aa r X i v : . [ phy s i c s . g e n - ph ] N ov Integrable Quantum Computation
Yong Zhang Abstract
Integrable quantum computation is defined as quantum computingvia the integrable condition, in which two-qubit gates are either nontrivialunitary solutions of the Yang–Baxter equation or the Swap gate (permutation).To make the definition clear, in this article, we explore the physics underlyingthe quantum circuit model, and then present a unified description on bothquantum computing via the Bethe ansatz and quantum computing via theYang–Baxter equation.
Key Words
Quantum Computing, Bethe Ansatz, The Yang–Baxter Equation
PACS numbers
Computers are physical objects, and computations are physical processes, whichmay be one of Deutsch’s famous quotes [1]. This statement gives rise to a naturalquestion: what is the physics underlying the quantum circuit model? Deutsch’sanswer may be that quantum computing votes for the many-worlds interpretationof quantum mechanics. Nielsen and Chuang’s answer to it stays the same in thetwo editions of their popular textbook [2]: “outside the scope of the presentdiscussion”. In our current understanding [3], the quantum circuit model canbe viewed as a generalization of the multi-qubit factorisation scattering model inintegrable systems [4], and the scalable quantum computer is thus supposed tobe an exactly solvable model satisfying the integrable condition [4].It becomes well known in quantum information science after DiVincenzo’swork [5] that an arbitrary N -qubit quantum gate can be expressed as a sequenceof products of some two-qubit gates. Hence a quantum circuit is described asa network of two-qubit gates, which may be called the locality principle of thequantum circuit, see Preskill’s online lecture notes [6]. On the other hand, it hasbeen widely accepted for a long time in integrable systems [4] that many-bodyfactorisable scattering can be expressed as a sequence of two-body scattering.As two-qubit quantum gates are considered as two-qubit scattering operators,the quantum circuit is thought of as the generalized factorisable scattering model defined as a generalization of the factorisation scattering model [4].Sutherland in his book [4] describes three types of the integrable conditionswhich are capable of yielding exactly solvable quantum many-body systems: the Institute of Physics, Chinese Academy of Sciences, Beijing 100190, P.R. China [email protected] integrable quantumcomputation as quantum computing via the integrable condition so that we canobtain a unified description on both quantum computing via the Bethe ansatz [3]and quantum computing via the Yang–Baxter equation [7]. Note that we do notdeal with quantum computing via the quantum Lax equation in this paper.Section 3 is a preliminary introduction on quantum computing via the Betheansatz [3]. The main reference is Gu and Yang’s paper [8] on the applicationof the Bethe ansatz to the N Fermion problem with delta-functional potential.Given a quantum many-qubit system, i.e., the Hamiltonian is known, if the Betheansatz is satisfied, then it is an integrable model to yield many-qubit factorisablescattering as a sequence products of two-qubit scattering matrices in which two-qubit gates satisfy the Yang–Baxter equation.Section 4 is a brief sketch on quantum computing via the Yang–Baxter equa-tion [7]. Given a nontrivial unitary solution of the Yang–Baxter equation, theHamiltonian of an integrable system can be constructed in principle. Addition-ally, the paper [7] is written in the way to emphasize how to derive unitarysolutions of the Yang–Baxter equation from unitary representations of the braidgroup. In view of the research [3], however, it has been distinct that integrablequantum computation is a research subject independent of quantum computingvia unitary braid representations. Hence this section is a refinement of the re-search [7] from the viewpoint of integrable quantum computation .In mathematics, integrable quantum computation specifies a type of quantumcircuit model of computation in which two-qubit gates are either the Swap gate(permutation) or nontrivial unitary solutions of the Yang–Baxter equation.
Let us consider the model of N qubits (spin-1/2 particles) in one dimensioninteracting via the delta-function potential, with the Hamiltonian given by H = − N X i =1 ∂ ∂x i + 2 c X ≤ i
0) means repulsive (attraction) interaction, see [8] for thedetail. Interested readers are also invited to refer to Bose and Korepin’s article[9] for the physical reason why this model is interesting in experimental quantuminformation and computation.For two qubits at x and x , the outgoing wave in the region x < x withrespective momenta k < k is given by α out e i ( k x + k x ) , α out = α , (2)the incoming wave in the region x < x is given by α e i ( k x + k x ) , α inc = α , (3)2o that the two-qubit gate (i.e., the scattering matrix) ˇ R ( k , k ) defined by α out = ˇ R α inc has the formˇ R ( k , k ) = i ( k − k )( − P ) + ci ( k − k ) − c , (4)where the permutation operator P exchanges the spins of two flying qubits.For three qubits at x , x and x , the outgoing wave in the region x < x < x with respective momenta k < k < k is given by α out e i ( k x + k x + k x ) , α out = α , (5)the incoming wave in the region x < x < x is given by α e i ( k x + k x + k x ) , α inc = α , (6)so that α out is determined by α inc in the way α out = ˇ R ( k , k ) ˇ R ( k , k ) ˇ R ( k , k ) α inc , (7)or in the other way α out = ˇ R ( k , k ) ˇ R ( k , k ) ˇ R ( k , k ) α inc , (8)which give rises to the consistency condition (the Yang–Baxter equation) on thethree-qubit scattering,ˇ R ( u ) ˇ R ( u + v ) ˇ R ( v ) = ˇ R ( v ) ˇ R ( u + v ) ˇ R ( u ) (9)with u = k − k , v = k − k and u + v = k − k .The outgoing wave of two qubits at x and x in the region x < x withrespective momenta k < k is given by (2), but the incoming wave in the region x < x is given by α inc e i ( k x + k x ) , α inc = ( − P ) α , (10)so that the scattering matrix of two qubits in different regions has the form α out = ˇ R ( k , k )( − P ) α inc . (11)Similarly, the outgoing wave of three qubits at x , x and x in the region x 0. At u = − c , the two-qubit gate ˇ R ( u ) is the √ Swap gate given byˇ R ( − π − p Swap, p Swap = P + + iP − (17)which is an entangling two-qubit gate but can not yield all unitary 4 × R ( ϕ ) has the form of the time evolutionaloperator U ( ϕ ) of the Heisenberg interaction ~S · ~S between two qubitsˇ R ( ϕ ) = − e − i ϕ U ( − ϕ ) , U ( ϕ ) = e − iϕ~S · ~S (18)modulo a global phase factor with the permutation P = 2 ~S · ~S + so that ˇ R ( ϕ )and U ( ϕ ) are two equivalent two-qubit gates in the quantum circuit model.As is realized by DiVincenzo et al. [11], universal quantum computation canbe set up only using the Heisenberg interaction when a logical qubit is encodedas a two-dimensional subspace of eight-dimensional Hilbert space of three qubits.Hence, the quantum circuit model in terms of the Swap gate P and the two-qubitgate ˇ R ( u ) (14) is also able to perform universal quantum computation, if and onlyif a logical qubit is chosen in a suitable way. A quantum circuit model in terms of both a nontrivial unitary solution ˇ R ( u )of the Yang–Baxter equation (9) and the Swap gate P is called the generalizedfactorisable scattering model in the paper. Once it is given, the problem becomeshow to find out the underlying physical system, which is thought of as the inverseproblem of quantum computing via the Bethe ansatz.4n the literature, the Yang–Baxter equation has the usual formalism,( ˇ R ( x ) ⊗ ) (11 ⊗ ˇ R ( xy )) ( ˇ R ( y ) ⊗ ) = (11 ⊗ ˇ R ( y )) ( ˇ R ( xy ) ⊗ ) (11 ⊗ ˇ R ( x )) (19)where the two-qubit gate ˇ R ( x ) is a linear operator on the Hilbert space of twoqubits, i.e., ˇ R : H ⊗ H → H ⊗ H , 11 denotes the 2 × x, y are called the spectral parameter. Take x = e u and y = e v , then the formalism ofthe Yang–Baxter equation (9) can be derived. As a two-qubit gate, the solutionˇ R ( x ) of the Yang–Baxter equation (19) has to satisfy the unitary conditionˇ R ( x ) ˇ R † ( x ) = ˇ R † ( x ) ˇ R ( x ) = ρ (20)with the normalization factor ρ .The construction of the Hamiltonian via a nontrivial unitary solution of theYang–Baxter equation (19) is a very flexible or subtle process with physical rea-soning, namely, there does not exist a universal law to guide such a construction,partly because the same solution of the Yang–Baxter equation can be yielded bymany different kinds of physical interactions.The two-qubit gate (14), as a rational solution of the Yang–Baxter equation(19), has the form ˇ R ( α ) = − 11 + α (11 + αP ) , α = i uc (21)which is a linear combination of the identity 11 and the permutation P . In termsof the variable ϕ , the two-qubit gate (14) can be expressed as an exponentialformalism of the permutation P given byˇ R ( ϕ ) = e i ( π − ϕ ) e iϕP , tan ϕ = uc , (22)hence the Hamiltonian for the generalized factorisable scattering model can bechosen as the Heisenberg interaction ~S · ~S between two qubits. On the otherhand, it is easy to examine the delta-function interaction (1) as a suitable Hamil-tonian underlying the two-qubit gate (14), but it is not so explicit to realize thedelta-function interaction (1) from the formalism of the two-qubit gate (14).In [7], an approach for the construction of the Hamiltonian is presented via atwo-qubit solution ˇ R ( x ) of the Yang–Baxter equation (19). With an initial state ψ , the evolution state ψ ( x ) determined by ˇ R ( x ) is given by ψ ( x ) = ρ − ˇ R ( x ) ψ .The Shr¨odinger equation has the form i ∂ψ ( x ) ∂x = H ( x ) ψ ( x ) (23)with the Hamiltonian H ( x ) given by H ( x ) = i ∂∂x ( ρ − ˇ R ) ρ − ˇ R − ( x ) (24)5hich is time-dependent of x .In terms of an appropriate parameter instead of x , the Hamiltonian H ( x ) (27)may be transformed to a time-independent formalism. For example, the Hamil-tonian H ( u ) (27) via the two-qubit gate (14) is obtained to be time-dependent ofthe parameter u after some algebra using the formula (24), but the Hamiltonianvia the the formalism (22) of this gate is found to be time-independent Heisenberginteraction.Consider the other solution of the Yang–Baxter equation (19),ˇ R ( x ) = x − x x − (1 − x ) 00 1 − x x − (1 − x ) 0 0 1 + x , (25)where the unitarity condition (20) requires x real with the normalization factor ρ = 2(1 + x ). In terms of the variable θ defined bycos θ = 1 √ x , sin θ = x √ x , (26)the two-qubit gate ˇ R ( x ) (25) has the formalism of θ , and the Hamiltonian H ( θ )(27) is calculated to be H ( θ ) = i ∂∂θ ( ρ − ˇ R ) ρ − ˇ R † = 12 σ x ⊗ σ y (27)with the conventional form of the Pauli matrices. The time evolutional operatorhas the form U ( θ ) = e − iHθ as well as the two-qubit gate (25) is given byˇ R ( θ ) = cos( π − θ ) + 2 i sin( π − θ ) H = e i ( π − θ ) H . (28)At θ = 0, the two-qubit gate ˇ R (0) given byˇ R (0) = / √ / √ 20 1 / √ − / √ / √ / √ − / √ / √ (29)is a unitary basis transformation matrix from the product base to the Bell states.Besides the ˇ R ( x ) matrix (25), in [7], the following type of nontrivial unitarysolutions ˇ R ( x ) of the Yang–Baxter equation (19),ˇ R ( x ) = ω ( x ) 0 0 ω ( x )0 ω ( x ) ω ( x ) 00 ω ( x ) ω ( x ) 0 ω ( x ) 0 0 ω ( x ) , (30)6ave been recognized as two-qubit gates modulo a phase factor as well as theassociated time-dependent Hamiltonians H ( x ) (24) also respectively calculated.The two-qubit gate ˇ R ( x ) (30) is determined by two submatrices U and U , U ( x ) = (cid:18) ω ( x ) ω ( x ) ω ( x ) ω ( x ) (cid:19) , U ( x ) = (cid:18) ω ( x ) ω ( x ) ω ( x ) ω ( x ) (cid:19) , (31)which are in the unitary group U (2).According to Terhal and DiVincenzo’s understanding [12] of Valient’s work onmatchgates, when the above unitary matrices U and U are elements of the spe-cial unitary group SU (2), quantum computations with the ˇ R ( x ) gate (30) actingon nearest-neighbor qubits can be efficiently simulated on a classical computer,whereas the ˇ R ( x ) gate (30) combined with the Swap gate acting on farther-neighbor qubits may be capable of performing universal quantum computation.Obviously, the two-qubit gate (25) has its two submatrices U and U in the SU (2) group.In terms of U and U in SU (2), the other two-qubit gate with six nonvanishingentries given byˇ R ( θ ) = sinh( γ − iθ )) 0 0 00 e iθ sinh γ − i sin θ − i sin θ e − iθ sinh γ 00 0 0 sinh( γ + iθ ) , (32)with the normalization factor ρ = sinh γ + sin θ , can be found in [7] as a unitarysolution of the Yang–Baxter equation (9) with the spectral parameter θ and realparameter γ , but which gives rise to a time-dependent Hamiltonian (27).However, the two-qubit gate (22) associated with the delta-function interac-tion (1) or Heisenberg interaction consists of the following submatrices U and U given by U = − (cid:18) (cid:19) , U = − e − iϕ (cid:18) cos ϕ i sin ϕi sin ϕ cos ϕ (cid:19) , (33)where U is in SU (2) but U is in U (2) because of the global phase. With theencoded logical qubit, it has been numerically verified [11] that the two-qubitgate (22) itself can achieve universal quantum computation by acting on nearest-neighbor qubits. Feynman is well known in the community of quantum information science due tohis pioneering work on both universal quantum simulation in 1982 and quantumcircuit model in 1986. Shortly before he passed away in early 1988, very unex-pectedly, Feynman wrote: “I got really fascinated by these (1+1)-dimensional7odels that are solved by the Bethe ansatz”, see Batchelor’s feature article [13]on the research history of the Bethe ansatz.A definition of integrable quantum computation in both physics and mathe-matics has been proposed in this article, in order to remove potential conceptualconfusions between the papers [3] and [7] as well as declare integrable quantumcomputation as an independent research subject in quantum information andcomputation. Note that integrable quantum computation has been also arguedfrom the viewpoint of the Hamiltonian formalism of quantum error correctioncodes [14].The asymptotic condition ˇ R ( x = 0) of the solution of the Yang–Baxter equa-tion (19) satisfies the braid group relation and hence ˇ R ( x = 0) can be viewedas a unitary braiding gate, which suggests similarities and comparisons between integrable quantum computation and quantum computing via unitary braid rep-resentations, see [7, 14] for the detail and [3] for comments. Acknowledgements The author wishes to thank Professor Lu Yu and Institute of Physics, ChineseAcademy of Sciences, for their hospitality and support during the visit in whichthis work was done and the part of the previous work [3] had been done. References [1] D. Deutsch, Quantum Computational Networks , Proc. R. Soc. Lond. A September 1989 vol. no. Quantum Computation and Quantum Infor-mation , pp. 203-204, Cambridge University Press, 2000 and 2011.[3] Y. 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