Scalable star-shape architecture for universal spin-based nonadiabatic holonomic quantum computation
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Scalable star-shape architecture for universal spin-based nonadiabatic holonomicquantum computation
Vahid Azimi Mousolou
1, 2 Department of Mathematics, Faculty of Science,University of Isfahan, Box 81745-163 Isfahan, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran
Nonadiabatic holonomic quantum computation as one of the key steps to achieve fault tolerantquantum information processing has so far been realized in a number of physical settings. However,in some physical systems particularly in spin qubit systems, which are actively considered for real-ization of quantum computers, experimental challenges are undeniable and the lack of a practicallyfeasible and scalable scheme that supports universal holonomic quantum computation all in a singlewell defined setup is still an issue. Here, we propose and discuss a scalable star-shape architec-ture with promising feasibility, which may open up for realization of universal (electron-)spin-basednonadiabatic holonomic quantum computation.
I. INTRODUCTION
Holonomic quantum computation [1–4] is recognisedamong key approaches to fault resistant quantum com-putation. Nonadiabatic holonomic quantum compu-tation [2–4] compared to its adiabatic counterpart [1]is more compatible with the short coherence time ofquantum bits (qubits). To achieve a feasible platform,nonadiabatic holonomic quantum computation has beenadapted and developed for different physical settings [2–10]. Nonadiabatic holonomic quantum computation hasalso been combined with decoherence free subspaces [21–28], noiseless subsystems [29], and dynamical decoupling[30] to further improve its robustness. Experimental re-alizations of nonadiabatic holonomic quantum computa-tion in various physical systems, such as NMR [11, 12],superconducting transmon [13–15], and NV centers in di-amond [16–20] have been carried out.Nevertheless, the implementation of nonadiabaticholonomic quantum computation in some physical sys-tems particularly in spin qubit system, which is one ofthe natural and promising candidates to built quantumcomputers upon, has been remained at the level of single-qubit gates. In fact, from practical perspectives, estab-lishing a scalable multipartite scheme, which possessesfull holonomic computational power for quantum pro-cessing, in these physical systems is still a challenge.In this paper, we aim to address this issue by proposinga scalable architecture for universal spin-based nonadia-batic holonomic quantum computation, which enjoys areasonable capability of being implemented with currenttechnologies. We consider a star-shape system, wherein principal an arbitrary number of register spin qubitsare arranged about and all coupled to an auxiliary spinqubit in the middle of architecture. Universal holonomicsingle-qubit gates are achieved by controlling the cou-pling between two computational basis states of regis-ter qubits through local transverse magnetic fields. Themiddle auxiliary spin qubit introduces an indirect bridge coupling between each pair of register qubits bringingabout a double Λ structure, which permits to implementholonomic entangling gates between selected pair.
II. SCALABLE ARCHITECTURE
The model system that we have in mind is a scalable n register spin qubits coupled in a star-shape architecturethrough an auxiliary spin qubit as depicted in Fig. 1. J k J l B k ⊥ B l ⊥ a FIG. 1. (Color online) Scalable star-shape architecture foruniversal spin-based nonadiabatic holonomic quantum com-putation. An arbitrary n number of register spin qubits ar-ranged about and all coupled to an auxiliary spin qubit de-ployed in the middle of architecture. Each register qubit isallowed to interact with a controllable local magnetic field. Since only universal single-qubit and two-qubit gatesare needed to achieve a universal quantum informationprocessing, the Hamiltonian adapted here is a collectivesingle-qubit and two-qubit Hamiltonians given by H = n X k =1 H k + n X k,l =1 H kl . (1)For single-qubit Hamiltonians we consider H k = B ⊥ k · S ( k ) , (2)which describes the interaction of the k th spin qubit, S ( k ) = ( S ( k ) x , S ( k ) y , S ( k ) z ), with a local transverse magneticfield B ⊥ k = ( B xk , B yk , H kl = J k H ( k ) XY + J l H ( l ) XY , (3)where H ( • ) XY = S ( • ) x S ( a ) x + S ( • ) y S ( a ) y , bullet stands forthe corresponding superscript k or l , and a representsthe auxiliary qubit. The H kl describes a three-bodyanisotropic interaction between the corresponding reg-ister qubits k, l , and the auxiliary qubit. The J k and J l are the exchange coupling strengths to the auxiliaryqubit. In fact, the two-qubit Hamiltonian in Eq. (3) in-troduces an indirect coupling between the selected tworegister qubits k and l through the auxiliary qubit. Thiscan be seen in Fig. 1 as well.In the following, we discuss the realization of a uni-versal family of single-qubit and two-qubit gates in thissetup. A. Singel-Qubit Gates
For a single-qubit gate on the given qubit k , we onlyturn on the single-qubit Hamiltonian, H k , in the collec-tive Hamiltonian given in Eq. (1) by exposing the qubit k to a local transverse magnetic field B ⊥ k . During thisimplementation, we assume that the other terms in Eq.(1) are kept off. Thus, our effective Hamiltonian in thiscase is H k = B ⊥ k · S ( k ) = B ~n · ~σ, (4)where B and ~n = (cos β, sin β, B ⊥ k in the xy plane. The ~σ = ( σ x , σ y , σ z )is the standard Pauli operators and ~ = 1 from now on.To achieve holonomic single-qubit gates, we considercyclic evolutions of an arbitrary qubit state | ψ i = cos θ | i + e iφ sin θ | i , (5)in which only geometric phases are relevant. Explicitly,we are interested in evolutions U ( τ , τ ) | ψ i = exp[ − i Z ττ H k ( s ) ds ] | ψ i (6), along which no dynamical phases occur, for instancethe condition h ψ | U † ( τ , t ) H k ( t ) U ( τ , t ) | ψ i = 0 (7) is satisfied at any time t ∈ [ τ , τ ] [34]. Considering alocal transverse magnetic field B ⊥ k = B~n with constantphase β , reduces the condition in Eq. (7) to one of thefollowing simplified conditions:( i ) φ − β = (2 m + 1) π , m = 0 , ± , ± , ... ( ii ) θ = 0 or π (8)This follows from the fact that [ H k ( t ) , U ( τ , t )] = 0 atany time t , when the phase β is fixed constant.In the light of the above simplified conditions, we carryout our cyclic evolution in the following three steps: • Step 1: We first evolve the general initial state | ψ i to the computational basis state | i by turn-ing on the local transverse magnetic field B ⊥ k = B~n for a time interval [0 , τ ] with constant phase β = φ − π and (time-dependent) strength B suchthat R τ Bdt = θ . Hence, we have U (0 , τ ) | ψ i = | i . (9) • Step 2: Next, we evolve the state | i , all theway along the meridian of the Bloch sphere cor-responding to the fixed azimuthal angle ˜ φ , to thestate e i ˜ φ | i by employing the constant magneticphase β = ˜ φ + π/ B for a time interval [ τ , τ ] suchthat R τ τ Bdt = π . Namely, U ( τ , τ ) | i = e i ˜ φ | i . (10) • Step 3: Finally, we run the Hamiltonian H k foranother time interval [ τ , τ ] with fixed magneticphase β = φ − π/ B such that R τ τ Bdt = π − θ . This wouldevolve the final state of step 2, i.e., the state e i ˜ φ | i ,into the final state e i ∆ φ | ψ i , where ∆ φ = ˜ φ − φ . Inother words (11) U ( τ , τ ) e i ˜ φ | i = e i ∆ φ | ψ i . (12)We illustrate the above three steps evolution on theBloch sphere in Fig. 2. As shown in Fig. 2, the firstand third steps evolve the qubit state along the meridianof the Bloch sphere corresponding to the fixed azimuthalangle φ . It is important to note that the parameters θ and φ are constant during the evolution and the magneticstrength B is the only allowed time-dependent controlvariable. At the completion of the three steps, we havea cyclic evolution U (0 , τ ) | ψ i = U ( τ , τ ) U ( τ , τ ) U (0 , τ ) | ψ i = e i ∆ φ | ψ i . (13) θ φ ∆φ ψ ɶ φ FIG. 2. (Color online) Cyclic evolution of a general qubitstate | ψ i on Bloch sphere is carried out in three steps: step1, which is illustrated in blue, evolves the state | ψ i to thestate | i along the meridian of the Bloch sphere correspondingto the fixed azimuthal angle φ ; step 2, which is shown inred, moves the north pole state | i all the way down to thesouth pole state e i ˜ φ | i along the meridian of the Bloch spherecorresponding to the fixed azimuthal angle ˜ φ ; finally, step3, which is depicted in black, evolves the state e i ˜ φ | i backinto the initial state with overall accumulated phase ∆ φ =˜ φ − φ , i.e., e i ∆ φ | ψ i , along the meridian of the Bloch spherecorresponding to the fixed azimuthal angle φ . The overallcyclic evolution introduces a parallel transport of the state | ψ i along an orange slice shaped path. The solid angle ∆ φ subtended by the orange slice shaped path, is the associatednon-adiabatic Abelian geometric phase. In fact, this three-step evolution introduces a cyclic evo-lution of the general qubit state | ψ i about an orange sliceshaped path on the Bloch sphere, where the two geodesicedges of the path differ as ∆ φ = ˜ φ − φ in their azimuthalangles.Note that the evolution in step 1 satisfies the condi-tion ( i ) of Eq. (8) and the evolutions in step 2 and3 satisfy the condition ( ii ) in Eq. (8), which indicatethat no dynamical phases occur along the three step evo-lutions. Therefore, the dynamical phase vanishs in thecyclic evolution of the general state | ψ i and the overallphase accumulated in this evolution, i.e., ∆ φ , is all ge-ometric. Strictly speaking, the phase ∆ φ , which is thesolid angle subtended by the orange slice shaped path,is the non-adiabatic Abelian geometric phase accompa-nying the parallel transport of the state | ψ i along thisorange slice shaped path [34].One may further note that the orthogonal counterpartstate of | ψ i , i.e., (cid:12)(cid:12) ψ ⊥ (cid:11) = sin θ | i − e iφ cos θ | i , (14)would accordingly evolve in a cyclic fashion giving rise to U (0 , τ ) (cid:12)(cid:12) ψ ⊥ (cid:11) = e − i ∆ φ (cid:12)(cid:12) ψ ⊥ (cid:11) . (15) with geometric phase − ∆ φ .Eqs. (13) and (15) imply that the final time evolutionoperator U (0 , τ ) has actually a geometric structure givenby U (0 , τ ) = U ( τ , τ ) U ( τ , τ ) U (0 , τ )= e i ∆ φ | ψ i h ψ | + e − i ∆ φ (cid:12)(cid:12) ψ ⊥ (cid:11) (cid:10) ψ ⊥ (cid:12)(cid:12) , (16)which takes the following form in the qubit computa-tional {| i , | i} basis U (0 , τ ) = R ~m (∆ φ ) = cos ∆ φ + i sin ∆ φ [ ~m · ~σ ] (17)with ~m = (sin θ cos φ, sin θ sin φ, cos θ ). The Eq. (17)indicates that the time evolution operator U (0 , τ ) is ac-tually a general SU (2) rotation about the rotation axis ~m with rotation angle given by non-adiabatic Abelian ge-ometric phase ∆ φ . Thus, the proposed evolution U (0 , τ )introduces a practical route to realize universal non-adiabatic geometric single-qubit gates. B. Two-Qubit Gates
A two-qubit gate on given register qubits k and l inthe system is achieved by the two-qubit Hamiltonian H kl identified in Eq. (3). Hamiltonian H kl embeds the com-putation system of two register qubits k and l into ahost three-qubit system via, as shown in Fig. 1, indirectcoupling of our register qubits k and l through a thirdauxiliary qubit a . The H kl in the computational basistakes the following double Λ structure H kl = J l | i h | + J k | i h | + J k | i h | + J l | i h | + h.c., (18)where 0 and 1 at each site in the basis states from left toright, respectively, represent the states of qubits k , a , l .Assume ( J k , J l ) = Ω(cos θ , sin θ , (19)where Ω = p J k + J l . If we fix the angle θ and turn onthe Hamiltonian H kl for a time interval [0 , τ ] such that R τ Ω dt = π then the double Λ structure of H kl leads tothe final time evolution operator U (0 , τ ) = e − i R τ H kl dt = U (0 , τ ) ⊕ U (0 , τ ) (20)in the ordered basis {| i , | i , | i , | i , | i , | i , | i , | i} , where U (0 , τ ) = θ − sin θ − sin θ − cos θ
00 0 0 − U (0 , τ ) = − − cos θ − sin θ − sin θ cos θ
00 0 0 1 . (21)We pursue with some remarks and properties of the sys-tem in Eq. (18) and its time evolution operator describedin Eq. (20): • Let us denote H q = span {| q i , | q i , | q i , | q i} (22) q = 0 ,
1. Each of the subspaces H q , q = 0 ,
1, indeedcorresponds to the four dimensional computationalsubspace of the two register qubits k and l , whenthe auxiliary qubit is fixed to the state | q i . Eq. (20)indicates that the subspaces H q , q = 0 ,
1, evolvein cyclic manners during the time interval [0 , τ ].Therefore the time evolution operator components U q (0 , τ ), q = 0 ,
1, introduce two-qubit gates on reg-ister qubits k and l , if the auxiliary qubit is initial-ized and measured in the same basis state | q i . • By evaluating the entangling powers [31, 35] e p [ U (0 , τ )] = e p [ U (0 , τ )] = 29 [1 − cos θ ] , (23)we obtain that both gates provide the same andfull entangling power controlled by the parameter θ . For each θ satisfying | cos θ | <
1, the gates U q (0 , τ ), q = 0 ,
1, are entangling two-qubit gatesand thus they allow for universal quantum infor-mation processing when accompanied with univer-sal single-qubit gates given in Eq. (17). • Moreover, we notice that the gates U q (0 , τ ), q =0 ,
1, have remarkable holonomic natures, which canbe verified from two points of views.
First : As mentioned in the first remark above, fromEq. (20) we have that the subspaces H q evolve incyclic fashions during the time interval [0 , τ ]. Thesecyclic evolutions actually take place in the Grass-manian G (8 , k , l , and a . We may call C q thecorresponding loops in the Grassmanian G (8 , q = 0 ,
1, one may observe that U (0 , t ) P q U † (0 , t ) H kl U (0 , t ) P q U † (0 , t ) = 0 , (24)at each time t ∈ [0 , τ ], where P q is the projec-tion operator on the subspace H q and U (0 , t ) =exp[ − i R t H kl ds ] is the evolution operator at time t . The Eq. (24) follows from [ H kl , U (0 , t )] = 0 ateach time t and that P q H kl P q = 0.Therefore, the Eqs. (20) and (24) imply that foreach q = 0 ,
1, the subspace H q evolves cycliclyabout the corresponding closed path C q in theGrassmanian G (8 , U q (0 , τ ), which is actually the pro-jection of the final time evolution operator U (0 , τ ) into the subspace H q , i.e., U q (0 , τ ) = P q U (0 , τ ) P q ,is the non-Abelian nonadiabatic quantum holon-omy associated with the parallel transport of H q about the loop C q in the Grassmanian G (8 ,
4) [36].
Second : Looking more carefully into the final timeevolution operator given by Eqs. (20, 21) and thedouble Λ coupling structure of Eq. (18), we seethat the two-qubit entangling gates U q (0 , τ ) pos-sess even richer holonomic structures. For the sakeof simplicity, in the following we restrict ourselvesto explain the further holonomic structure of thegate U (0 , τ ), however the same type of explana-tion would exists for the gate U (0 , τ ).We shall rewrite the two-qubit computational space H given in Eq. (22) in the following directsumform H = H ⊕ H ⊕ H , (25)where H = span {| i} , H = span {| i , | i} and H = span {| i} . Accordingly, we may putthe gate operator U (0 , τ ) in a directsum form as U (0 , τ ) = (1) ⊕ (cid:18) cos θ − sin θ − sin θ − cos θ (cid:19) . ⊕ ( − . (26)The state | i does not contribute into the Hamil-tonian given in Eq. (18) and thus it is kept un-changed during the time evolution of the system.In other words, the evolution of the subspace H would be stationary with associated trivial phaseduring any time interval. This explains the first el-ement, (1), in the right hand side directsum of Eq.(26).However, the double Λ structure of Eq.(18) implies that the evolutions of H and H are non-stationary and, respectively,take place in the three dimensional invari-ant subspaces span {| i , | i , | i} andspan {| i , | i , | i} . Explicitly speaking,for each d = 1 ,
2, the evolution of H d specifies anon-trivial path in the Grassmanian G (3 , d ), whichwe may here call C d . Moreover, the block diagonalform of the final time evolution operator, U (0 , τ ),in the corresponding ordered basis given belowEq. (20) further implies that each of the subspaces H d , d = 1 ,
2, undergoes a cyclic evolution duringthe time interval [0 , τ ] and thus the correspondingpath C d is a closed path in G (3 , d ). If we assume P d to be the projection operator on the subspace H d then from P d H kl P d = 0 and [ H kl , U (0 , t )] = 0we obtain U (0 , t ) P d U † (0 , t ) H kl U (0 , t ) P d U † (0 , t ) = 0 (27)at each time t , for d = 1 ,
2, which indicates nodynamical phases occur along the cyclic evolutions C d , d = 1 , H d , d = 1 ,
2, are actually parallel trans-ported about the loops C d giving rise to the follow-ing nonadiabatic quantum holonomies [36] U ( C ) = P U (0 , t ) P = ( − U ( C ) = P U (0 , t ) P = (cid:18) cos θ − sin θ − sin θ − cos θ (cid:19) . (28)As a result, we see in Eq. (26) that the two-qubitentangling gate U (0 , τ ) is indeed composed of theholonomies in Eq. (28), namely U (0 , τ ) = (1) ⊕ U ( C ) ⊕ U ( C ) . (29)In conclusion, our analysis above reveals the richholonomic nature of the two-qubit entangling gate U (0 , τ ) by demonstrating that the gate U (0 , τ ) notonly as a whole is a nonadiabatic holonomy but alsoeach of its nonzero block constitutes is a nonadia-batic holonomy. III. DISCUSSION AND SUMMARY
As the nonadiabatic holonomies became an importantapproach for implementation of fast fault-tolerant quan-tum gates, experimental implementation of nonadiabatic holonomic quantum computation with spin qubits, as anatural and suitable platform for realization of quantumcomputers, caught increasing interests in recent years[16–20]. Despite a number of significant efforts in thisarea, only single-qubit gates have been addressed. There-fore, still a lack of feasible scheme, which supports scala-bility as well as universal single-qubit and two-qubit en-tangling gates all in the same configuration is felt. Com-pared to the existing works, our scheme above is consistof arbitrary n register spin qubits arranged in a star-shape architecture about a shared single auxiliary spinqubit in the middle (see Fig. 1). In addition to the scal-ability, the proposed star-shape configuration permits foruniversal nonadiabatic holonomic quantum computation,where an arbitrary holonomic single-qubit gate on eachregister qubit is achieved by a local transverse magneticfield and a two-qubit entangling gate between a given pairof register qubits is performed in a double Λ structure asdemonstrated by indirect bridge coupling between the se-lected register qubits through the auxiliary qubit. Whilesingle-qubit gates are realized through Abelian nonadia-batic holonomies [34], the proposed entangling two-qubitgates obey a rich holonomic description associated withnon-Abelian as well as Abelian nonadiabatic holonomies[36]. All holonomic universal computations take placein the subspace of the system, where the state of auxil-iary qubit is fixed to one of its computational basis states(say for instance the basis state | i ). A universal circuitcorresponding to our nonadiabatic holonomic scheme isdepicted in Fig. 3. M ea s u r e d ou t pu t s t a t e o f n -r e g i s t e r qub it s I npu t s t a t e o f n -r e g i s t e r qub it s Ancila qubit Ancila qubit Input state
Measured output state B ⊥ B l − ⊥ B l + ⊥ B n ⊥ B k ⊥ B k + ⊥ ψ f ψ i FIG. 3. (Color online) Schematic digram of a universal holonomic quantum computation. Holonomic single-qubit gates areimplemented by local transverse magnetic fields, which introduce transverse coupling between two computational basis statesof qubits. The auxiliary qubit, which is illustrated in blue, only contributes in two-qubit gates. Two register qubits are coupledthrough the auxiliary qubit in a double Λ structure allowing for implementation of two-qubit entangling gates. The auxiliaryqubit is initialized, and measured at the end of computation in the same computational basis state, which here we selected tobe the state | i . In summary, we have proposed a scalable spin-based setup for universal nonadiabatic holonomic quantumcomputation. We hope the present scheme helps to over-come practical challenges and establish a feasible plat-form for realization of scalable universal nonadiabaticholonomic quantum computation particularly with spinqubits. The discussion for the holonomic nature of thegates would further improve our understanding of theconcept of quantum holonomy in solid state systems and its relation to quantum computation.
IV. ACKNOWLEDGMENT
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