Abstract
The imaginarytime evolution method is a wellknown approach used for obtaining the ground state in quantum manybody problems on a classical computer. A recently proposed quantum imaginarytime evolution method (QITE) faces problems of deep circuit depth and difficulty in the implementation on noisy intermediatescale quantum (NISQ) devices. In this study, a nonlocal approximation is developed to tackle this difficulty. We found that by removing the locality condition or local approximation (LA), which was imposed when the imaginarytime evolution operator is converted to a unitary operator, the quantum circuit depth is significantly reduced. We propose twostep approximation methods based on a nonlocality condition: extended LA (eLA) and nonlocal approximation (NLA). To confirm the validity of eLA and NLA, we apply them to the maxcut problem of an unweighted 3regular graph and a weighted fully connected graph; we comparatively evaluate the performances of LA, eLA, and NLA. The eLA and NLA methods require far fewer circuit depths than LA to maintain the same level of computational accuracy. Further, we developed a “compression” method of the quantum circuit for the imaginarytime steps to further reduce the circuit depth in the QITE method. The eLA, NLA, and compression methods introduced in this study allow us to reduce the circuit depth and the accumulation of error caused by the gate operation significantly and pave the way for implementing the QITE method on NISQ devices.
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Introduction
Quantum computers, initially proposed by Feynmann^{1}, were reported by Benioff^{2}, Deutsch^{3,4}, Grover^{5}, and Shor^{6} to have great potential that could overwhelmingly surpass that of classical computers. Furthermore, Google experimentally demonstrated quantum supremacy, which is the refutation of the extended Church–Turing thesis, proving the feasibility of quantum computers and raising the expectation for solving practical problems that a classical computer cannot address^{7}. Quantum computers can efficiently solve problems in the BQP (boundederror quantum polynomial) complexity class^{8} and verify an answer to a problem in the QMA (quantum Merlin–Arthur) complexity class^{9}. One of the actively researched problems for quantum computers is combinatorial optimization, which is an NPhard problem^{10}. Combinatorial optimization problems are closely related to our daily lives, and they include the traveling salesman problem^{11}, scheduling problem^{12}, and SAT (satisfiability problem) solver^{13}. Although combinatorial optimization problems are NPhard, some quantum algorithms were shown to be superior to the classical ones. Grover’s algorithm is already known to improve the computational cost with quadratic speedup when compared with classical computers^{14,15}. It has been reported that quantum annealing is faster than simulated annealing in several cases^{16,17,18,19,20}. Recently, quantum approximate optimization algorithm (QAOA) has been researched owing to its superiority over classical algorithms, which was demonstrated at the time of its proposal^{21}. However, with the development of classical algorithms^{22}, the quantum advantage of QAOA is now an open question.
Under these circumstances, it is challenging for researchers all over the world to employ existing or nearfuture quantum computers to achieve tasks that are very difficult or impossible using classical computers. Currently available quantum computers are noisy intermediatescale quantum (NISQ) devices^{23}. Further, conventional quantum algorithms, such as Grover’s algorithm, require many gate operations and they cannot be implemented on NISQ devices with no error correction due to short coherence time. Recently, classicalquantum hybrid algorithms called variational quantum eigensolver (VQE)^{24,25}, and QAOA^{21,26,27,28,29,30} have been proposed for NISQ devices. In these methods, ansatz states with parameters are implemented on quantum circuits, and the parameters included in the ansatz states are optimized on a classical computer. While VQE and QAOA can be realized with a limited number of quantum operations and have good noise tolerance, it is difficult to determine the ansatz states properly and converge highdimensional parameters^{31}.
For quantum manybody problems, an imaginarytime evolution method is a known computational method to identify the ground state. The imaginarytime evolution method selectively extracts the groundstate component by performing time evolution in the direction of imaginary time. Various combinatorial optimization problems are converted to a Hamiltonian format, and their corresponding Hamiltonian is derived^{32}. Thus, it is possible to solve the combinatorial optimization problem using the imaginarytime evolution method.
The implementation of the imaginarytime evolution method on a quantum computer involves a critical problem in that the imaginarytime evolution operator is a nonunitary operator, and therefore, it cannot implement the imaginarytime evolution method on a quantum computer in its current state. To overcome this challenge, two quantum imaginarytime evolution (QITE) methods—one that assumes an ansatz state and another that does not—were proposed in previous studies^{33,34}. The method that assumes the ansatz state traces the imaginarytime evolution of the parameters contained in the ansatz state^{33,35,36}. The other method introduces a unitary operation to reproduce the state on which the imaginarytime evolution operator has acted accurately without assuming an ansatz state^{34,37,38}.
We focus on the QITE method without the ansatz assumption and apply it to the optimization problems. The QITE method requires defining a domain size, which determines the accuracy of reproducing the imaginarytime evolution operator. The quantum circuit of an imaginarytime step scales exponentially with respect to this domain size. Besides, as an additional quantum circuit is added at each imaginarytime step, the quantum circuit becomes deeper in proportion to the lapse of imaginary time^{34}. These two features make it difficult to implement the QITE method on NISQ devices.
Therefore, we propose two approximations and one computational technique to overcome this difficulty. We succeeded in significantly reducing the quantum circuit depth of the QITE method, and we applied the developed algorithms to the maxcut problem, which is an NPhard problem. For the maxcut problem, we chose an unweighted 3regular graph and a weighted fully connected graph. The latter is a problem known as the classification problem in the context of unsupervised machine learning^{39,40}.
Results
Unitarization of imaginarytime evolution operators
Consider a scenario wherein a Hamiltonian \(\hat{H}\) is given for the optimization problem considered in this study. The Hamiltonian \(\hat{H}\) is expressed as the summation of some partial Hamiltonians \(\hat{h}[m]\) as \(\hat{H}=\mathop{\sum }\nolimits_{m = 1}^{{N}_{{\rm{ham}}}}\hat{h}[m]\), where N_{ham} is the number of the partial Hamiltonians. The maxcut problem, which is a computational target of this work, is represented by the Hamiltonian in the form of Ising spins and can be mapped to the Paulioperator representation for qubits in a straightforward manner. In the case of the Hamiltonian of quantum chemistry, each partial Hamiltonian can be mapped to the Paulioperator representation on qubits via the Bravyi–Kitaev representation^{41} or Jordan–Wigner representation^{42}.
For a given Hamiltonian, the ground state is obtained by using the imaginarytime evolution method. We apply the imaginarytime evolution operator defined by \({{\rm{e}}}^{\tau \hat{H}}\), where τ is the imaginary time to reach the initial (τ = 0) state of the system, \(\left{{\Psi }}(\tau =0)\right\rangle\); and \({{\rm{e}}}^{\tau \hat{H}}\left{{\Psi }}(\tau =0)\right\rangle\). The imaginarytime evolution operator is decomposed by a firstorder Suzuki–Trotter decomposition into ones with a small imaginarytime step Δτ (τ ≡ Δτ × N_{step}) of the individual partial Hamiltonians \(\hat{h}[m]\).
Because the operators of the imaginarytime evolution are nonunitary, they cannot be directly implemented as a gate operation on a quantum computer. In the QITE method, the unitary operator \({e}^{i{{\Delta }}\tau {\hat{A}}_{n}[m]}\) is defined such that it reproduces the state \({{\rm{e}}}^{{{\Delta }}\tau \hat{H}}\left{{{\Psi }}}_{n}\right\rangle\) for a given state \(\left{{{\Psi }}}_{n}\right\rangle \equiv \left{{\Psi }}(\tau =n{{\Delta }}\tau )\right\rangle\). We determine the Hermitian operator \({\hat{A}}_{n}[m]\) that minimizes the following residual norm.
Nonlocal condition for imaginarytime evolution operators
We express the Hermitian operator \({\hat{A}}_{n}[m]\) as a linear combination of the Dth order tensor products of Pauli operators \(\{{\hat{I}}_{l},{\hat{\sigma }}_{X,l},{\hat{\sigma }}_{Y,l},{\hat{\sigma }}_{Z,l}\}\) acting on the lth qubit as
where the prime on the first summation symbol indicates removing the double counting of the repeated tensors. We defined \({{\mathbb{L}}}_{m}\) as the set of \({N}_{{{\mathbb{L}}}_{m}}\) qubits, each of which is directly connected with those acted on by the partial Hamiltonian \(\hat{h}[m]\); however, \({{\mathbb{L}}}_{m}\) does not contain the qubits acted on by \(\hat{h}[m]\) [see Fig. 1a]. The parameter D, which is called the domain size, satisfies \(k\,\leqq \,D\,\leqq \,k+{N}_{{{\mathbb{L}}}_{m}}\), where we assumed the partial Hamiltonian \(\hat{h}[m]\) to be written by a tensor product of the kth order. {l_{1}(m),⋯,l_{k}(m)} is the set of qubits contained in the partial Hamiltonian \(\hat{h}[m]\). The summation in Eq. (3) is taken over all combinations of D − k qubits, {l_{k + 1}(m),⋯, l_{D}(m)}, and chosen from \({{\mathbb{L}}}_{m}\). D is an input parameter that represents the level of approximation; a larger D indicates that the imaginarytime evolution operator is expressed using higherorder tensor products and the residual norm in Eq. (2) shows a smaller value, which leads to a better approximation. Note that for D = N_{bit}, with N_{bit} being the number of qubits, the residual norm in Eq. (2) vanishes when minimized, yielding the exact imaginarytime evolution operator. In this context, the parameter D represents a truncation level. We consider a scenario where the domain size D incorporates all elements in \({{\mathbb{L}}}_{m}\), namely \(D=k+{N}_{{{\mathbb{L}}}_{m}}\), and then Eq. (3) reproduces the operator A_{n}[m] introduced in reference^{34}. This implies that Eq. (3) is a natural extension of the approximation introduced in reference^{34}. We call the method for determining the operator A_{n}[m] defined in reference^{34} local approximation (LA) for comparison with later approximation. Then, we refer to the method defined in Eq. (3) as extended localapproximation (eLA). The following notation is used to indicate the domain size D: e.g., LA with D = 6 is denoted by LAD6. Note that, for LA, it is a welldefined approximation only when the domain size \(D=k+{N}_{{{\mathbb{L}}}_{m}}\), and the value of D that can be taken is limited by the Hamiltonian. With an illdefined domain size D in LA, we found that the calculation accuracy decreased, which is called “Inexact QITE" in reference^{34}. Note that eLA can remove such constraints on the Hamiltonian and flexibly determine the parameter D by considering the linear combination for qubits. This flexibility is obvious in the maxcut problem of the fully connected graph. Solving the minimization problem in Eq. (2) to determine the coefficients \({a}_{\{i,l\}}^{(n)}[m]\) results in the linear equation S^{(n)}a^{(n)}[m] = b^{(n)}[m], which can be solved using a classical computer. Here, \({S}_{\{i,{l}_{i}\}\{j,{l}_{j}\}}^{(n)}=\left\langle {{{\Psi }}}_{n}\right{\hat{\sigma }}_{\{i,{l}_{i}\}}^{\dagger }{\hat{\sigma }}_{\{j,{l}_{j}\}}\left{{{\Psi }}}_{n}\right\rangle\) and \({b}_{\{i,{l}_{i}\}}^{(n)}[m]=\left\langle {{{\Psi }}}_{n}\right{\hat{\sigma }}_{\{i,{l}_{i}\}}^{\dagger }\hat{h}[m]\left{{{\Psi }}}_{n}\right\rangle\). Figure 1b shows a schematic of the quantum circuit representing one imaginarytime step of LA. In LA, the operator of the imaginarytime evolution is approximated by the tensor products of Pauli operators up to the Dth order; therefore, 4^{D} gate operations are required for each partial Hamiltonian. The total number of gate operations for one step of the imaginarytime evolution is N_{ham}4^{D}. Table 1 summarizes the size of the linear equation of the LA per step of the imaginarytime evolution and the number of gate operations per qubit, where N_{bit} is the total number of qubits.
Furthermore, this study proposes another approximation method for \({\hat{A}}_{n}\) in the following form:
The difference from Eq. (3) is that we remove the restriction on the set {l_{1}(m),⋯ l_{k}(m)} and extend the summation over qubits to incorporate all possible combinations of D qubits {l_{1}(m),⋯ l_{D}(m)}. We call this an NLA. As per this definition, we expand the Hermitian operator, \({\hat{A}}_{n}\), using tensor products of Pauli operators over all qubit combinations. Moreover, in LA and eLA, the tensor product space describing \({\hat{A}}_{n}[m]\) is different depending on m, which is the partial Hamiltonian. The NLA has a notable feature in that the tensor product space that describes \(\hat{A}[m]\) is the same for all m. Table 1 lists the size of the linear equations of the NLA per step of the imaginarytime evolution and the number of gate operations per qubit, where the NLA requires only 4^{D} unitary operators in \({}_{{N}_{{\rm{bit}}}}{C}_{D}\) combinations for the quantum circuit in the first step of the imaginarytime evolution. Figure 1c shows the schematic of the quantum circuit of the NLA for one step of the imaginarytime evolution (for D = 2).
Reduction effect of circuit depth
To clarify the accuracy and effectiveness of NLA, we applied it to the maxcut problem, which is an NPhard problem. The Hamiltonian of the maxcut problem in qubit representation is given in the following form containing secondorder tensor products^{32}.
As for the maxcut problem, we considered typical graphs such as 3regular and fully connected graphs. The 3regular graphs have three connected edges at every vertex, where E is the set of edges contained in the graph and d_{i,j} is the weight of the edges connecting the ith and jth vertices.
The circuit depths, when LA and NLA are applied to the maxcut problem, are shown in Fig. 1d for the 3regular graph and Fig. 1e for the fully connected graph because different graphs of the maxcut problem change the number of the partial Hamiltonian N_{ham}; the necessary circuit depths for each approximation change correspondingly. In Fig. 1d, e, the circuit depth calculated using Qiskit^{43} is plotted with points, and the plotted points are extrapolated. In the case of kregular graphs, the number of the partial Hamiltonians is given by N_{ham} = kN_{bit}/2. It increases linearly with the number of vertices N_{bit} so that the number of gate operations per qubit does not depend on the number of qubits, as listed in Table 1. Thus, we extrapolated using \(y={\rm{const.}}\). In NLA, regardless of the structure of the Hamiltonian, the number of gate operations per qubit is scaled by \({\mathcal{O}}({{N}_{{\rm{bit}}}}^{D1})\) with respect to the number of qubits N_{bit} because all combinations of \({}_{{N}_{{\rm{bit}}}}{C}_{D}\) are taken for gate operations including the Dth order tensor product. In Fig. 1d, the circuit depth of the NLA is extrapolated by the function fitted by f(x) = x^{D − 1}.
Note that in LA, D = 3, 4, and 5 are not welldefined in the 3regular graph. Thus, D = 6 is required, and 4^{6} = 4096 gate operations are necessary for the imaginarytime evolution of one partial Hamiltonian, which leads to a deeper circuit depth and difficulty in implementation on NISQ devices. In addition, the circuit depth required for LAD6, compared to NLAD2, NLAD3, etc., is considerably higher in the region with a small number of qubits. The circuit depth of the NLA becomes deeper than that of LA in the region where the number of qubits increases.
In Fig. 1e, LAD2 and eLAD3 are not shown for the fully connected graph (\({N}_{{\rm{ham}}}{ = }_{{N}_{{\rm{bit}}}}{C}_{2}\)) because the circuit depth of LAD2 is equal to that of the NLAD2, and that of eLAD3 is equal to that of NLAD3. In addition, because the domain size has to be D = N_{bit} in LA, which is the exact imaginarytime evolution in a fully connected graph, and the circuit depth increases exponentially with respect to the number of qubits. In NLA, it can be scaled down to the linear or quadratic function with respect to the number of qubits. This result indicates that the NLA and eLA are efficient in reducing the circuit depth, especially when the number of partial Hamiltonians increases; further, these algorithms are effective for NISQ devices.
Calculation accuracy
Simulations were performed after modifying the code provided in reference^{34}. As an initial state, we adopted a state in which all states were superimposed with equal a priori weights. We adopt a figure of merit to discuss the accuracy of the QITE method.
The first target of the maxcut problem is an unweighted 3regular graph with ten vertices, where E_{GS} is the energy of the ground state, and it is obtained from the exact diagonalization. The energy of the ground state is E_{GS} = −12. It is known that designing a classical algorithm that achieves r > 331/332 for an unweighted 3regular graph is an NPhard problem^{44}. Further, the approximation accuracy of the current classical algorithm is r ≈ 0.9326^{45}. Figure 2a shows the imaginarytime dependence of the energy. The imaginarytime step was set to Δτ = 0.01. In LAD2, as the imaginarytime τ increased, the energy decreased exponentially in the beginning and converged to around −9, which is higher than the exact solution by about 3. Another important point is that the energy does not monotonically decreases along the imaginarytime evolution. This behavior indicates that the conversion of the operator of the imaginarytime evolution to the unitary operators is less accurate in expanding it in the space of LAD2. Furthermore, the LAD6 calculation result shows E = −11.99, which is the energy almost equal to the exact solution. We found that an approximation accuracy in the eLAD3 is E = − 11.17 (r = 0.93) (the lowest value is E = −11.33 (r = 0.94)); in NLAD2, E = −11.42 (r = 0.95); and in NLAD3, E = −12.00 (r = 1.00). We found that eLAD3 had an approximation accuracy similar to that of the classical algorithm, and NLAD2 had already exceeded the approximation accuracy of the classical algorithm. NLAD3 shows better accuracy than NLAD2 and can reach a nearly exact solution. Note that eLA and NLA monotonically decrease the energy along the imaginarytime evolution with sufficiently good accuracy compared to LAD2. This behavior was confirmed not only for NLAD2 but also for NLAD3 and others. As can be seen from Fig. 1d, in LAD6, the circuit depth of one imaginarytime step is 369757, while the circuit depth in the NLAD2 is 789. This implies the circuit depth of NLA can be significantly shallower than that of LA.
While NLAD3 has extremely high accuracy, its circuit depth increases with a quadratic function with respect to the number of qubits. Then, we developed NLAD2.5 to keep the scaling of the circuit depth as linear as NLAD2 while maintaining the accuracy of NLAD3, which is an approximation to expand the space of \({\hat{A}}_{n}\) to the space involving the secondorder tensor products incorporated by NLAD2 and the thirdorder tensor products by eLAD3. Thus, by incorporating some portions of bases of eLAD3 into those of NLAD2, computational scaling can be made linear with respect to the number of qubits, which makes it applicable even in regions with a large number of qubits. Figure 1d shows that the circuit depth is almost the same as that of NLAD2 for 50 qubits or more, which means that the circuit depth can be significantly reduced compared to that of NLAD3. In addition, the calculation result of NLAD2.5 is E = −11.95 and r = 0.99, which gives a good approximation accuracy with a small circuit depth.
Here, for further consideration, we decomposed the state \(\left{{\Psi }}(\tau )\right\rangle ={{\rm{e}}}^{\tau \hat{H}}\left{{\Psi }}(\tau =0)\right\rangle\) into the eigenstate components of the Hamiltonian, and the calculated n(E) ≡ ∑_{i}∣〈i∣Ψ(τ)〉∣^{2}δ(E − E_{i}) as a function of energy E at each imaginarytime step τ is plotted in Fig. 2b where \(\lefti\right\rangle\) is the eigenstate of \(\hat{H}\) and E_{i} is the eigen energy of \(\lefti\right\rangle\). Here, we note that the ground state of the eigenstate component n(E_{GS}) is equal to the socalled fidelity defined as F = ∣〈Ψ(τ)∣Ψ_{GS}〉∣^{2}. The ground state can be observed with probabilities of n(E_{GS}) = 0.60 for eLAD3 (at maximum, n(E_{GS}, τ = 2.87) = 0.65), n(E_{GS}) = 0.69 for NLAD2, n(E_{GS}) = 0.97 for NLAD2.5, and n(E_{GS}) = 1.00 for NLAD3. The imaginarytime dependence of the probability of the first excited state is also plotted. For the first excited state, it is observed that the probability is amplified up to τ = 1, and it starts to decrease, which increases the groundstate probability.
Next, we deal with another computational model called a weighted fully connected graph (classification problem). The coupling constants d_{i,j} were given by random numbers. The groundstate energy is E_{GS} = −57.993. In addition, the imaginarytime step is set to Δτ = 0.01. In the classification problem, as shown in Fig. 2e, each graph vertex is colored red or blue. In LAD2, as in the 3regular graph, we observed that the energy does not necessarily decrease monotonically. The energy of eLAD3 is lower than that of NLAD2; E = −57.504 (r = 0.99) for eLAD3, E = −57.026 (r = 0.98) for NLAD2, and E = −57.985 (r = 0.99) for NLAD3 (Fig. 2c). From the viewpoint of the component analyses of the states, the ground state and the first excited state are pseudodegenerate (Fig. 2e), and therefore, the probability of the first excited state remains at the same level as the ground state even around τ = 2 when the energy converges sufficiently (Fig. 2d). In NLA, the first excited state gradually decays along with the imaginarytime evolution; however, a sufficiently long imaginarytime evolution is necessary. In particular, NLAD2 behaves similarly to NLAD3, and NLAD2 is sufficiently accurate to obtain the ground state in actual applications.
We now consider why the accuracy of eLA and NLA is better than that of LA with a relatively small domain size D. From the actual application results of eLAD3, we found that the \({b}_{\{i,{l}_{i}\}}^{(n)}[m]=\left\langle {{{\Psi }}}_{n}\right{\hat{\sigma }}_{\{i,{l}_{i}\}}^{\dagger }\hat{h}[m]\left{{{\Psi }}}_{n}\right\rangle \approx 0\) when the Pauli operator \({\hat{\sigma }}_{\{i,{l}_{i}\}}^{\dagger }\) and \(\hat{h}[m]\) do not intersect each other. With a rough approximation for such cases, \({b}_{\{i,{l}_{i}\}}^{(n)}[m]=\left\langle {{{\Psi }}}_{n}\right{\hat{\sigma }}_{\{i,{l}_{i}\}}^{\dagger }\hat{h}[m]\left{{{\Psi }}}_{n}\right\rangle =0\), a sparsity in the coefficients of A_{n} can be deduced, which eLA highlight and leverage. This fact means that the terms that would require a large domain size D in LA can be efficiently captured with a smaller domain size D in eLA, leading to its high accuracy. Furthermore, by considering that the definition of NLA is expanded from that of eLA, NLA can improve further the accuracy of eLA.
Compression of imaginarytime steps
The approximation accuracy of the NLA and its circuit depth have been discussed. The “compression of imaginarytime steps” is introduced in this section for further reduction of the number of gate operations in NLA. Figure 3a shows a schematic of the compression technique. When the imaginarytime step Δτ is sufficiently small, the timeevolution operators can be compressed into a single exponential form via the reverse Suzuki–Trotter decomposition
where N_{comp} is the number of compressed steps. It is necessary to choose an appropriate N_{comp} within the range that guarantees sufficient accuracy for the Suzuki–Trotter decomposition because its accuracy decreases if the N_{comp} becomes large. To determine the specific N_{comp} in this work, we increased the N_{comp} parameter by one at every timeevolution step until the total energy increases. In actual QITE calculations, N_{comp} is not necessarily a constant throughout the calculation. This method enables the reduction of quantum circuits to as small as 1/N_{comp}. We discussed the error of the second order for Δτ in the compression method in Supplementary Note 1.
The graph used for the calculation is the same as that in Fig. 2a, b, which is a 3regular graph with ten vertices. Figure 3 shows the results of the compression technique for the QITE. In Fig. 3b, the time the compression ended is plotted as a blue circle. In the case of Fig. 3b, the quantum circuit depth is significantly reduced by the compression technique to four compressed imaginarytime steps, and the energy at τ = 10 is E = −11.43 (r = 0.95) without and E = −11.59 (r = 0.97) with the compression technique. We found that sufficient accuracy was achieved regardless of the compression, which indicates that compression does not affect the results. It may be assumed that the compressed technique has a lower energy than that of the uncompressed calculation; a detailed investigation revealed that this was attributed to the accidental acceleration of the convergence by compression. Figure 3c plots the component analyses of the wavefunctions during the imaginarytime evolution with and without the compression method. Finally, the probability of obtaining a ground state is \(n({E}_{\min })=0.76\) with and \(n({E}_{\min })=0.73\) without the compression technique.
The “compression of imaginarytime steps” is effective in reducing the circuit depth, and simultaneously, it reduces the noise associated with the gate operations. We discuss the results of the simulation with noise. The actual qubits are currently connected only with neighboring sites; however, in this study, we simulated a fully connected model. For implementation on an actual quantum computer, in which only adjacent sites are connected, a SWAP gate can be used with an overhead of \({\mathcal{O}}(\sqrt{{N}_{{\rm{bit}}}})\)^{46}. For example, QAOA uses a SWAP network^{47,48} to implement a \({\mathcal{O}}({N}_{{\rm{bit}}})\) overhead^{49}. We describe the quantum circuit of NLAD2 of the QITE method for an adjacentcoupling circuit using the SWAP network in Supplementary Note 2. Figure 3d shows the simulation results of the maxcut problem for an unweighted graph with four vertices. The coefficients \({a}_{\{i,l\}}^{(n)}\) in Eq. (4) for the noisy calculation are the same as those for the nonnoisy calculation. The noiseless condition without compression results in E = −3.94, which is close to the exact solution E = −4.00 around τ = 5. However, the circuit depth is 922 (Δτ = 0.5), and the simulation result with noise is E = −3.13, which is far from the exact solution. This gap was attributed to the accumulation of errors caused by an increase in circuit depth. The result with compression is E = −3.85 in the case without noise; however, the circuit depth is 163, and the effect of noise is expected to be less sensitive. In fact, the simulation result with noise is E = −3.63, which shows that the noise can be reduced with compression. Thus, it has been shown that the “compression” method of quantum circuits has the advantage of reducing the accumulation of errors.
Discussion
In this study, we proposed twostep approximation methods based on nonlocality: eLA and NLA. We applied them to the Maxcut problem of an unweighted 3regular graph and a weighted fully connected graph, and comparatively validated the performances of LA, eLA, and NLA. We found that NLA requires significantly less circuit depth than LA while maintaining the same level of computational accuracy. For example, when we request the classical approximation limit in the QITE calculations, the circuit depth required for a single imaginarytime step can be significantly reduced from 369,757 for LA to 789 for NLA when applying it to a 3regular graph, and from about 314,000 for LA to 789 for NLA when applying it to a fully connected graph. Further, we developed a “compression” technique of the imaginarytime evolution steps to further reduce the circuit depth in the QITE method. With this compression method, we succeeded in further reducing the circuit depth. We showed that the reduction in circuit depth using this compression method has a secondary effect of reducing the accumulation of error caused by the gate operation. Thus, it is an effective method for realization on NISQ devices. The eLA, NLA, and compression methods introduced in this study enable us to significantly reduce the circuit depth and the accumulation of error caused by the gate operation and have paved the way for the realization of the QITE method on NISQ devices.
Methods
Noisy simulation of QITE method
Our numerical simulations were performed after implementing the eLA, NLA, and compression method on the code provided in reference^{34}. The simulation of the quantum noise’s presence is performed with the implementation of the QITE method at the level of the NLAD2 on the IBM Qiskit quantum simulator. Although almost all actual quantum devices’ qubit connectivity is restricted, we simulated the QITE method based on the fully connected coupling. For implementation on a device connected only with neighboring qubits, we provide a circuit of the QITE method using the SWAP network in Supplementary Note 2. The error model of the gate was constructed from the thermal relaxation time (T_{1}, T_{2}) = (100 μs, 80 μs), and the gate time (T_{g1}, T_{g2}) = (0.02 ns, 0.1 ns). The noise simulation was performed by introducing the readout errors (p_{00}, p_{01}, p_{10}, p_{11}) = (0.995, 0.005, 0.02, 0.98). These parameters were assumed to be close to the actual values of IBMQ^{50}.
Note added to proof
During our review of this paper, we noticed an independent workrelated “compression method” being done in parallel^{51}.
Data availability
The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.
Code availability
The code depeloped for the current study is available from the corresponding author on reasonable request.
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Acknowledgements
This research was supported by MEXT as an Exploratory Challenge on PostK computer (Frontiers of Basic Science: Challenging the Limits) and by GrantsinAid for Scientific Research (A) (Grant Number 18H03770) from JSPS (Japan Society for the Promotion of Science).
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H.N. and Y.M. conceived the general idea. H.N. modified the code provided in prior work. H.N. and T.K. developed the code for noisy simulation. Numerical simulations were performed by H.N. All authors contributed equally to the manuscript preparation and presentation of results.
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Nishi, H., Kosugi, T. & Matsushita, Yi. Implementation of quantum imaginarytime evolution method on NISQ devices by introducing nonlocal approximation. npj Quantum Inf 7, 85 (2021). https://doi.org/10.1038/s4153402100409y
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DOI: https://doi.org/10.1038/s4153402100409y
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