Abstract
Quantum annealing is an innovative idea and method for avoiding the increase of the calculation cost of the combinatorial optimization problem. Since the combinatorial optimization problems are ubiquitous, quantum annealing machine with high efficiency and scalability will give an immeasurable impact on many fields. However, the conventional quantum annealing machine may not have a high success probability for finding the solution because the energy gap closes exponentially as a function of the system size. To propose an idea for finding high success probability is one of the most important issues. Here we show that a degenerate twolevel system provides the higher success probability than the conventional spin1/2 model in a weak longitudinal magnetic field region. The physics behind this is that the quantum annealing in this model can be reduced into that in the spin1/2 model, where the effective longitudinal magnetic field may open the energy gap, which suppresses the Landau–Zener tunneling providing leakage of the ground state. We also present the success probability of the Λtype system, which may show the higher success probability than the conventional spin1/2 model.
Introduction
Quantum annealing is an interesting approach for finding the optimal solution of combinatorial optimization problems by using the quantum effect^{1,2,3,4}. The combinatorial optimization problems are ubiquitous in the real social world, therefore the spread of quantum annealing machine with high efficiency and high scalability will give impacts and benefits on many fields, such as an industry including drug design^{5}, financial portfolio problem^{6}, and traffic flow optimization^{7}. After the commercialization of superconducting quantum annealing machine by DWave Systems inc.^{8}, several hardwares have been investigated and developed^{9,10,11,12,13}.
However, there are bottlenecks for implementing scalable quantum annealing machine; for the conventional and scalable quantum annealing machine may not have a high success probability for finding the solution of a combinatorial optimization problem because of the emergence of the first order phase transition, where the energy gap between ground state and the first exited state closes exponentially as a function of the system size^{2}. In this case, it necessitates an exponentially long annealing time for finding the solution of the problem^{14,15,16}. In the case of the second oder phase transition, on the other hand, an annealing time for finding the solution may scales polynomially as a function of the system size^{17}.
To propose an idea for finding high success probability is one of the most important and challenging issue in the field of quantum annealing. One of the approaches for obtaining the high success probability is to engineer the scheduling function for the driving Hamiltonian and the problem Hamiltonian, such as a monotonically increasing scheduling function satisfying the local adiabatic condition^{18}, the reverse quantum annealing^{19} implemented in Dwave 2000Q^{20}, inhomogeneous sweeping out of local transverse magnetic fields^{21,22}, and a diabatic pulse application^{23}. Another is to add an artificial additional Hamiltonian for suppressing the emergence of the excitations with avoiding the slowing down of annealing time, which is called shortcuts to adiabaticity by the counterdiabatic driving^{24,25,26,27}, and to add an additional Hamiltonian for avoiding the first order phase transition^{17,28,29}. In this paper, we study the possibility of other approach: to employ a variant spin, such as a qudit, in the quantum annealing architecture.
Recently, two of the authors have studied the quantum phase transition in a degenerate twolevel spin system, called the quantum Wajnflasz–Pick model, where an internal spin state is coupled to all the same energy internal states with a single coupling strength, and to all the different energy internal states with the other single coupling strength^{30}. In the earlier study, this model is found to show a several kinds of phase transition while annealing; single or double firstorder phase transitions as well as a single secondorder phase transition, depending on an internal state coupling parameter^{30}, which suggests that the quantum annealing of this model may be controlled by an internal state tuning parameter. However, the study is based on the static statistical approach using the meanfield theory, because only the order of the phase transition has been interested in. Therefore, the enhancement of the success probability for quantum annealing based on degenerate twolevel systems is not clear yet. Furthermore, they employed a fullyconnected uniform interacting system, and it is unclear whether their idea works that a double (or evennumber of) firstorder phase transition while annealing would bring the system back into the ground state at the end of the annealing, where the even number of the Landau–Zener tunneling may happen with respect to the ground state.
In the present paper, we clarify the success probability of the quantum annealing in the quantum Wajnflasz–Pick model, focusing on (i) the Schrödinger dynamics, (ii) eigenenergies, and (iii) nonuniform effects of the spininteraction as well as the longitudinal magnetic field. We find that the quantum Wajnflasz–Pick model is more efficient than the conventional spin1/2 model in the weak longitudinal magnetic field region as well as in the strong coupling region between degenerate states. We also find that the quantum Wajnflasz–Pick model is reducible into a spin1/2 model, where effect of the transverse magnetic field in the original Hamiltonian emerges in the reduced Hamiltonian not only as the effective transverse magnetic field but also as the effective longitudinal magnetic field. As a result, this model may provide the higher success probability in the case where the effective longitudinal magnetic field opens the energy gap between the ground state and the first excited state. We also evaluate the success probability in another variant spin, a Λtype system^{31,32,33,34,35,36,37,38,39,40}, which has three internal levels. This model also shows the higher success probability than the conventional spin1/2 model in the weak magnetic field region.
A multilevel system is ubiquitous, which can be seen, for example, in degenerate twolevel systems in atoms^{41,42}, Λtype atoms^{31,32,34}, Λ, V, Θ and Δtype systems in the superconducting circuits^{33,35,36,37,38,39,40,43} as well as Λtype systems in the nitrogenvacancy centre in diamond^{44}. We hope that insights of our results in the degenerate twolevel system and knowledge of their reduced Hamiltonian inspire and promote further study as well as future engineering of quantum annealing.
Quantum Wajnflasz–Pick Model
A conventional quantum annealing consists of the spin1/2 model, where the time dependent Hamiltonian is given by^{1}
where \({\hat{H}}_{z,x}\) are a problem and driver Hamiltonian, respectively, and \(s\equiv t/T\) is the time \(t\in [0,T]\) scaled by the annealing time T. The problem Hamiltonian \({\hat{H}}_{z}\) with the number of spins N, which encodes the desired optimal solution, has a nontrivial ground state. In contrast, the driver Hamiltonian \({\hat{H}}_{x}\) has a trivial ground state, where the driver Hamiltonian \({\hat{H}}_{x}\) must not be commutable with the problem Hamiltonian \({\hat{H}}_{z}\). A problem Hamiltonian and driver Hamiltonian are typically given by
where \({\hat{\sigma }}^{x,z}\) are the Pauli matrices, \({J}_{ij}\) is the coupling strength between spins, \({h}_{i}^{z}\) is the local longitudinal magnetic field, and \({h}_{i}^{x}\) is the local transverse magnetic field. The timedependent total Hamiltonian \(\hat{H}(s)\) gradually changes from the driver Hamiltonian \({\hat{H}}_{x}\) to the problem Hamiltonian \({\hat{H}}_{z}\). If the Hamiltonian changes sufficiently slowly, the quantum adiabatic theorem guarantees that the initial quantum ground state follows the instantaneous ground state of the total Hamiltonian^{45}. We can thus finally obtain a nontrivial ground state of the problem Hamiltonian starting from the trivial ground state of the deriver Hamiltonian making use of the Schrödinger dynamics.
The quantum Wajnflasz–Pick model is a quantum version of the Wajnflasz–Pick model^{46}, which can describe one of the interacting degenerate twolevel systems. In the language of the quantum annealing, the problem Hamiltonian and the driver Hamiltonian are respectively given by^{30}
(Schematic picture of this model is shown in Fig. 1). The Hamiltonian of this model can be simply obtained by replacing the Pauli matrices \({\hat{\sigma }}^{x,z}\) in Eqs. (2) and (3) with the spin matrices of the quantum WajnflaszPick model \({\hat{\tau }}^{x,z}\). The spin operator \({\hat{\tau }}^{z}\) is given by^{30}
where \({g}_{{\rm{u}}(l)}\) is the number of the degeneracy of the upper (lower) states. The spinoperator \({\hat{\tau }}^{x}\) in the driver Hamiltonian is given by
where \({\bf{A}}(l)\) is a \((l\times l)\) matrix with the offdiagonal term \(\omega \), given by
Here, \(\omega \) is a parameter of the internal transition between the degenerated upper/lower states. The matrix \({\bf{1}}(m,n)\) is the \((m\times n)\) matrix, where all the elements is unity, which gives the transition between the upper and lower states. The constant \(c\) is the normalization factor, where the maximum eigenvalue is normalized to be +1, so as to be equal to the maximum eigenvalue of \({\hat{\tau }}^{z}\).
In the following, for the consistency to the earlier work^{30}, we consider a uniform transverse field \({h}_{i}^{x}\equiv 1\), and also take the parameter of the internal transition to be real \(\omega ={\omega }^{\ast }\) with \(\omega > \,1\). In the case where \(({g}_{{\rm{u}}},{g}_{{\rm{l}}})=(2,1)\), we have
with \(c=(\omega +\sqrt{8+{\omega }^{2}})/2\), which is a kind of the Δtype system^{38}.
In this paper, we employ the common sets of parameters in both quantum Wajnflasz–Pick model and the conventional spin1/2 model, including the coupling strength \({J}_{ij}\), the magnetic fields \({h}_{i}^{z,x}\), and the annealing time \(T\). By using these parameters, we can obtain the same spin configuration (+1 or −1) in the ground state of the problem Hamiltonian. We thus compare efficiency of these models from the success probability.
Schrödinger Dynamics
In order to numerically calculate the success probability of the quantum annealing, we employ the Crank–Nicholson method^{47} for solving the Schrödinger equation
In this method, the timeevolution of the wave function is calculated by using the Cayley’s form^{47}
Although the inverse matrix is needed, this method conserves the norm of the wave function and is secondorder accurate in time^{47}.
We first consider the fully connected model, where the spinspin coupling is ferromagnetic and the longitudinal magnetic field is uniform \({h}_{i}^{z}\equiv h\), which is consistent with the earlier work^{30}. For example, in the case where \((\omega ,h)=(0.8,0.02)\) and \((\,\,0.8,\,0.02)\) for \(({g}_{{\rm{u}}},{g}_{{\rm{l}}})=(2,1)\), the timedependence of the ground state population of the problem Hamiltonian, given by \({n}_{0}\equiv \langle \Psi (t){\Psi }_{0}(T)\rangle {}^{2}\), clearly shows that this quantity in the quantum Wajnflasz–Pick model is greater than that in the conventional spin1/2 model (Panels (a) and (b) in Fig. 2). Here, \({\Psi }_{0}(T)\rangle \) is the ground state of the problem Hamiltonian, and \(\Psi (T)\rangle \) is the wave function obtained from the timedependent Schrödinger equation. In the case where \((\omega ,h)=(0.8,\,0.1)\) and \((\,\,0.8,0.1)\) for \(({g}_{{\rm{u}}},{g}_{{\rm{l}}})=(2,1)\), on the other hand, the ground state population of the problem Hamiltonian in the quantum Wajnflasz–Pick model is less than that in the spin1/2 model (Panels (c) and (d) in Fig. 2).
Compare the success probability of the quantum Wajnflasz–Pick model, \(P\equiv \langle \Psi (T){\Psi }_{0}(T)\rangle {}^{2}\), with that of the conventional spin1/2 model denoted as \({P}_{\mathrm{1/2}}\), where \(\Psi (T)\rangle \) is the final state obtained from the timedependent Schrödinger equation. In almost all regions in the \(\omega \)h plane, efficiencies of both models are almost the same, where the ratio of the success probability of the quantum Wajnflasz–Pick model and that of the conventional spin1/2 model is almost unity (Fig. 3). On the other hand, in the regime of the weak longitudinal magnetic field h, we can find higher or lower efficiency regions in the quantum Wajnflasz–Pick model, compared with the spin1/2 model. In the spin glass model, a nontrivial state may emerge in the weak longitudinal magnetic field limit^{48}. In a pspin model where \(p=3,5,7,\ldots \), the energy gap is known to close exponentially and the firstorder phase transition emerges in the absence of the longitudinal magnetic field^{15}. In this sense, it is of interest that the quantum Wajnflasz–Pick model may provide the high efficiency in the weak longitudinal magnetic field region.
In the case where \(({g}_{{\rm{u}}},{g}_{{\rm{l}}})=(2,2)\), where the numbers of upper and lower states are equal, the success probability of the quantum Wajnflasz–Pick model is almost equal to that of the conventional spin1/2 model (Panel (a) in Fig. 4). In the case where \(({g}_{{\rm{u}}},{g}_{{\rm{l}}})=(3,2)\), the success probability of the quantum Wajnflasz–Pick model is almost equal to that of the case where \(({g}_{{\rm{u}}},{g}_{{\rm{l}}})=(2,1)\), where the differences between the number of the upper states and that of lower states are the same in both cases (Fig. 3 and Panel (b) in Fig. 4).
Eigenvalues
Eigenvalue spectrum of the instantaneous Hamiltonian may help to understand these higher or lower success probabilities of the quantum Wajnflasz–Pick model than that of spin1/2 model, although eigenvalues of the instantaneous Hamiltonian shows tangled spaghetti structures (Fig. 5). For example, in the case where \((\omega ,h)=(0.8,\,0.1)\), the energy gap between the ground state and the first excited state clearly closes once, which causes the low success probability (Panel (c) in Fig. 5). In the case where \((\omega ,h)=(0.8,0.02)\), the ground state and the first excited state are finally merged at the annealing time, where the degeneracy would cause the high success probability (Panel (a) in Fig. 5). However, according to the following discussion, it will be found that the latter explanation would not be correct in the case where \((\omega ,h)=(0.8,0.02)\). From panels (b) and (d) in Fig. 5, many crossings of eigenvalues are found to emerge. It suggests that there are no matrix elements in some states, and we may find symmetry behind the present quantum Wajnflasz–Pick model, where the Hamiltonian would be block diagonalized by a unitary operator \(\hat{U}\). Since the energy spectrum of the original quantum WajnflaszPick model shows very complicated behavior, it would be better to find out the reason of the high/low success probability from the reduced Hamiltonian, which are truly relevant for the efficiency of the quantum annealing.
For example, in the case where \(({g}_{{\rm{u}}},{g}_{{\rm{l}}})=(2,1)\), the singlespin Hamiltonian in the quantum Wajnflasz–Pick model is decomposable, where the irreducible representation is given by
for arbitrary values of s, by using the unitary operator
where \({h}^{\pm }(s)\equiv {h}^{z}s\pm 2\omega h^{\prime} (s)\), and \(h^{\prime} (s)\equiv (1s){h}^{x}/(2c)\). As a result, we may reduce a quantum annealing problem in the singlespin quantum Wajnflasz–Pick model into that of the spin1/2 model, the Hamiltonian of which is given in the form
Since the initial ground state of the singlespin Hamiltonian is given by \(\Psi (s=0)\rangle \propto {(c/2,c/2,1)}^{{\rm{T}}}\) in the original quantum Wajnflasz–Pick model, this state can be mapped to \(\hat{U}\Psi (s=0)\rangle \propto {(c/\sqrt{2},0,1)}^{{\rm{T}}}\). It indicates that the initial ground state \(\hat{U}\Psi (s=0)\rangle \) can be also projected to the Hilbert space of the reduced Hamiltonian \(\hat{\mathscr H}(s)\).
This reduction of the singlespin problem in the case where \(({g}_{{\rm{u}}},{g}_{{\rm{l}}})=(2,1)\) can be generalized to an interacting Nspin problem (Fig. 6). A quantum annealing problem of the original quantum Wajnflasz–Pick model is reduced into that of the spin1/2 model, given in the form
where
with
As in the singlespin case, the initial ground state of the original Nspin quantum Wajnflasz–Pick model can be also projected to the Hilbert space of the reduced Hamiltonian (15). The coupling \({J}_{ij}\) in the reduced Hamiltonian is the same as that of the original Wajnflasz–Pick model. The effective longitudinal magnetic field \({h}_{{\rm{eff}},i}^{z}\) in the reduced Hamiltonian also reaches the same value as that of the original Wajnflasz–Pick model at the end of the annealing: \({h}_{{\rm{eff}},i}^{z}(s=1)={h}_{i}^{z}\). Eigenvalues of the reduced spin1/2 model exactly trace eigenvalues in the original Wajnflasz–Pick model (Fig. 5). The timedependence of the ground state population of the problem Hamiltonian is confirmed to show the completely same behavior between the reduced model and the original model.
This effective model clearly explains behavior of success probability of the quantum Wajnflasz–Pick model shown in Fig. 3. Note that the coefficient \(c\) is a positive real number such that the maximum eigenvalue of \({\tau }^{x}\) is unity, and we take \({h}_{i}^{x}=1\). Then, \({h^{\prime} }_{i}(s)\ge 0\) always holds during the annealing time \(0\le s\le 1\). In the case where the longitudinal magnetic field \({h}_{i}^{z}\) is very large, \({h}_{i}^{z}\gg \omega {h^{\prime} }_{i}(0)\), the effect of the original longitudinal magnetic field \({h}_{i}^{z}\) is dominant compared with the effective additional term \(\omega {h^{\prime} }_{i}(s)\) except at the very early stage of the annealing \(s\ll \omega {h^{\prime} }_{i}(0)/{h}_{i}^{z}\). In this case, the problem Hamiltonian in the reduced model is almost the same as that in the conventional spin1/2 model in Eq. (2). As a result, the success probability of the quantum Wajnflasz–Pick model is almost the same as that of the conventional spin1/2 model, which provides \(P\simeq {P}_{\mathrm{1/2}}\).
In the case where the original longitudinal magnetic field \({h}_{i}^{z}\) is not large, the effective additional longitudinal magnetic field \(\omega {h^{\prime} }_{i}(s)\) cannot be neglected compared with \({h}_{i}^{z}\). When the effective additional field is in the same direction as the original longitudinal field, the total effective longitudinal magnetic field \({h}_{{\rm{eff}},i}^{z}(s)\) is enhanced, which opens the energy gap between the ground state and the first excited state (Panels (a) and (b) in Fig. 7). This region is given by the condition \(\omega {h}_{i}^{z} > 0\), which is consistent with the result shown in Fig. 3. As a result, the success probability of the quantum Wajnflasz–Pick model become superior to that of the conventional spin1/2 model. When the effective additional field is in the opposite direction to the original longitudinal field, the total effective longitudinal magnetic field \({h}_{{\rm{eff}},i}^{z}(s)\) is diminished, which closes the energy gap between the ground state and the first excited state (Panels (c) and (d) in Fig. 7). This region is given by the condition \(\omega {h}_{i}^{z} < 0\), which is consistent with the result shown in Fig. 3. As a result, the success probability of the quantum Wajnflasz–Pick model become inferior to that of the conventional spin1/2 model.
Behavior of success probability is also explained by the reference of the annealing time^{49}
where
Here, \({\Psi }_{0(1)}(s)\rangle \) and \({E}_{\mathrm{0(1)}}(s)\) are the wave functions and eigenenergies of the ground (firstexcited) state with respect to the instantaneous Hamiltonian, respectively. Annealing machine needs the annealing time \(T\) much larger than \({\mathscr T}\). Let \({T}^{\ast }\equiv b(s)/{\Delta }^{2}(s)\) be an instantaneous reference time of the annealing. The maximum value of this time \({T}^{\ast }\) in the reduced Wajnflasz–Pick model given in (15) is suppressed compared with that of the conventional spin1/2 model, where the effective additional field \(\omega {h^{\prime} }_{i}(s)\) is in the same direction as the original longitudinal field \({h}_{i}^{z}\) (Panels (a) and (b) in Fig. 8). It is consistent with the case where the quantum Wajnflasz–Pick model is more efficient than the conventional spin1/2 model in the region where \(\omega {h}_{i}^{z} > 0\) (Fig. 3). The maximum value of \({T}^{\ast }\) in the effective Wajnflasz–Pick model has larger values than that of the spin1/2 model, where the effective additional field \(\omega {h^{\prime} }_{i}(s)\) is in the opposite direction to the original longitudinal field \({h}_{i}^{z}\) (Panels (c) and (d) in Fig. 8). It is consistent with the case where the quantum Wajnflasz–Pick model is less efficient than the conventional spin1/2 model in the region where \(\omega {h}_{i}^{z} < 0\) (Fig. 3).
In order to perform the scaling analysis of the minimum energy gap \({\Delta }_{{\rm{\min }}}\equiv \,{\rm{\min }}\,[{E}_{1}(s){E}_{0}(s)]\), we consider the \(p\)spin model in the absence of the longitudinal magnetic field:
where the transverse magnetic field is homogeneous. Replacement of \({\tau }_{i}^{x,y}\) with \({\sigma }_{i}^{x,y}\) provides the conventional \(p\)spin model, where the first order phase transition emerges, and the minimum energy gap is known to close exponentially as \(N\) increases in the case where \(p\) is odd^{15}. After mapping to the subspace spanned by the spin1/2 model, the reduced Hamiltonian of the quantum Wajnflasz–Pick model with \(({g}_{{\rm{u}}},{g}_{{\rm{l}}})=(2,1)\) can be given by
up to the constant energy shift, where \({\Gamma }^{z}\equiv \omega {h}^{x}/(2c)\), \({\Gamma }^{x}\equiv \sqrt{2}{h}^{x}/c\), and \({\hat{M}}^{z,x}\equiv \mathop{\sum }\limits_{i}^{N}\,{\hat{\sigma }}_{i}^{z,x}\). By using the commutation relation \([{\hat{\sigma }}_{i}^{x},{\hat{\sigma }}_{j}^{z}]=2i{\hat{\sigma }}_{i}^{z}{\delta }_{ij}\) and by following the standard argument of the angular momentum, where the total spin \({\hat{{\bf{M}}}}^{2}\equiv {({\hat{M}}^{x})}^{2}+{({\hat{M}}^{y})}^{2}+{({\hat{M}}^{z})}^{2}\) conserves, the Hilbert space can be spanned by states \(J,M\rangle \), where \({\hat{{\bf{M}}}}^{2}J,M\rangle =J(J+2)J,M\rangle \) and \({\hat{M}}^{z}J,M\rangle =MJ,M\rangle \) with \(M=\,J,\,J+2,\cdots ,J2,J\). The diagonal elements of this Hamiltonian is given by \({\mathscr H}_{MM}=\,s{M}^{p}/({N}^{p1})(1s){\Gamma }^{z}M\), and the offdiagonal elements are \({\mathscr H}_{M,M\pm 2}=\,(1s){\Gamma }^{x}\sqrt{J(J+2)M(M\pm 2)}/2\). Since the ground state of this model is given by the case \(J=N\), we diagonalize the \((N+\mathrm{1)}\times (N+\mathrm{1)}\) matrix of the reduced Hamiltonian. We compare the minimum energy gap of this model reduced from the quantum Wajnflasz–Pick model with that of the conventional \(p\)spin model composed of the spin1/2 system (Eq. (24) with \({\Gamma }^{z}=0\) and \({\Gamma }^{x}={h}^{x}\)). Figure 9 clearly shows that the minimum energy gap closes exponentially in the conventional spin1/2 model, and the gap closes polynomially in the model reduced from the quantum Wajnflasz–Pick model. This polynomial gap closing originates from the emergence of the effective longitudinal magnetic field in the reduced model: \({\Gamma }^{z}=\omega {h}^{x}/(2c)\ne 0\).
Random Coupling
In the random spinspin coupling case, where \({J}_{ij}\) are randomly generated by the gaussian distribution function^{50}
the density plot of the meanvalue of the success probability is similar to the uniform coupling case. The maximum (minimum) value of the success probability is, however, suppressed (increased) compared with the uniform coupling case (Fig. 10). The variances of the success probability of the quantum Wajnflasz–Pick model are almost ranged from 0.03 to 0.06 in the first and third orthants in the \(\omega \)\(h\) plane, where the higher success probability may be obtained than the conventional spin1/2 model. They are almost ranged from 0.02 to 0.15 in the second and forth orthants in the \(\omega \)\(h\) plane, where the lower success probability may be obtained. In the spin1/2 model, the variance of the success probability is almost within the range from 0.03 to 0.06 in all the orthants.
The discussion above is in the case for a uniform longitudinal magnetic field. In the following, we discuss the case of random longitudinal magnetic fields \({h}_{i}^{z}\) in addition to the random interactions \({J}_{ij}\). The success probabilities \(P\) and \({P}_{\mathrm{1/2}}\) are almost equal in the weak internal state coupling case (\(\omega =\pm \,0.1\) in Fig. 11). In the strong internal state coupling case (\(\omega =\pm \,1\) in Fig. 11), the distribution is broaden. Although we can find cases where the conventional spin1/2 model is superior to the quantum Wajnflasz–Pick model, we can also find many cases where the quantum Wajnflasz–Pick model is superior to the conventional spin1/2 model, where the success probability is close to the unity compared with the conventional spin1/2 model.
In these random coupling cases, it may not be definitely concluded that the quantum Wajnflasz–Pick model is always more efficient than the conventional spin1/2 model. The variance is relatively large, and there are cases where the quantum Wajnflasz–Pick model is inferior to the conventional spin1/2 model (Fig. 11). However, we can find many cases where the quantum Wajnflasz–Pick model is possibly more efficient than the conventional spin1/2 model. In the quantum Wajnflasz–Pick model and its reduced model, we have chances to find a better solution of the combinatorial optimization problem. In real annealing machines, we can extract a better solution after performing many sampling experiments by tuning \(\omega \).
Discussion
In the case where \(({g}_{{\rm{u}}},{g}_{{\rm{l}}})=(2,1)\), the spin matrix in the quantum Wajnflasz–Pick model is represented by a (3 × 3)matrix, which suggests that the quantum Wajnflasz–Pick model in this case may be mapped into the model represented by the spin1 matrices given by
Indeed, after we interchange elements of second and third rows in the spin matrices defined in Eq. (9) in the case where \(({g}_{{\rm{u}}},{g}_{{\rm{l}}})=(2,1)\), as well as we interchange elements of second and third columns, simultaneously, we find the following maps
where we have introduced quadrupolar operators^{51,52}
and \(\Re \omega \) (\(\Im \omega \)) is the real (imaginary) part of \(\omega \). Since \([{\hat{q}}^{z},{({\hat{S}}^{x})}^{2}]=0\) and \([{\hat{q}}^{x},{({\hat{S}}^{x})}^{2}]=i(\Im \omega /c){\hat{S}}^{x}\) hold, we find that \({({\hat{S}}^{x})}^{2}\) is the operator of the conserved quantity in the case where the parameter \(\omega \) is a real number. The coupling of \({\hat{\tau }}_{i}^{z}{\hat{\tau }}_{j(\ne i)}^{z}\) is mapped into the interaction \({\hat{q}}_{i}^{z}{\hat{q}}_{j(\ne i)}^{z}\), which is a kind of the biquadratic interaction with respect to the spin. In short, the interacting quantum Wajnflasz–Pick model with \(({g}_{{\rm{u}}},{g}_{{\rm{l}}})=(2,1)\) can be mapped into the spin1 model with an artificial biquadratic interaction. In particular, in the case where \(\omega \in {\mathbb{R}}\), there is the hidden symmetry related to \({({\hat{S}}^{x})}^{2}\), which indicates that the quantum Wajnflasz–Pick model is reducible in this case.
It is general that an interacting quantum Wajnflasz–Pick model is reducible to the conventional spin1/2 model. It holds for an arbitrary number of the degeneracy \(({g}_{{\rm{u}}},{g}_{{\rm{l}}})\) and at an arbitrary time s, which can be proven in the case where the parameter \(\omega \) is a real number and the condition \(\omega > \,1\) holds. In Supplementary Information, we show that the Hamiltonian of the interacting quantum Wajnflasz–Pick model with arbitrary \(({g}_{{\rm{u}}},{g}_{{\rm{l}}})\) can be projected to the spin1/2 model, and the initial ground state in the original quantum Wajnflasz–Pick Hamiltonian is also projected to the reduced Hilbert space. It indicates that the quantum annealing in the quantum Wajnflasz–Pick model can be always described by the reduced Hamiltonian.
As shown in Supplementary Information, this projection holds not only in the 2body interacting quantum Wajnflasz–Pick model, but also in the \(N\)body interacting model. It indicates that if the quantum Wajnflasz–Pick model is embedded into the Lechner–Hauke–Zoller (LHZ) architecture^{53,54}, it can be also projected into the LHZ architecture composed of the spin1/2 model, where the effective additional magnetic fields may emerge. The present quantum Wajnflasz–Pick model is a degenerate twolevel system in the presence of the transverse magnetic field. The possibility of the implementation of the degenerate twolevel system has been discussed for the \({D}_{2}\) line of ^{87}Rb^{41,42}. The quantum Wajnflasz–Pick model is also similar to the Δtype cyclic artificial atom in the superconducting circuit^{38,43}. In the Δtype artificial atom, the population is controllable by making use of the amplitudes and/or phases of microwave pulses, where the amplitudes alone controls the population in the conventional threelevel system (Λtype system)^{43}. However, the Δtype system in the superconducting circuit is not an exactly degenerate twolevel system. With this regard, it may be difficult to directly implement our model in the Δtype cyclic artificial atom in the superconducting circuit. Actually, it may be feasible to employ the spin1/2 model with the scheduling function inspired by the quantum Wajnflasz–Pick model, in the case where the Schrödinger dynamics without the dissipation holds.
The quantum Wajnflasz–Pick model is one of the qudit models, which is a kind of the artificial Δtype system^{38,43} in the case where \(({g}_{{\rm{u}}},{g}_{{\rm{l}}})=(2,1)\). The question naturally arises whether the Λtype system also shows the higher success probability than the conventional spin1/2 model. The spin matrix of the Λtype system we employ here is given by
where we take \(\varepsilon \le 1\), and the coefficient \(c\equiv \sqrt{1+{\kappa }^{2}}\) is a normalization factor so as the maximum eigenvalues of \({\hat{\tau }}^{x,z}\) are unity. The Hamiltonian of the quantum annealing with the Λtype system is given by Eqs. (1), (4) and (5), where \({\hat{\tau }}^{x,z}\) are replaced with those given in (32). The success probability in the Λtype system is found to be higher than that in the conventional spin1/2 model, in the case where \(\varepsilon \) is small in the weak longitudinal magnetic field region, which is similar to the case of the quantum Wajnflasz–Pick model (Panels (a) and (b) in Fig. 12). When \(\varepsilon \) is large, on the other hand, the success probability is drastically suppressed (Panel (c) in Fig. 12). In the case of a single Λspin system with \(\varepsilon =0\), which corresponds to a degenerate twolevel system, the unitary transformation
can map the Hamiltonian \(\hat{H}(s)=\,s{h}^{z}{\hat{\tau }}^{z}(1s){h}^{x}{\hat{\tau }}^{x}\) to the following block diagonal form:
As a result, after exchanging the first and second columns and also the first and second rows, we may reduce a quantum annealing problem in this Λspin model into that of the spin1/2 model, the Hamiltonian of which is given by \(\hat{\mathscr H}(s)=\,s{h}^{z}{\sigma }^{z}/2(1s){h}^{x}{\sigma }^{x}s{h}^{z}/2\). Although the Λtype system may provide the higher success probability than the conventional spin1/2 model, the effect of dark states (never employed states) on the quantum annealing in the general Λspin case and its reduction to the spin1/2 model in the manyspin system would be important issues for future study.
To summarize, we have demonstrated that qudit models, such as the quantum Wajnflasz–Pick model as well as the Λtype system, may provide the higher success probability than the conventional spin1/2 model in the weak magnetic field region. We have analytically shown that the quantum Wajnflasz–Pick model can be reduced into the spin1/2 model, where effect of the transverse magnetic field in the original Hamiltonian emerges as the effective additional longitudinal magnetic field in the reduced Hamiltonian, which possibly opens the energy gap between the ground state and the first excited state in the reduced Hamiltonian. Since qubits have experimental advantages for the manipulation, the direct implementation of the reduced spin1/2 model may be convenient for the quantum annealing. On the other hand, the reduction to the subspace in terms of the spin1/2 model is useful only in the case where we focus on the Schrödinger dynamics. If we consider the dissipation as a realistic system, the transition between the subspaces emerges. The efficiency of the quantum annealing in this system is open for further study.
Conclusions
We studied the performance of the quantum annealing constructed by one of the degenerate twolevel systems, called the quantum Wajnflasz–Pick model. This model shows the higher success probability than the conventional spin1/2 model in the region where the longitudinal magnetic field is weak. The physics behind this is that the quantum annealing of this model can be reduced into that of the spin1/2 model, where the effective longitudinal magnetic field in the reduced Hamiltonian may open the energy gap between the ground state and the first excited state, which gives rise to the suppression of the Landau–Zener transition. The reduction of the quantum Wajnflasz–Pick model to the spin1/2 model is general at an arbitrary time as well as in an arbitrary number of degeneracies. We also demonstrated that the Λtype system also shows the higher success probability than the conventional spin1/2 model in the weak magnetic field regions. We hope that studying quantum annealing with variant spins, and utilizing the insight of their reduced model will promote further development of high performance quantum annealer.
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Acknowledgements
We thank R. van Bijnen, W. Lechner, Y. Matsuzaki, T. Ishikawa, T. Yamamoto, and T. Nikuni for fruitful discussions and comments. Two of the authors (S.W. and S.K.) were supported by Nanotech CUPAL, Japan Science and Technology Agency (JST). Y.S. and S.K. were supported by the New Energy and Industrial Technology Development Organization (NEDO), Japan.
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S.W., Y.S. and S.K. designed the study, S.W. and Y.S. contributed to theoretical calculations, S.W. performed numerical simulation and S.W., Y.S. and S.K. contributed to writing the manuscript.
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Watabe, S., Seki, Y. & Kawabata, S. Enhancing quantum annealing performance by a degenerate twolevel system. Sci Rep 10, 146 (2020). https://doi.org/10.1038/s41598019567584
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