 5, but morepractical architectures may be possible.To evaluate the efficiency of our approximate CNOT we can define a normalized operationtime by summing the maximum magnitude of the coefficients on the exchange interactions ineach exponential and dividing by π/ n cycles time fidelity leakage3 39 8.5 0.99136 0.005525 63 12.5 0.99888 0.000709 111 20.5 0.99989 0.00007 TABLE I. Clock cycles, normalized time (such that one unit equals the duration of one swap),entanglement fidelity (Eq. (73)), and leakage (Eq. (74)) for various iterations n of the spin-independent CNOT approximation given by Eq. (72). The listed values for fidelity and leakageassume a total spin of 1. The values for cycles and time may be compared with that of the fiducialexact solution of [16], with 13 clock cycles and a normalized time of 12.3. entanglement fidelity [36], F ( G, CNOT ) = (cid:18)
14 tr(Π G † CNOT ) (cid:19) . (73)We also calculate the contribution from leakage out of the computational subspace to 1 − F : L ( G, CNOT ) = 14 tr(Π G † CNOT Π ⊥ CNOT † G ) , (74)where G is the gate under consideration and CNOT has been extended by identity to the com-plement of the computational subspace. Then for various iterations n of the approximationabove we obtain the values in Table I, where the fidelity and leakage values assume a totalspin of 1, this being the worst case (heuristically because its higher dimensional subsystemhas more cross-terms to decouple). If negative exchange interactions are canceled in theoptimal way described in Section V, the normalized time is increased by 1.3 in every casebut the fidelities and leakages are unchanged. The above times are comparable to that ofone of the most efficient exact solutions [16], which has a normalized time of 12.3 (includingthe local unitaries required for consistency with our computational basis) or 14.8 if negativeexchange interactions are canceled. D. Spin 1 optimized CNOT
Alternatively the qubits can be prepared with a total spin of 1 by the use of magneticfields [3], in which case spin-independence is not a concern, and the IX term in the abovederivation of CNOT can be expressed in terms of local interactions instead of cross-blockinteractions. In combination with the noncommuting cross-block interactions in the ZX term, this replacement breaks irrep-independence, since the two kinds of interactions trans-form differently across irreps. However the resulting approximation is made slightly moreefficient, as we show below.Let N ≡ √
34 ((56) − (46) + 3((34) − (35))) . (75)76 ✁✁✁✁✁✁✁ ❂❂❂❂❂❂❂ ❂❂❂❂❂❂❂ ✁✁✁✁✁✁✁
31 2
FIG. 3. Qubits arranged so that exchange interactions in the spin-dependent CNOT are betweenneighbors. Solid lines denote exchange interactions in the Hamiltonian yielding the CNOT subma-trix while dotted lines denote local interactions used only for the decoupling procedure. n cycles time fidelity leakage2 21 9 . . TABLE II. Clock cycles, normalized time (such that one unit equals the duration of one swap),entanglement fidelity (Eq. (73)), and leakage (Eq. (74)) for various iterations n of the spin 1 CNOTapproximation given by Eq. (77). The above values for cycles and time may be compared withthat of the fiducial exact solution of [16], with 13 clock cycles and a normalized time of 12.3. Then for spin 1 we have Π ρ ( N )Π = CNOT . (76)Unlike the previous spin-independent expression, N commutes with U b N U † b and similarly U a N U † a commutes with U a U b N U † b U † a . This simplifies the Suzuki-Trotter approximation forexp (cid:16) iα ( N + U a N U † a + U b N U † b + U b U a N U † a U † b ) (cid:17) and we findexp (cid:16) − i π I + (12)) (cid:17) ( T / U a T / U † a U b T U a T U † b T / U † a T / ) n = CNOT + O ( δt ) , (77)where T ≡ exp( iδtN ) and δt = π/ n . Again this can be arranged as neighboring exchangeinteractions in a plane (Fig. 3), where (12) commutes with N and so can be removed from U a .The smaller truncation error that results from this expression yields the results shownin Table II. Again if subtracted exchange interactions are canceled the normalized time isincreased by 1.3 in every case.8 V. DISCUSSION
For the case of two logical qubits encoded on a decoherence-free subsystem of threephysical spins each, we have presented an exchange-only method to decouple the resultingcomputational subspace from leaked states. Using Trotterization we showed how this methodcan be implemented to any desired fidelity. We then applied this procedure to explicitlyconstruct Pauli Hamiltonians from exchange interactions. Building on this we provided atransparent method to construct two-qubit gates from exchange interactions, including alarge family of spin-independent gates. In particular we have shown that an exchange-onlyCNOT gate can be implemented with a computational speed comparable to or better thanprevious solutions, with good fidelity, provided the exchange interactions can be performedin parallel. The fidelities for these constructions exceed 0 .
99, which is comparable to fidelitiesdemonstrated for one logical qubit in three quantum dots [10], and for two-qubit gates withother quantum devices [37–39]. Fidelities exceeding the conservative requirements for fault-tolerance [40–42] can be achieved with a slow down of about a factor of two.We note that for quantum dots the negative coefficients on exchange interactions inmany of the expressions above would seem to require negative charging energies [43]. This ispossible by coupling to magnetic fields or other means [19–24], but if it proves impractical,the negative coefficients can be removed by various methods. Of course for an exponentialof a single negative exchange interaction one may simply add an integer multiple of 2 π toits coefficient.To address the more general case of multiple negative exchange interactions, we observethat the sum of all transpositions in S acts as a constant on the irreps, ρ λ X j = k =1 ( jk ) ! = c λ I, (78)where c = 3 and c = 5. Therefore adding the sum of all exchange interactions toan exchange-only Hamiltonian, with a coefficient equal to the largest magnitude of thenegative coefficients in that Hamiltonian, will cancel that negative coefficient and ensurethat all other resulting coefficients are nonnegative. In general this results in an overall phasedifference between the spin 0 and spin 1 cases, due to the different constants effectively beingadded, but our encoding is independent of such a phase. Alternatively for negative localexchange interactions we can add an appropriate multiple of the sum of all local exchangeinteractions, for one or both blocks as needed, as that sum acts as zero on the computationalsubspace (Eqs. (36-39)). It follows that for negative cross-block interactions we can add anappropriate multiple of the sum of all cross-block interactions, as that sum acts as 3 I or5 I on the computational subspace depending again on whether the total spin is 0 or 1. Tosummarize in more physical terms, the sums discussed above commute with the logical two-qubit observables, which are in the group algebra generated by S × S , and can thereforebe added to the Hamiltonian as needed without changing relevant expectation values.We end with some open questions for future work. Aside from the Pauli and CNOT gates,it may be of interest to find efficient expressions for other gates and evaluate their efficienciesas we did for CNOT. Regarding CNOT, we sought to minimize the time required for givenfidelities, but one might instead seek to minimize the clock cycles or circuit depth. Relatedto this, it may be of practical interest to approximate our parallel interaction expressions bystrictly serial expressions. Finally, our decoupling procedure employed a first order Suzuki-9Trotter approximation, which is optimal for the fidelities considered here, but higher orderapproximations might be advantageous if higher fidelities are desired. Although the firstorder approximation can achieve any fidelity with sufficiently many iterations, the higherorder approximations require fewer iterations and for large enough fidelity they are moreefficient. ACKNOWLEDGMENTS
JRvM wishes to thank M. Ly, K. Mayer, and N. Thiem for helpful discussions. Thiswork includes contributions of the National Institute of Standards and Technology, whichare not subject to U.S. copyright. The identification of any product or trade names is forinformational purposes and does not imply endorsement or recommendation by the Na-tional Institute of Standards and Technology. This work was performed under the financialassistance award 70NANB18H006 from U.S. Department of Commerce, National Instituteof Standards and Technology. JRvM also acknowledges the support of the ProfessionalResearch Experience Program at the National Institute of Standards and Technology. [1] P. Zanardi. Stabilizing quantum information.
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