Nodal free geometric phases: Concept and application to geometric quantum computation
aa r X i v : . [ qu a n t - ph ] J a n Nodal free geometric phases: Concept andapplication to geometric quantumcomputation
Marie Ericsson a , David Kult b , Erik Sj¨oqvist b , ∗ , Johan ˚Aberg a a Centre for Quantum Computation, Department of Applied Mathematics andTheoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB30WA, United Kingdom b Department of Quantum Chemistry, Uppsala University, Box 518, Se-751 20Uppsala, Sweden
Abstract
Nodal free geometric phases are the eigenvalues of the final member of a paralleltransporting family of unitary operators. These phases are gauge invariant, alwayswell-defined, and can be measured interferometrically. Nodal free geometric phasescan be used to construct various types of quantum phase gates.
Key words:
Geometric phase; Quantum gates; Interferometry
PACS:
Geometric phases in noncyclic evolution [1] are undefined if the initial and finalstates are orthogonal. Similarly, off-diagonal geometric phases [2] are undefinedfor cyclic evolution. Thus, it appears to be a general fact that geometric phaseshave nodal points at which they are undefined. Nevertheless, these phasesprobe the geometry of the ray space, which is free from any singularities.This observation raises the question if the concept of geometric phase canbe modified so that it reflects the absence of singularities in ray space. Here,we propose such a modified concept, which we shall call nodal free geometricphases.Recently, geometric phases have been suggested to play a role in the design ofquantum gates that may be robust to certain kinds of errors. This idea was ∗ Corresponding author.
Email addresses:
[email protected] (Marie Ericsson), [email protected] (David Kult), [email protected] (Erik Sj¨oqvist),
[email protected] (Johan ˚Aberg).
Preprint submitted to Elsevier 29 October 2018 rst put forward in the case of cyclic adiabatic evolution [3], and was sub-sequently implemented in the nonadiabatic [4] and noncyclic [5,6] contexts.The desired geometric properties may also occur when the dynamical phasebecomes proportional to the geometric phase, leading to the notion of uncon-ventional geometric quantum computation [7]. Here, we propose the nodal freegeometric phases as another conceptual basis for the construction of quantumgates.Let U ( s ), s ∈ [0 , N -dimensional Hilbert space H . We require that U (0) = ˆ1, i.e.,the identity operator on H . This family is said to be parallel transporting ifthere exists an orthonormal basis ψ = {| ψ k i} k of H such that each U ( s ) | ψ k i satisfies the standard parallel transport condition [8,9,10] h ψ k | U † ( s ) ˙ U ( s ) | ψ k i = 0 . (1)Not all U ( s ) has this property, e.g., for a qubit ( N = 2), e − isσ z is parallel trans-porting (any pair of orthogonal vectors that represent states on the equatorof the Bloch sphere will do), while e − isσ x e − isσ z is not.Now, let U k ψ ( s ), s ∈ [0 ,
1] and U k ψ (0) = ˆ1, be a family of unitary operators thatparallel transport the orthonormal basis ψ = {| ψ k i} k . We are interested in theproperties of U k ψ (1). Consider the N × N matrix U k ψ = σ . . . σ N ... . . . ... σ N . . . σ NN , (2)where σ kl = h ψ k | U k ψ (1) | ψ l i , i.e., U k ψ is the matrix representation of U k ψ (1) withrespect to ψ . Since U k ψ is a unitary matrix, it has N unit modulus eigenvalues.These eigenvalues coincide with the standard geometric phase factors in cyclicevolution [11,12,13] if U k ψ is diagonal, i.e., when all U k ψ ( s ) | ψ k i undergo cyclicevolution. In the noncyclic case, U k ψ has nonzero off-diagonal elements. Thecorresponding eigenvalues are still gauge invariant phase factors, i.e., invari-ant under transformations of the form | ψ k i → e iα k | ψ k i , but differ from thestandard geometric phase factors in noncyclic evolution [1]. In particular, theeigenvalues are always well-defined, i.e., there are no nodal points where theybecome undefined. These eigenvalues are the nodal free geometric phase fac-tors of the set of paths { Π[ U k ψ ( s ) | ψ k i ] } k in ray space, Π being the projectionmap [11].To prove the gauge invariance of the nodal free geometric phase factors, we2ote that σ kl → σ kl e − i ( α k − α l ) , under | ψ k i → e iα k | ψ k i . Thus, U k ψ → V U k ψ V † ,where V = diag[ e − iα , e − iα , . . . , e − iα N ], from which it follows that the eigen-values are preserved.Similarly as for the standard [1] and off-diagonal [2] geometric phase factors thenodal free geometric phase factors can be defined in terms of gauge-invariantquantities. The eigenvalues λ of U k ψ are solutions of the secular equationdet( U k ψ − λ ) = 0, being the N × N unit matrix. Expanding the deter-minant yields an equation that involves γ (1) j = σ jj and γ ( l ) j ...j l ≡ σ j j l · · · σ j j , l = 2 , . . . , N , which are the gauge invariant quantities that define the stan-dard and off-diagonal geometric phase factors Φ[ γ ( l ) ], where Φ[ z ] = z/ | z | forany nonzero complex number z . For instance, for N = 3 we have − λ + (cid:16) γ (1)1 + γ (1)2 + γ (1)3 (cid:17) λ − (cid:16) γ (1)1 γ (1)2 + γ (1)2 γ (1)3 + γ (1)3 γ (1)1 − γ (2)12 − γ (2)23 − γ (2)31 (cid:17) λ + γ (3)123 + γ (3)132 − γ (2)12 γ (1)3 − γ (2)23 γ (1)1 − γ (2)31 γ (1)2 + γ (1)1 γ (1)2 γ (1)3 = 0 . (3)The solutions of this equation are fully determined by all the γ ’s.One may derive some results for the off-diagonal geometric phases using theabove nodal free scenario. We first note that a central motivation for Maniniand Pistolesi [2] to introduce the concept of off-diagonal geometric phaseswas to retain geometric phase information of the evolution in cases where thestandard geometric phase factors were undefined. In the present context, wecan indeed see that this must always be the case since if, for a given Hilbertspace dimension N , all γ ’s happened to vanish and thereby all Φ[ γ ] wereundefined, the secular equation would reduce to λ N = 0, which contradictsthe fact that | λ | = 1 (see also Ref. [14]). Furthermore, one may see that the λ coefficient of the eigenvalue equation has the structure γ ( N ) + γ ( N − γ (1) + γ ( N − h γ (1) i + γ ( N − γ (2) + . . . + h γ (1) i N . This term must be nonzero, since ifit would vanish, then λ = 0 would be a solution of the eigenvalue equation. Itfollows, in particular, that γ ( l = n ) = 0 for some fixed n is only possible for the‘extremal’ n = 1 and n = N .Next, we consider the relation between the nodal free phase factors and thecyclic geometric phase factors [11,12,13]. Let | φ k i be orthonormal eigenvectorsof U k ψ (1). These are cyclic states, i.e., U k ψ (1) | φ k i = λ k | φ k i , where λ k are thenodal free geometric phase factors. Note that the eigenvectors are in generalnot parallel transported by U k ψ ( s ). The nodal free phase factors λ k differ fromthe standard cyclic geometric phase factors e iβ k [11,12,13] associated with thepaths Π[ U k ψ ( s ) | φ k i ] in ray space. To see this we first note that given the family3f unitaries U k ψ ( s ) the Schr¨odinger equation uniquely determines the familyof Hamiltonians H ( s ) = i ˙ U k ψ ( s ) U k ψ † ( s ) that generates U k ψ ( s ). If we let τ k bethe phase associated with the nodal free phase factor, i.e., λ k = e iτ k , we canuse the technique in Ref. [11] (see Eq. (3) in Ref. [11]) to find that the cyclicgeometric phases β k of | φ k i are β k = τ k + Z h φ k ( s ) | H ( s ) | φ k ( s ) i ds, (4)which we can rewrite as e iβ k = λ k e i R h φ k | H ( s ) | φ k i ds . (5)This establishes an explicit relation between the nodal free geometric phasefactors and the geometric phase factors associated with the states that undergocyclic evolution for the family U k ψ ( s ). As can be seen, these two differ by thedynamical phase factors of the eigenvectors | φ k i . It is to be noted that, sincethe family of Hamiltonians is uniquely determined, these dynamical phasefactors do not introduce any ambiguity in the definition of the nodal freegeometric phases.As a final note on the general properties of the nodal free geometric phases, wepoint out that these can be measured in interferometry. A beam of particleswith some internal degree of freedom (e.g., spin or polarization) prepared inthe internal state | ϕ i is splitted by a 50-50 beam-splitter into two beams. Inone of the resulting beams, the internal state is transformed by the unitaryoperators U k ψ ( s ) , s ∈ [0 , ψ = {| ψ k i} k , anda variable U(1) phase shift e iχ is applied to the other beam. The two beams arebrought back to interfere at a second beam-splitter. The resulting interferencepattern is determined by the complex-valued quantity F ( ϕ ) = h ϕ | U k ψ (1) | ϕ i inthat the intensity measured in one of the output beams reads [15] I ∝ (cid:12)(cid:12)(cid:12) F ( ϕ ) (cid:12)(cid:12)(cid:12) cos h χ − arg F ( ϕ ) i . (6)Explicitly, F ( ϕ ) = X k |h φ k | ϕ i| λ k , (7)where P k (cid:12)(cid:12)(cid:12) h φ k | ϕ i (cid:12)(cid:12)(cid:12) = 1, i.e., the interference function F ( ϕ ) is a convex combi-nation of the nodal free geometric phase factors. It follows that the interferenceoscillations are shifted by the nodal free geometric phases arg λ k if and onlyif | ϕ i coincide with | φ k i , in case of which (cid:12)(cid:12)(cid:12) F ( ϕ ) (cid:12)(cid:12)(cid:12) = 1. Thus, the nodal freegeometric phases can be obtained by varying the input internal state ϕ untilunit visibility is attained. 4o illustrate the theory with a specific example, let us analyze the nodal freegeometric phases in the qubit case in some detail. Let η = (cid:12)(cid:12)(cid:12) h ψ | U k ψ (1) | ψ i (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) h ψ | U k ψ (1) | ψ i (cid:12)(cid:12)(cid:12) measure the degree of cyclicity and let Ω be the solid an-gle enclosed by the path traced out by the Bloch vector and the shortestgeodesics connecting the end-points. In terms of these quantities, we have γ (1)1 = ( γ (1)2 ) ∗ = ηe − i Ω / and γ (2)12 = η −
1. Thus, if η = 0 then the geometricphase factors Φ[ γ (1)1 ] and Φ[ γ (1)2 ] are undefined, but the off-diagonal phase fac-tor Φ[ γ (2)12 ] = −
1. If η = 1 then the off-diagonal geometric phase factor Φ[ γ (2)12 ]is undefined and Φ[ γ (1)1 ] = (cid:16) Φ[ γ (1)2 ] (cid:17) ∗ = e − i Ω / . On the other hand, the nodalfree geometric phase factors, i.e., the eigenvalues of U k ψ , are always defined.The secular equation reads λ − ( γ (1)1 + γ (2)2 ) λ + γ (1)1 γ (1)2 − γ (2)12 = 0, which givesthe eigenvalues λ ± = η cos(Ω / ± i q − η cos (Ω /
2) (8)with corresponding eigenstates | φ ± i . We note that 1 ≥ η cos (Ω /
2) and | λ ± | =1. For ‘bit flip’ evolution ( η = 0), the geodesically closed solid angle Ω isundefined since there is an infinite number of ways to close the path by ageodesics, and all these closures yield different solid angles. However, whatevergeodesic closure we choose, the nodal free geometric phase factors are givenby ± i . For cyclic evolution ( η = 1), we obtain the two eigenvalues λ ± =cos(Ω / ± i | sin(Ω / | . Hence, as expected we find the two cyclic geometricphase factors e i Ω / and e − i Ω / . More precisely, λ + , − = Φ[ γ (1)2 , ] for sin(Ω / > λ + , − = Φ[ γ (1)1 , ] for sin(Ω / <
0, i.e., the labeling of the eigenstatesdepends on the sign of sin(Ω /
2) (note that λ + = λ − when sin(Ω /
2) = 0,which allows for the ’flip’ of the labeling).The cyclic geometric phases arising from parallel transport have been consid-ered for implementations of phase gates [16]. In such applications, the paral-lel transported and computational bases coincide. The nodal free geometricphases provide an alternative implementation of phase gates, if we let thecomputational basis coincide with the eigenstates of U k ψ (1). Here, the compu-tational basis in general does not coincide with the parallel transported basis.In the single qubit case, considered above, we choose the computational basis | i ≡ | φ − i and | i ≡ | φ + i . The nodal free geometric phases may thus be in-terpreted as a realization of the one-qubit phase gate U = λ − | ih | + λ + | ih | .Such a phase gate is fully determined by the quantities γ (1)1 , γ (1)2 , and γ (2)12 viathe eigenvalue equation. In the special case of cyclic evolution of the paralleltransported basis, U reduces to the standard nonadiabatic geometric phasegate γ (1)1 | ih | + γ (1)2 | ih | = e − i Ω / | ih | + e i Ω / | ih | , which is sensitive tochanges in the solid angle Ω. On the other hand, the off-diagonal geometricphase factor Φ[ γ (2)12 ] is either undefined (for cyclic evolution η = 1) or − ≤ η < ig. 1. Paths on Bloch sphere for phase shift gate (left panel) and phase flip gate(right panel), based on the standard and nodal free geometric phases, respectively.These paths can be realized for instance by applying magnetic fields sequentially inthree different directions to a spin − particle. The phase shift gate is determinedby the solid angle Ω = ϕ . Here, the computational and parallel transported basescoincide. The phase flip gate depends only on the off-diagonal geometric phasearg γ (2)12 . This phase equals half the solid angle enclosed by the loop sequentiallytraced out by the path starting at | ψ i , followed by the path starting at | ψ i ,i.e., arg γ (2)12 = π [2]. Thus, the nodal free geometric phase factors ± q γ (2)12 = ± i that define this gate are ϕ independent. Note, though, that the precise locationof the computational basis indicated by ‘0’ and ‘1’, on the equator of the Blochsphere, depends on ϕ . One can show that | i = √ (cid:0) | ψ i + e − i ( ϕ − π ) / | ψ i (cid:1) and | i = √ (cid:0) | ψ i + e − i ( ϕ + π ) / | ψ i (cid:1) for this phase flip gate. that always encloses the solid angle 2 π . This property has been demonstratedexperimentally for neutron spin [17] and suggests that the off-diagonal geo-metric phases may be a useful component for the design of quantum gates.Since γ (2)12 = γ (2)21 , it seems at first sight that only a trivial phase gate can beimplemented based on the off-diagonal geometric phase. However, in the nodalfree geometric phase scenario with η = 0, we obtain iZ = q γ (2)12 | ih | − q γ (2)12 | ih | = i | ih | − i | ih | , (9)which, up to the unimportant overall phase factor i , is a nontrivial phase flipgate. Thus, the nodal free geometric phase concept makes it possible to use off-diagonal geometric phases in the design of quantum gates. The robustness withrespect to changes in Ω suggests that this gate implementation would have aninherent stability against noise in this parameter. Realizations of paths on theBloch sphere leading to the phase shift and phase flip gates using the standardand nodal free geometric phases, respectively, are shown in Fig. (1).6o develop this idea further, we consider two-qubit geometric phase gates, incase of which the relevant Hilbert space is four-dimensional. One may seekfor a similar path independence as in the above phase flip gate by lookingfor implementations that only involve the fourth order ( l = 4) off-diagonalgeometric phases. To this end, we assume that U k ψ (1) takes the matrix form U k ψ (1) = σ σ
00 0 0 σ σ (10)in the parallel transported basis. The eigenvalue equation reads λ − γ (4)1234 = 0,yielding the nodal free geometric phase factors λ k = e ikπ/ (cid:16) γ (4)1234 (cid:17) / , k =0 , . . . ,
3. The parallel transporting nature of the family U k ψ ( s ) implies that U k ψ (1) ∈ SU(4), i.e., det U k ψ (1) = 1. Thus, γ (4)1234 = − λ k = e i (2 k +1) π/ , k = 0 , . . . ,
3. Let us assume that there is a physically natural tensor productdecomposition of the Hilbert space such that the eigenvectors of U k ψ (1) coincidewith the computational product basis. For instance, if | φ i = | i , | φ i = | i , | φ i = | i , | φ i = | i , we obtain the conditional gate B = e i π | ih | + e i π | ih | + e i π | ih | + e i π | ih | . (11)On the other hand, if | φ i = | i , | φ i = | i , | φ i = | i , | φ i = | i , weobtain the product gate Z ⊗ S = (cid:16) | ih | − | ih | (cid:17) ⊗ (cid:16) e iπ/ | ih | − e − iπ/ | ih | (cid:17) , (12)where S is the phase or π/ π/ γ (8)12 ... . This π/ References [1] J. Samuel, R. Bhandari, Phys. Rev. Lett. 60 (1988) 2339.[2] N. Manini, F. Pistolesi, Phys. Rev. Lett. 85 (2000) 3067.[3] P. Zanardi, M. Rasetti, Phys. Lett. A 264 (1999) 94.[4] W. Xiang-Bin, M. Keiji Phys. Rev. Lett. 87 (2002) 097902.[5] A. Friedenauer, E. Sj¨oqvist, Phys. Rev. A 67 (2003) 024303.[6] D. Kult, J. ˚Aberg, E. Sj¨oqvist, Phys. Rev. A 74 (2006) 022106.[7] S.-L. Zhu, Z.D. Wang, Phys. Rev. Lett. 91 (2003) 187902.[8] J. Anandan, L. Stodolsky, Phys. Rev. D 35 (1987) 2597.[9] J. Anandan, Phys. Lett. A 129 (1988) 201.[10] J. Anandan, Y. Aharonov, Phys. Rev. D 38 (1988) 1863.[11] Y. Aharonov, J. Anandan, Phys. Rev. Lett. 58 (1987) 1593.[12] J. Anandan, Phys. Lett. A 133 (1988) 171.[13] L.-A. Wu, Phys. Rev. A 50 (1994) 5317.[14] D. Kult, J. ˚Aberg, E. Sj¨oqvist, Europhys. Lett. 78 (2007) 60004.[15] A.G. Wagh, V.C. Rakhecha, Phys. Lett. A 197 (1995) 107.[16] X.-Q. Li, L.-X. Cen, G. Huang, L. Ma, Y. Yan, Phys. Rev. A 66 (2002) 042320.[17] Y. Hasegawa, R. Loidl, M. Baron, G. Badurek, H. Rauch, Phys. Rev. Lett. 87(2001) 070401.[18] P.O. Boykin, T. Mor, M. Pulver, V. Roychowdhury, F. Vatan, Inform. Process.Lett. 75 (2000) 101.[1] J. Samuel, R. Bhandari, Phys. Rev. Lett. 60 (1988) 2339.[2] N. Manini, F. Pistolesi, Phys. Rev. Lett. 85 (2000) 3067.[3] P. Zanardi, M. Rasetti, Phys. Lett. A 264 (1999) 94.[4] W. Xiang-Bin, M. Keiji Phys. Rev. Lett. 87 (2002) 097902.[5] A. Friedenauer, E. Sj¨oqvist, Phys. Rev. A 67 (2003) 024303.[6] D. Kult, J. ˚Aberg, E. Sj¨oqvist, Phys. Rev. A 74 (2006) 022106.[7] S.-L. Zhu, Z.D. Wang, Phys. Rev. Lett. 91 (2003) 187902.[8] J. Anandan, L. Stodolsky, Phys. Rev. D 35 (1987) 2597.[9] J. Anandan, Phys. Lett. A 129 (1988) 201.[10] J. Anandan, Y. Aharonov, Phys. Rev. D 38 (1988) 1863.[11] Y. Aharonov, J. Anandan, Phys. Rev. Lett. 58 (1987) 1593.[12] J. Anandan, Phys. Lett. A 133 (1988) 171.[13] L.-A. Wu, Phys. Rev. A 50 (1994) 5317.[14] D. Kult, J. ˚Aberg, E. Sj¨oqvist, Europhys. Lett. 78 (2007) 60004.[15] A.G. Wagh, V.C. Rakhecha, Phys. Lett. A 197 (1995) 107.[16] X.-Q. Li, L.-X. Cen, G. Huang, L. Ma, Y. Yan, Phys. Rev. A 66 (2002) 042320.[17] Y. Hasegawa, R. Loidl, M. Baron, G. Badurek, H. Rauch, Phys. Rev. Lett. 87(2001) 070401.[18] P.O. Boykin, T. Mor, M. Pulver, V. Roychowdhury, F. Vatan, Inform. Process.Lett. 75 (2000) 101.