Quantum computing of fluid dynamics using the hydrodynamic Schr\"odinger equation

Simulating fluid dynamics on a quantum computer is intrinsically difficult due to the nonlinear and non-Hamiltonian nature of the Navier-Stokes equation (NSE). We propose a framework for quantum computing of fluid dynamics based on the hydrodynamic Schr\"odinger equation (HSE), which can be promising in simulating three-dimensional turbulent flows in various engineering applications. The HSE is derived by generalizing the Madelung transform to compressible/incompressible flows with finite vorticity and dissipation. Since the HSE is expressed as a unitary operator on a two-component wave function, it is more suitable than the NSE for quantum computing. The flow governed by the HSE can resemble a turbulent flow consisting of tangled vortex tubes with the five-thirds scaling of energy spectrum. We develop a prediction-correction quantum algorithm to solve the HSE. This algorithm is implemented for simple flows on the quantum simulator Qiskit with exponential speedup.

Quantum computing is not only for simulating quantum systems [16,[18][19][20][21][22][23][24], but also possible to simulate classical systems [25][26][27][28].Fluid dynamics, described by the Navier-Stokes equation (NSE), is notoriously difficult to be fully simulated on a classical computer at a large Reynolds number (Re), because the high-Re turbulent flow involves length and time scales over a wide range of orders of magnitude.The computational cost with O(Re 3 ) operations for the direct numerical simulation (DNS) of turbulence [29] is unaffordable in engineering applications [30,31].Therefore, the combination of computational fluid dynamics (CFD) and quantum computing can be promising for the next-generation simulation method [32].
To date, quantum computing has been demonstrated to be effective to handle some linear problems [13,[33][34][35][36], but remains intrinsically difficult in solving nonlinear differential equations [37][38][39] due to the linear nature of quantum mechanics.Thus, it appears to be challenging to efficiently solve the highly nonlinear NSE on a quantum computer.The current studies on the quantum computation of fluid dynamics can be divided into three categories.
First, quantum computing was performed for a specific simplified problem, e.g., 1D steady inviscid Laval nozzle [40], 1D steady channel flow [41], 1D Burgers equation [42], and 2D thermal convection [43], to avoid dealing with the intractable full 3D NSE.These works demonstrated the feasibility of quantum computing in CFD, but cannot be simply extended to complex 3D flows.
Second, quantum algorithms were applied to solving linear systems [13,44], e.g., the quantum linear solver [45][46][47] and Poisson solver [48], to replace a part of a classical CFD algorithm.These hybrid quantum-classical algorithms involve frequent data exchanges between classical and quantum hardware.Since the conversion can take even much longer time than the computational time for solving the equations [49], only steady problems were considered in these works to avoid data exchange.Third, fluid dynamics was described by the approaches that are more suitable than the NSE for quantum computing, e.g., the Madelung transform [50], the generalized Koopmanvon Neumann (KvN) representation [51], the lattice Boltzmann method [52][53][54], and the tensor network-based method inspired from quantum many-body physics [55,56].On the other hand, each of these methods has certain limitations, e.g., the Madelung transform can only describe inviscid potential flows, and the KvN representation encounters the non-closure problem of the probability density function.
The present study adopts the third approach to describe fluid dynamics using the hydrodynamic Schrödinger equation (HSE).The HSE is derived by generalizing the Madelung transform to compressible/incompressible flows with finite vorticity and dissipation.It can be expressed as a unitary operator on a two-component wave function, so it is more natural than the NSE for quantum computing.We develop a quantum algorithm for solving the HSE with a notable speedup, and implement the algorithm for simple flows on IBM's quantum simulator [57].
The outline of the present paper is as follows.Section II introduces the HSE.Section III compares the flows governed by the HSE and NSE.Section IV develops and validates the quantum algorithm.Some conclusions are drawn in Section V.

A. Madelung transform
In quantum mechanics, the probabilistic current for a wave function ψ(x, t) is defined as with the momentum operator p and particle mass m, where f denotes the complex conjugate of f , and f denotes an operator.In the coordinate representation, we have p = −i ∇ with the imaginary unit i and Planck constant .Considering a particle moving in a potential field V ∈ R, its motion satisfies the Schrödinger equation [59] i Without loss of generality, we set m = 1.From Eqs. ( 1) and ( 2), the conservation of the probability density ρ ≡ ψψ reads The form of Eq. ( 3) is identical to the continuity equation in fluid mechanics, with a "velocity" u ≡ J /ρ.The Madelung transform [60] shows an analogy between quantum mechanics and fluid mechanics.Table I explains the physical meanings of the same symbol in different contexts.
Using the Madelung transform, the momentum equation of a fluid flow is obtained from Eq. ( 2), where the fluid velocity is with ψ = √ ρe iφ/ .Equation ( 4) corresponds to the Euler equation for a potential flow with vanishing vorticity.It has very limited applications for general viscous flows with finite vorticity [61][62][63].

B. Schrödinger flow
To introduce the finite vorticity into the hydrodynamic representation of the Schrödinger equation, we use a two-component wave function [64][65][66] represented by a quaternion as with the basis vectors {i, j, k} of the imaginary part of the quaternion and real-valued functions a, b, c, and d.This quaternion facilitates deriving governing equations of the fluid flow below, and it is essentially the same as the two-component spinor [67][68][69][70].
The probabilistic current in Eq. ( 1) is generalized to Similarly, the fluid mass density and velocity become ρ ≡ ψψ and respectively.Then, we obtain after some algebra.With identities iψ = −ψi and i∇ 2 ψ = −∇ 2 ψi, we derive a sufficient condition with a real-valued potential V for the continuity equation Note that Eq. ( 10) is the Schrödinger-Pauli equation (SPE) in a quaternion form, which describes the motion of a spin-1/2 particle without an external electromagnetic field in the non-relativistic limit [71][72][73].
After some algebra (detailed in Appendix A), we derive the momentum equation for ψ, along with an equation of state Here, is a nonlinear potential, where V F corresponds to conservative body forces and denotes a spin vector.The last term in the right-hand side (RHS) of Eq. ( 12) does not appear in the momentum equation of practical fluid flows.It can be considered as an external body force involving a dissipation effect, and it degenerates to the "Landau-Lifshitz force" (LLF) [74,75] for constant ρ.
In sum, we convert the compressible flow with finite vorticity in Eqs.(11), (12), and (13), into a hydrodynamic Schrödinger equation (HSE) The HSE can be considered as a SPE with a specific potential in Eq. ( 14).The fluid flow governed by the HSE is then called the Schrödinger flow (SF).Comparing with the Gross-Pitaevskii equation [76,77] i with an external potential V (x) and a coupling constant g, which is a well-known model equation describing the dynamics of the Bose-Einstein condensate, the HSE (16) has a more complex nonlinear potential and incorporates the spin effect of a particle.
Since the real-valued Hamiltonian of the SF is Hermitian, the evolutionary operator is unitary, with a time increment ∆t.We are able to use it to obtain ψ(x, t) at a given time from an initial wave function in quantum computing.The procedure of the simulation and measurement for the SF is sketched in Fig. 1.This simulation of the SF only involves the wave function and its derivatives without fluid quantities, so it is equivalent to a Hamiltonian simulation for the motion of a particle.

C. Incompressible Schrödinger flow
We consider a special SF for a constant-density incompressible flow with The wave function on the sphere S 3 with radius √ ρ 0 and the spin vector in Eq. ( 15) on S 2 with radius ρ 0 for this flow are linked by the Hopf fibration [78].Taking the material Setting (Dψ/Dt)ψ = f ψ as an pure quaternion in Eq. ( 21) yields After some algebra, the momentum equation in an incompressible flow is obtained, where f s ≡ ψf ψ ψ is also a pure quaternion.Note that the flow governed by Eqs.(20) and (23) has some physically interesting properties, such as the helicity conservation [79][80][81] and the Lagrangian-like evolution of vortex surfaces [82,83]. Letting and substituting it into Eqs.( 22) and ( 23), we obtain the Euler equation and the corresponding nonlinear Schrödinger equation However, the non-Hermitian Hamiltonian in Eq. ( 26) can inhibit an efficient quantum computation.
To make the Hamiltonian Hermitian, Eq. ( 24) is modified to Then we have the modified momentum equation and the incompressible hydrodynamic Schrödinger equation (IHSE) The physical meaning of the last term in the RHS of Eq. ( 29), i.e., the LLF [74,75], is further discussed in Appendix B. The flow governed by Eq. ( 29) with ρ 0 = 1 has been called the incompressible Schrödinger flow (ISF) [74,75] and studied by numerical simulations [84].
The Hamiltonian of the ISF is Hermitian, so the ISF is suitable for quantum computing.Moreover, the form of the IHSE can be obtained by setting ρ = ρ 0 in the HSE.
Similar to the incompressible Navier-Stokes equation (INSE), the pressure p in the IHSE ( 30) is coupled with Eq. ( 29) to ensure the divergence-free velocity.Thus, the mathematical natures of the HSE ( 16) with non-constant ρ and IHSE (30) are very different, which is similar to the difference between compressible and incompressible NSEs, so they have to be solved by different methods.In the present study, we focus on the physical property and quantum algorithm for the ISF.

III. TURBULENT ISFS
We investigate two ISFs, the Taylor-Green (TG) flow and decaying homogeneous isotropic turbulence (HIT), to illustrate the similarities and differences between the ISF and the real viscous flow.Additionally, the evolution of vortex knots was investigated in the ISF [84].
Since quantum hardware and algorithms for simulating such complex flows are still under development, the DNS of the ISF was carried out to solve Eq. ( 30) with ρ 0 = 1 on a classical computer.The standard pseudo-spectral method [83,85,86] was adopted in a periodic cube of side L = 2π on 512 3 uniform grid points.The numerical implementation was described in detail in Ref. [84].
The vortex surface [82] in the fully developed turbulent ISF at t = 5 is visualized in Fig. 5 using the isosurface of s 1 s 2 s 3 = 0.18.We observe a network of entangled vortex tubes and sheets, which can be mapped to closed curves, the intersection of s 2 1 + s 2 2 + s 2 3 = 1 and s 1 s 2 s 3 = 0.18, on the unit sphere S 2 (or the Bloch sphere) via the vorticity Clebsch mapping [74,75,84].The geometry of vortex surfaces in the ISF is in between the vortex filaments in quantum turbulence [87][88][89] and the tangle of spiral vortex tubes and sheets in classical turbulence [85,90,91].Therefore, the turbulent ISF manifests the features of both quantum and classical turbulent flows.energy spectrum in the ISF exhibits the −5/3 scaling law in the inertial range as in classical turbulence [29], and it decays with time due to energy dissipation.In addition, the total kinetic energy decays with time in the turbulent ISF as in the classical HIT (not shown).

IV. QUANTUM ALGORITHM FOR THE ISF A. Prediction-correction approach
We develop a quantum algorithm for simulating the ISF.As sketched in Fig. 1, the algorithm can be executed on a quantum processor with measurements only at the end of the simulation, so it does not involve frequent information exchanges between classical and quantum hardware as in existing hybrid quantum-classical methods [43,45,46,92].
This quantum algorithm can have significant advantages over classical and hybrid ones in terms of computational speedup, memory saving, and reduction of noises introduced by measurements.
We apply a prediction-correction approach to bypass handling the nonlinear potential in Eq. ( 14) in the IHSE.As in classical algorithms [93][94][95] for simulating incompressible flows, the pressure is not solved using the pressure-Poisson equation because the RHS in Eq. ( 33) is difficult to encode on a quantum computer.First, we perform a prediction to obtain a temporary wave function using Eq. ( 30) with ignoring p− 2 |∇s| 2 /8.

State preparation
Step 1

Prediction
Step 2

Phase calculation
Step Second, we apply a divergence-free projection of the temporary wave function.The flowchart of this quantum algorithm the ISF is illustrated in Fig. 7. Next, we elaborate each step in the algorithm in Fig. 7 using a 1D problem, and it is straightforward to extend the algorithm to 3D problems.
In this way, the quaternionic wave function ψ(x, t) is approximated by the state vector with 2 )∆x, j = n−1 i=0 j i 2 i , and Hence, the wave function is reconstructed by C. Quantum algorithm for solving the IHSE Step 1: Prediction In step 1, we treat the particle governed by the IHSE as a free Pauli particle.Namely, Eq. (30 The motion of such a particle is described by a temporary solution [19,[96][97][98] |ψ with the 2 1 × 2 1 identity matrix I 2 , the quantum Fourier transform (QFT) [99][100][101] QFT : because the kinetic energy H 0 = | p| 2 /2 and V F are not commute, i.e., [H 0 , V F ] = 0.The time stepping ∆t should be small enough to ensure accuracy.
An efficient quantum implementation of P(f ) in Eq. ( 37) is important.The variable in V (x) is discretized with constants c 0 = ∆x and c 1 = (−d + ∆x/2)/(n∆x), and thus is computable.
Here, Z j denotes a phase-shift gate at the j-th qubit and φ = ∆t/ is a small phase shift on a time step.The quantum circuit for calculating a Hamiltonian with the form of H = exp(i∆t Z 2 ⊗ Z 1 ⊗ Z 0 ) with a given ∆t is shown in Fig. 9, using an ancilla qubit [2].
Taking, e.g., n = 3 qubits and ignoring the global phase e iφ/8 , Eq. ( 43) becomes which can be realized by a quantum circuit in Fig. 10.Thus, only O(n 2 ) quantum gates are sufficient to calculate the momentum operator in Eq. ( 43).
11. Conceptual quantum circuit for calculating U N in Eq. ( 46).The dashed boxes mark that This normalization step appears to be difficult to implement on a quantum computer.
As illustrated in Fig. 11, we can only conceptually decompose the operator with scale transformations Here, each Q i may be non-unitary with Q † i Q i = I and thus it is not realizable using a quantum gate, whereas their product Eq.( 46) is unitary.Therefore, an effective quantum algorithm for calculating U N with the complexity O(poly(n)) remains an open problem.
Step 3: Phase calculation After step 2, the divergence of the velocity u * * = (∂ x ψ * * iψ * * − ψ * * i∂ x ψ * * )/2 can be non-zero.A divergence-free projection of u * * , as a gauge transformation u → u − ∇q, is applied, where the phase q is solved from a Poisson equation 12. Quantum circuit for calculating P(f (x)) in Eq. ( 50) with n = 4 qubits.Hollow and solid circles denote control qubits with values 0 and 1, respectively.The operation F j on the target qubit |j 0 is active only when the control qubits are in the state |j = |j 3 j 2 j 1 with j = 2 i=0 j i+1 2 i .
This projection corresponds to the gauge transformation ψ → e −iq/ ψ for the wave function [103].
Step 4: Gauge transformation In the final step, we take a gauge transformation from the temporary state |ψ * * at t to the state |ψ at t + ∆t.The diagonal unitary transformation of a function f (x) can be implemented by O(2 n ) generalized controlled-phase shift gates.
They apply the single qubit gate F j to a target qubit |j 0 only when the other n−1 controlled qubits are in the state |j = |j n−1 j n−2 • • • j 1 [96].An example with n = 4 qubits is shown in Fig. 12.
However, such an implementation is inefficient because its complexity scales exponentially with n.A more efficient quantum circuit can be designed for a specific form of f (x), e.g., only O(n 2 ) basic quantum gates are used for f (k) = k 2 ∆t in Eq. ( 37).In general, if f (x) has a specific form, e.g., f (x) for the harmonic oscillator, square well, and quantum tunneling, Eq. ( 50) can be calculated with the complexity O(poly(n)).

Algorithm complexity
We estimate the total complexity of the quantum algorithm for solving the IHSE in Eq. (30).The overall quantum circuit with the prediction, normalization, phase calculation, and gauge transformation is illustrated in Fig. 13.The prediction is a standard simulation of a potential-free Pauli particle, using only O(n 2 ) basic quantum gates.The normalization is unconventional in quantum computing.The upper and lower bounds of operations are O(2 n ) and O(poly(n)), respectively.The gauge transformation, currently, can only be realized through the generic diagonal unitary transformation in Fig. 12, using O(2 n ) basic quantum gates.
The complexities of each step and the entire algorithm are summarized in Table II.The present quantum algorithm can achieve exponential speedup in steps 1 and 3, and possible speedup overall.The bottlenecks for the computational efficiency in steps 2 and 4 need to be tackled in the future work.
Besides the spatial complexity, the temporal and spatial steps are related by the Courant-Friedrichs-Lewy (CFL) condition in both the quantum and classical implementations.This implies that the number of time iterations is N t = O(N ), where N = 2 n is the total number of grid points.Thus, the numerical error decreases polynomially with the number of grid points.Assuming that the unitary operators are smooth enough and the norm of the exponential operators are bounded by one [108,109], the error after N t iterations scales as for the second-order Trotter decomposition.Given an error tolerance, the upper and lower bounds of the number of gates scale as N gate ∼ −1 poly(log 2 −1 ) and , respectively, for a quantum algorithm.In the classical algorithm, the number of operations scales as N op ∼ −1 ( −1 log 2 −1 +poly( −1 )).Therefore, even if the CFL condition constrains the time stepping, the quantum algorithm has a possible exponential speedup Step 1 Step  Step 1 Step 2 Step 3 Step 4 Total We provide a simple 1D example to go through the entire algorithm, which is implemented on a quantum computer with exponential speedup.The initial wave function is and the corresponding spin vector is s(x, 0) = (0, 0, 1).The solution of this steady ISF satisfies a Helmholtz equation This simplified IHSE with V F = 0 with the initial condition in Eq. ( 53) has a steady solution u = 1, where the nonlinear potential in Eq. ( 30) is simplified to p = −1/2.Thus, we only need to perform steps 1 and 4 as The complexity for calculating a time step of Eq. ( 55) is O(n 2 ).Compared to O(poly(2 n n)) for the classical algorithm, the exponential quantum speedup is achieved.
We validate the algorithm in Eq. ( 55) using IBM's Qiskit [57].The Qiskit is an opensource software development kit for quantum computers at the level of pulses, circuits, and application modules.We used a simulator with the quantum assembly language (QASM) on a classical computer, which mimics a quantum computer by adding small noises to the result [110].The QASM also operates by running the quantum circuit multiple times and storing the number of times when an outcome occurs, similar to the procedure on a practical quantum computer.
The reconstruction of the probability distribution |ψ(x, t)| 2 with a small statistical error needs to repeat the quantum simulation a large number of times.After an outcome x j is obtained M j times in M runs, |ψ(x j , t)| 2 ≈ N 2 M j /M is estimated, with a normalization factor N in Eq. (35).Moreover, it is also possible to reconstruct the entire ψ(x j , t) using a Ramseytype quantum interferometry method [96,111], the quantum-state tomography [112][113][114], or the direct weak tomography [115].
In Fig. 14(a), the result of the simple example described in Eq. ( 55) from the Qiskit simulation with M = 10 6 runs agrees with the theoretical distribution Re [ψ 0 (x j , t)] 2 = cos 2 (x j / )/2.Figures 14(b, c  dissipation, is governed by the HSE in Eq. ( 16) of a two-component wave function or by the continuity and momentum equations in Eqs.(11) and (12).Since the Hamiltonian of the SF is Hermitian, we are able to obtain ψ(x, t) from an initial wave function in the quantum computing of the HSE (see Fig. 1).
In particular, we develop a prediction-correction quantum algorithm for the ISF, a constant-density incompressible SF governed by the IHSE (30).This algorithm can be executed on a quantum processor with measurements only at the end of the simulation (see Fig. 7).Thus, it does not involve frequent information exchanges as in existing hybrid quantum-classical methods, which brings a significant advantage in the computational speedup over classical methods and in the reduction of noises introduced by measurements.
We estimate the complexity of the quantum algorithm for solving the IHSE.The overall quantum circuit contains four steps of the prediction, normalization, phase calculation, and gauge transformation in the algorithm (see Fig. 13).The breakdown of the algorithm complexities is summarized in Table II With the development of hardware and algorithms for quantum computing, the HSE framework, involving the quantum unitary evolution and characterizing 3D turbulent statistics and structures, can be promising in CFD applications.In the future work, the bottlenecks for the efficient quantum algorithm will be tackled in the steps of normalization and gauge transformation.Moreover, the difference between the SF and real flows can be reduced by introducing further modifications and models in the HSE.The hybrid quantum-classical simulation is governed by Eq. ( 30) with ρ 0 = 1.In each time step, we first implement step 1 in the algorithm in Section IV C using n = 10 qubits, where n depends on the computational resource.We set five qubits in each of the x-and y-directions, equivalent to using 2 5 ×2 5 = 32 2 grid points.Then, we measure the state vector |ψ * (t) in Eq. ( 37) to obtain the corresponding a * , b * , c * , and d * .Note that the algorithm for the 2D or higher-dimensional problem is essentially the same as that for the 1D problem in Section IV.Finally, we implement steps 2-4 in Section IV C using 32 2 grid points using the methods for the classical computer, and then prepare the state vector corresponding to the final output wave function.The above process is iterated for time marching.Since step 1 dominates the computational complexity in the classical simulation (see Table II), the hybrid simulation itself can be a temporal method for simulating 3D ISFs. between the two simulations is smaller than 6% at t < 1 for three independent runs in Fig. 17  [8] M. Zhang, L. Feng, M. Li, Y. Chen, L. Zhang, D. He, G. Guo, G. Guo, X. Ren, and D. Dai,

Figure 3
Figure3shows similar large-scale vortical structures with = 1 and 0.1, whereas much more small-scale tube-like structures emerge for smaller = 0.1.In general, the length scale of vortices is proportional to via the vorticity Clebsch mapping[74,75,84], and the flow stability depends on the value of .In Fig.3, the flow with smooth large-scale structures does not have a transition for = 1, whereas the flow breaks down into turbulence with numerous chaotic vortex tubes for = 0.1.The energy spectrum E k (k) of a turbulent ISF in Fig.4exhibits a −5/3 scaling law in the inertial range as in classical turbulence[29].As decreases, the inertial range broadens with a more pronounced −5/3 scaling.

5 FIG. 5 .
FIG. 5. Visualization of the tangled vortex tubes for the decaying HIT in the ISF.(a) Isosurface of s 1 s 2 s 3 = 0.18 color-coded by |ω| at t = 5.(b) Close-up view of the region marked by the blue box in (a).Some vortex lines (blue) are integrated and plotted on the isosurface in (b).

Figure 6
Figure 6 shows the evolution of E k (k) for the decaying HIT in the ISF with = 0.1 and the INSF with ν = 0.0005 (or Re = 2000 for unity length and velocity scales).The

3 FIG. 6 .
FIG. 6. Evolution of the energy spectra for the decaying HIT in the ISF with = 0.1 and the INSF with Re = 2000.
FIG. 7. Flowchart of the quantum algorithm for simulating the ISF.The dashed boxes highlight the bottlenecks of quantum speedup in the present algorithm.

B.
Quantum encoding of the IHSEIn an n+1-qubit quantum register, the state of a "Pauli particle"[71][72][73], whose motion is governed by the IHSE(30) with ρ = 1, can be encoded as follows.We use n qubits with state vectors |j 0 , |j 1 , • • • , |j n−1 to encode a particle location.The last qubit |j n = |s stores the spin state of the particle.The state of each qubit j 0 , j 1 , • • • , j n−1 , j n takes the value 0 or 1.The domain −d x d for the particle location is discretized into 2 n segments with the spacing ∆x = 2d/2 n .These segments can be represented by the computational basis ) show that the reconstructed mass density and velocity have slight deviations from the theoretical values ρ = 1 and u = 1 due to the statistical errors and the noises introduced by the QASM simulator.Moreover, we performed a hybrid quantum-classical simulation of a 2D unsteady TG ISF in Appendix D to demonstrate the capability of simulating high-dimensional ISFs illustrated in Section III.The obstacles (steps 2-4) in the quantum algorithm were tentatively treated on the classical computer.V. CONCLUSIONS We develop a framework for the quantum computing of fluid dynamics based on the HSE with a generalized Madelung transform.The SF, a flow with finite vorticity and
. The present quantum algorithm can achieve exponential speedup in the steps of prediction and phase calculation, and possible O((n2 n + poly(2 n ))/poly(n)) speedup overall.The quantum algorithm is implemented using IBM's Qiskit for a simple 1D flow.The result agrees with the theoretical solution with finite noises, and demonstrates an exponential speedup on a quantum computer.Note that the HSE without a viscous term and with an external LLF term is different from the NSE, but the SF resembles the viscous flow in terms of the similar flow statistics and structures.We use the TG vortex and decaying HIT to demonstrate the similarities between the ISF and the viscous flow.The role of the parameter in the HSE is similar to the kinetic viscosity.The flow stability depends on the value of in the TG vortex, and the inertial range with the −5/3 scaling broadens with decreasing in the HIT.The tangle of vortex tubes and sheets is observed in the turbulent ISF as in the classical turbulent flow.

FIG. 15 .
FIG.15.Isosurfaces of (a) s 1 = −0.9 and (b) s 3 = −0.9 in the TG initial field.Some vortex lines are integrated and plotted on the isosurfaces.

Figure 16
Figure 16 shows evolution of the contour of ω/|ω| max in the 2D TG ISF with = 1, where ω denotes the z-component of ω and |ω| max denotes the instantaneous maximum |ω| in the computational domain.The hybrid simulation result shows a good agreement with the classical one, and only has slight oscillations due to the statistical error and noises introduced by the QASM simulator.The discrepancy E(t) ≡ |ω| hybrid −|ω| classical 2 2between the two simulations is smaller than 6% at t < 1 for three independent runs in Fig.17.In . In

FIG. 16 .FIG. 17 . 2 2
FIG. 16.Comparison of the instantaneous contours of the normalized vorticity in the hybrid quantum-classical simulation and classical simulation of the 2D TG ISF with = 1 and n = 10 qubits.

TABLE I .
Meanings of the same symbol in different contexts.
(38)tum circuit for the QFT in Eq.(38)with n qubits, where the SWAP gates for reversing the order of qubits in the end of the QFT are not shown.
[2] Hermitian transpose †, and the diagonal unitary transformation P(f ) ≡ e −if / .The QFT can be implemented by O(n 2 ) quantum gates in Fig.8, which achieves an exponential acceleration compared to O(n2 n ) operations of the fast Fourier transform.Equation (37) is an approximation based on the second-order Trotter decomposition[2] 1 may be non-unitary, so they are not realizable quantum gates.Hollow and solid circles denote control qubits with values 0 and 1, respectively.The operation Q j on the target qubit |s is active only when the control qubits are in the state |j 2 Steps 3 & 4 FIG.13.Overall quantum circuit for solving the IHSE, from time t to t + ∆t.The bottom qubit is an auxiliary one for implementing the momentum operator.

TABLE II .
Breakdown of algorithm complexities of quantum and classical algorithms.