Physics Letters A 350 (2006) 27–30 www.elsevier.com/locate/pla
Reducing execution time of quantum algorithms by additional permutation gates Mikio Nakahara a,∗ , Juha J. Vartiainen b , Yasushi Kondo a , Shogo Tanimura c , Kazuya Hata a a Department of Physics, Kinki University, Higashi-Osaka 577-8502, Japan b Low Temperature Laboratory, POB 3500, FIN-02015 HUT, Helsinki University of Technology, Finland c Graduate School of Engineering, Osaka City University, Osaka 558-8585, Japan
Received 16 April 2005; received in revised form 29 September 2005; accepted 2 October 2005 Available online 7 October 2005 Communicated by P.R. Holland
Abstract For a given quantum algorithm the time-optimal implementation is attained using the Cartan decomposition. We extend this approach and show that the execution time of a quantum algorithm can be further reduced by adding an extra gate to it. We apply this method to Grover’s algorithm and show that the execution time is halved. 2005 Elsevier B.V. All rights reserved. PACS: 03.67.Lx; 82.56.Jn Keywords: Time-optimal control; Cartan decomposition; Grover’s algorithm; NMR quantum computer
1. Introduction Quantum computing is an emerging discipline based on encoding information into a quantum-mechanical system [1,2]. There a quantum algorithm is expressed in a form of a unitary matrix T i Ualg = T exp − (1) H γ (t) dt , h¯ 0
where T stands for time-ordered product, γ (t) denotes collectively the control parameters of the Hamiltonian H at time t and T is the time required to implement the algorithm. It has been shown [3–5] that the set of U(2) gates and the CNOT gate are universal and a quantum circuit is often implemented with these so-called elementary quantum gates. * Corresponding author.
E-mail addresses:
[email protected] (M. Nakahara),
[email protected] (J.J. Vartiainen),
[email protected] (Y. Kondo),
[email protected] (S. Tanimura),
[email protected] (K. Hata). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.10.002
It should be noted, however, that the universality theorem does not address the issue of optimality of the implementation at all. A quantum computer is always under the threat of decoherence. Furthermore it contains analogue parameters such as amplitudes, phases and widths of pulses and therefore there always exist errors inherent in such gate operations. It is certainly desirable to reduce the execution time and the number of gates to protect a quantum computer from these threats. It has been shown recently [6,7] that the Cartan decomposition of a unitary matrix gives a time-optimal control of an NMR quantum computer, in which a single-gate operation takes negligible time compared to two-qubit entangling operations. This strategy is successfully demonstrated recently using an NMR quantum computer, whose pseudopure state is generated by cyclic permutations of state population [8]. It is the purpose of the present Letter to show that an execution time of a quantum algorithm can be further reduced by adding an extra gate W following a given algorithm Ualg . This seems to be somewhat counterintuitive from a classical viewpoint: if a classical algorithm which takes time τ2 to execute is combined serially after an algorithm which takes τ1 , it will take time τ1 + τ2 to execute the combined algorithm. This is not nec-
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essarily the case in quantum algorithms. By adding W to Ualg , the modified algorithm W Ualg may be mapped to a point closer to the unit matrix I than Ualg so that it takes less time to execute W Ualg than Ualg . In the next section, we briefly review the Cartan decomposition and its application to NMR quantum computation. In Section 3, we introduce extra permutation gates, which are shown to reduce the execution time. We also obtain optimized NMR pulse sequences with and without a permutation gate. Section 4 is devoted to summary and discussion. 2. Cartan decomposition and time-optimal control In an NMR quantum computer, a one-qubit operation may be carried out in a short time on the order 10 µs while a two-qubit entangling operation takes time typically ∼ 10 ms. Therefore one-qubit operation time may be ignored in estimating the overall execution time of a quantum algorithm [6]. Let us consider a molecule with two heteronucleus spins for definiteness, whose Hamiltonian, in the rotating frame with respective Larmor frequency, is H(γ ) = −ω11 cos φ1 (σx ⊗ I2 /2) + sin φ1 (σy ⊗ I2 /2) − ω12 cos φ2 (I2 ⊗ σx /2) + sin φ2 (I2 ⊗ σy /2) + 2πJ σz ⊗ σz /4,
(2)
where ω1i and φi are control parameters denoting the amplitude and the angle in the xy-plane of the external rf fields, respectively [9]. This Hamiltonian generates a unitary matrix Ualg ∈ SU(4) via the time-evolution equation (1). Above, J denotes the spin–spin coupling constant. Typically we have ω1i J , which justifies the above assumption of negligible one-qubit operation time compared to two-qubit operation time. Neglecting one-qubit operation time in evaluating the execution time amounts to identifying the matrices U1 , U2 ∈ SU(4) which differ by an element of K ≡ SU(2) ⊗ SU(2). Thus the relevant space for evaluating the time-optimal path is the coset space SU(4)/SU(2) ⊗ SU(2). To find the time-optimal path connecting the unit matrix I and the matrix Ualg , therefore, amounts to finding the time-optimal path connecting cosets [I ] and [Ualg ], where [U ] ≡ {kU | k ∈ K}. The Lie algebra su(4) of SU(4) is decomposed as su(4) = k ⊕ p [10], where k = Span {iI ⊗ σj /2, iσj ⊗ I /2} (j = x, y, z), (3) ⊥ p = k = Span {iσj ⊗ σk /4} (j, k = x, y, z). (4) Note that they satisfy the commutation relations [k, k] ⊂ k,
[p, k] ⊂ p,
[p, p] ⊂ k.
(5)
Decomposition of a Lie algebra g into k and p, satisfying the above commutation relations, is called the Cartan decomposition. The Cartan subalgebra h = Span({iσj ⊗ σj /4}) ⊂ p plays an important role in the following construction. A general theorem of Lie algebras proves that any element Ualg ∈ SU(4) has a KP decomposition Ualg = kp with k ∈ K ≡ exp k and p ∈ P ≡ exp p. Moreover, any matrix p ∈ P is rewritten in the
conjugate form p = k1† hk1 , where k1 ∈ K and h is an element of the Cartan subgroup H of SU(4),
αj H ≡ exp h = exp i (6) σj ⊗ σj αj ∈ R . 4 j =x,y,z
Therefore we have a corresponding Cartan decomposition for a group element as Ualg = k2 hk1 , where ki ∈ K and h ∈ H . Then the quantum algorithm Ualg is decomposed into one-qubit operations k1 , k2 and the two-qubit entangling operation h. This decomposition determines an optimized pulse sequence of the NMR quantum computer as discussed in [6–8]. The Cartan decomposition of an arbitrary U ∈ SU(4) is carried out explicitly as follows. We take the Bell basis [11] √ |Ψ0 = 1/ 2 |00 + |11 , √ |Ψ1 = i/ 2 |01 + |10 , √ |Ψ2 = 1/ 2 |01 − |10 , √ |Ψ3 = i/ 2 |00 − |11 . (7) The transformation rule of a matrix U with respect to the standard binary basis |00 , |01 , |10 , |11 into that with the Bell basis |Ψi is U → UB ≡ Q† U Q, where 1 0 0 i 1 0 i 1 0 . Q= √ (8) 0 i −1 0 2 1 0 0 −i The matrix Q defines an isomorphism between K = SU(2) ⊗ SU(2) and SO(4) and is used to classify two-qubit gates [7,11]. Namely, it is easy to verify that Q† kQ is an element of SO(4) for k ∈ K. Moreover, Q diagonalizes elements of the Cartan subgroup, viz. Q† hQ = diag(eiθ0 , eiθ1 , eiθ2 , eiθ3 ) for h ∈ H . Therefore we find for U = k2 hk1 that UB = Q† U Q = Q† k2 Q · Q† hQ · Q† k1 Q = O2 hD O1 , where Oi ≡ Q† ki Q is an element of SO(4) and hD ≡ Q† hQ is a diagonal matrix. From UBT UB = O1T h2D O1 , we notice that UBT UB is diagonalized by O1 and its eigenvalues form the diagonal elements of h2D . Finally O2 is found as O2 = UB (hD O1 )−1 . As a concrete example, let us consider implementing twoqubit Grover’s database search algorithm Uij with an NMR quantum computer. The data is encoded in one of the basis vectors |00 , |01 , |10 , |11 and the gate Uij picks out a particular binary basis vector |ij as a “target file” upon acting on |00
[12,13]. Here we consider U10 which picks out the file |10
with a single step. The unitary matrix representing this algorithm takes the form 0 1 0 0 0 0 0 −1 . U10 = (9) −1 0 0 0 0 0 −1 0 We apply the above strategy and find the Cartan decomposition
M. Nakahara et al. / Physics Letters A 350 (2006) 27–30
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Table 1 Time-optimal pulse sequences for Grover’s algorithm U10 and the combined algorithm W4 U10 . The number 1 (2) denotes the first (second) qubit. Here X (Xm) and Y (Ym) denote π/2-pulse around x (−x) and y (−y) axis, respectively. The symbol Pi(θ) denotes a π -pulse around a vector (cos θ, sin θ, 0) in the Bloch sphere. The symbol (1/2J ) indicates the length of the idle time, during which no external pulses are applied. The number of pulses is reduced from 10 to 4 and the execution time is halved by adding the extra gate W4 Gate
Pulse sequence
U10
1: 2:
W4 U10
1: 2:
X X
(1/2J ) (1/2J )
Execution time
Xm Xm
Y Ym
(1/2J ) (1/2J )
X Y
Xm
Pi(−π/4)
(1/2J ) (1/2J )
X Pi(π)
Ym Pi(π/4)
1/J 1/2J
Fig. 1. Conceptual diagram showing the effect of an extra gate W in a compact space SU(4). The curves connecting these matrices represent time-optimal paths. (a) The time-optimal path connection I and Ualg . (b) The extra gate W is added following Ualg . The combined path is not time-optimal in general. (c) The combined gate W Ualg may be reachable from I in a shorter time than Ualg if the time-optimal path is taken.
of U10 = k2 hk1 as k1 = I 2 ⊗ I 2 ,
h = ei(π/4)(σx ⊗σx −σy ⊗σy ) , √
k2 = e−i(π/4)σz ⊗ ei(π/2
2 )(σx +σy )
.
(10)
Actually, the decomposition is not unique and we choose a solution that minimizes the execution time of an NMR quantum computer. To implement this decomposition with the NMR, such terms as ei(π/4)(σx ⊗σx ) must be rewritten in favour of the subset of generators of SU(4) contained in the Hamiltonian (2). We verify, for example, that ei(π/4)(σx ⊗σx ) = e−i(π/4)σy ⊗ ei(π/4)σy × e−i(π/4)(σz ⊗σz ) · ei(π/4)σy ⊗ e−i(π/4)σy . (11) Table 1 shows the pulse sequence to implement U10 with an NMR quantum computer. The time-optimal path requires the execution time of 1/J , which happens to be the same as that for the conventional pulse sequence [14]. 3. Reduction of execution time using extra gate
SU(2n ), there always exist such accelerating gates which reduce the execution time. In the present work, we choose the permutation matrices of the binary basis vectors as candidates for such gates. Although there are 4! = 24 permutation matrices, we are not required to examine all of 24 permutations since 18 of them are obtained by applying one-qubit rotations on the following six permutations: W 0 = I4 ,
12 W1 = UCNOT ,
W3 = USWAP ,
12 21 W4 = UCNOT UCNOT ,
21 12 UCNOT , W5 = UCNOT
(12) ij
where USWAP represents the swap gate and UCNOT is the CNOT gate whose control bit is i while the target bit is j . We have optimized the execution time of the matrices Wi U10 (0 i 5) by utilizing the Cartan decomposition and found that the time-optimal execution time is 1/J for i = 0, 1 and 2 while 1/2J for i = 3, 4 and 5. Therefore W3,4,5 are accelerating gates. A time-optimal control for the combined gate W4 U10 = k2 hk1 is √
k1 = ei(π/2 Suppose we are to implement an algorithm Ualg with a timeoptimal path connecting the unit matrix I and Ualg in Fig. 1(a). By introducing an extra gate W , it may be possible to have an algorithm W Ualg such that the time-optimal path for W Ualg takes shorter time to execute than that for Ualg as shown in Fig. 1(c). We call such a gate “accelerating”. The extra gate W must be simple enough so that we can deduce the output of Ualg efficiently from that of W Ualg using classical computation only. Since a matrix Ualg is an element of a compact group
21 W2 = UCNOT ,
2 )(σx −σz )
k2 = I ⊗ eiπσx /4 .
⊗ ei(π/4)σx ,
h = ei(π/4)σy ⊗σy , (13)
The execution time T satisfies πJ T /2 = π/4, yielding T = 1/2J . Since the decomposition (13) contains generators which do not exist in the Hamiltonian, a trick similar to (11) must be employed. Table 1 shows the actual NMR pulse sequences derived from the Cartan decompositions of U10 and W4 U10 . In spite of the extra gate W4 , the latter requires less pulses and half of the execution time compared to the former
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algorithm. The permutation W4 maps the binary basis vectors as W4 : |10 → |11 , and accordingly the output state of W4 U10 upon acting on the initial state |00 is |11 . We have executed the pulse sequences given in Table 1 with an NMR spectrometer and observed that the desired output states were produced. 4. Summary and discussion In summary, we have demonstrated that a quantum algorithm may be accelerated by adding an extra gate to it. We employed the Cartan decomposition of SU(4) to find the time-optimal path for any two-qubit algorithm. Next we introduced an extra gate W . If properly chosen, the combined gate W Ualg may be executed in a shorter time than Ualg . When applied to twoqubit Grover’s algorithm in the NMR experiment the execution time will be halved along with a reduction in the number of the gate pulses from 10 to 4. It is certainly desirable to find the best possible accelerating gate W without operating it on Ualg and optimizing the path using Cartan decomposition. We suppose that W partially disentangles entanglement created by Ualg . In view of this, proper quantification of entanglement, such as Makhlin invariants [11] for two-qubit gates, may be helpful. Although W3,4,5 halve the other two-qubit Grover’s algorithms, it should not be expected that these gates reduce the execution time of an arbitrary twoqubit gate: clearly, such a universal acceleration gate does not exist. In view of the compactness of the unitary group, however, there exists a finite set of simple unitary transformations, such that any quantum algorithm is accelerated by properly chosen elements of the set. We expect that the set of permutation matrices, or certain subset thereof, is such a universal acceleration set. It remains a challenging problem to analyze how the reductions in the execution time and in the number of pulses scale with the number of qubits involved. In addition to our method, various methods to use extra permutations to reduce execution time of quantum algorithms have been proposed. For example, Fu et al. [15] took advantage of the fact that a quantum Fourier transform (QFT) gate may be designed so that it has a special swap gate which renumbers qubits in a reverse order in the final layer. Therefore it is clear that neglecting this permutation gate accelerates a QFT gate. However, this reduction of execution time heavily depends on the particular structure of a QFT gate. In contrast, our general prescription is applicable to any algorithm by including general permutation gates, which are not necessarily decomposable into swap gates. Similarly Blais [16] has shown that one can modify a quan-
tum circuit for QFT to be amenable for parallel processing by inserting swap gates in the circuit. We point out again that this approach assumes qubit swaps while our permutations are more general and applicable to any algorithms. Furthermore, we optimize NMR pulse sequence for the entire combined algorithm which results in speedup which is difficult to notice in quantum circuit picture. In Marx et al. [17] they altered the output state, in which irrelevant qubits take different states than their original states. The modification of the algorithm is, however, ad hoc and lacks geometrical viewpoints that we stressed in our work. Acknowledgements We thank Akio Hosoya for helpful discussions. We also thank the referees for their useful suggestions and drawing our attention to [15–17]. M.N. would like to thank partial supports of Grant-in-Aids for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan, Grant No. 13135215 and from Japan Society for the Promotion of Science (JSPS), Grant No. 14540346. J.J.V. would like to thank Nokia Foundation for support. S.T. is partially supported by JSPS, Grant Nos. 15540277 and 17540372. References [1] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press, Cambridge, UK, 2000. [2] A. Galindo, M.A. Martin-Delgado, Rev. Mod. Phys. 74 (2002) 347. [3] A. Barenco, et al., Phys. Rev. A 52 (1995) 3457. [4] J.J. Vartiainen, M. Möttönen, M.M. Salomaa, Phys. Rev. Lett. 92 (2004) 177902. [5] M. Möttönen, J.J. Vartiainen, V. Bergholm, M.M. Salomaa, Phys. Rev. Lett. 93 (2004) 130502. [6] N. Khaneja, R. Brockett, S.J. Glaser, Phys. Rev. A 63 (2001) 032308. [7] J. Zhang, J. Vala, S. Sastry, K.B. Whaley, Phys. Rev. A 67 (2003) 042313. [8] M. Nakahara, Y. Kondo, K. Hata, S. Tanimura, Phys. Rev. A 70 (2004) 052319. [9] L.M.K. Vandersypen, I.L. Chuang, Rev. Mod. Phys. 76 (2004) 1037. [10] A.W. Knapp, Lie Groups Beyond an Introduction, second ed., Birkhäuser, Boston, 2002. [11] Y. Makhlin, Quantum Inform. Process. 1 (2002) 243. [12] L.K. Grover, in: Proceedings of the 28th Annual ACM Symposium on the Theory of Computation, ACM Press, New York, 1996, p. 212. [13] L.K. Grover, Phys. Rev. Lett. 79 (1997) 325. [14] I.L. Chuang, N. Gershenfeld, M. Kubince, Phys. Rev. Lett. 80 (1998) 3408. [15] L. Fu, J. Luo, L. Xiao, X. Zeng, Appl. Magn. Reson. 19 (2000) 153. [16] A. Blais, Phys. Rev. A 64 (2001) 022312. [17] R. Marx, A.F. Fahmy, J.M. Myers, W. Bermel, S.J. Glaser, Phys. Rev. A 62 (2000) 012310.
