ABSTRACT
A new approach called Reference Input Generation Algorithm (RIGA) is introduced for quantum gate generation. Let be the trajectory that is obtained in step
, with
. In step
,
is right-translated in order to displace
to
. The translated trajectory is used as a reference for a Lyapunov-based tracking control law, generating
, and so on. A proof of the exponential convergence of
to zero is provided for
large enough. Two examples present numerical experiments regarding N coupled qubits. The first example considers N = 3 with a known minimum time
. It presents excellent results for
. The second example is a benchmark for the comparison between RIGA and GRAPE, considering a Hadamard gate for the systems with N = 2, 3, …, 10 qubits. The runtime of RIGA could be improved, and GRAPE was implemented in a faster CPU. However, RIGA presents results that are similar to GRAPE, with faster runtime in some cases, showing that the RIGA is indeed a promising algorithm.
Disclosure statement
No potential conflict of interest was reported by the authors.
Notes
1 Gradient Ascent Pulse Engineering (GRAPE).
2 It is shown in Silveira, Pereira da Silva, and Rouchon (Citation2014) that is a positively invariant set for the time-varying closed-loop system (Equation6
(6)
(6) )–(Equation7
(7a)
(7a) ).
3 The closed-loop system is also equivalently written as (Equation1(1)
(1) )–(Equation5
(5)
(5) )–(Equation7
(7a)
(7a) ).
4 Phase 2 only works when is finite, that is,
(see (Equation3
(3)
(3) )). When this is not the case, it is possible to apply the two-step procedure presented in Silveira et al. (Citation2016) that uses the square root of R in each iteration.
5 This is the Algorithm 2 in the next subsection.
6 Recall that , where
. For this define
. Since
, then such p always exists.
7 It is shown in Silveira et al. (Citation2016) that is indeed a compact set in
.
8 Since is non-decreasing and
is compact, then
is well-defined and remains in
for
.
9 Proposition A.2 of Appendix 3 implies that this is true almost surely when M is large enough.
10 This high order of brackets indicates that the control problem is rather nontrivial.
11 According to the erratum associated to Khaneja et al. (Citation2002) (available at doi:10.113/PhysRevA.68.049903), the range of θ is , and not
as it was in the original paper.
12 See the supplementary material for the values of .
13 The value J=1 is considered in Leung et al. (Citation2017) but the absolute value of inputs are bounded by , which is equivalent to consider
,
and
,
.
14 The data of runtime of GRAPE that is considered in Figure is the one of Figure 7 of Leung et al. (Citation2017) after correction according to their Figure 3 (since Figure 7 present the best result between GPU and CPU).
15 The value of λ may depend on the radius c.
16 It is invertible because its derivative is everywhere positive, and given by .
17 As the Lyapunov function is nonnegative it follows that .
18 See Proposition A.10 of Appendix 6 for the definition of and,
.
19 Recall that this means that .