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Emulation of complex open quantum systems using superconducting qubits

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Abstract

With quantum computers being out of reach for now, quantum simulators are alternative devices for efficient and accurate simulation of problems that are challenging to tackle using conventional computers. Quantum simulators are classified into analog and digital, with the possibility of constructing “hybrid” simulators by combining both techniques. Here we focus on analog quantum simulators of open quantum systems and address the limit that they can beat classical computers. In particular, as an example, we discuss simulation of the chlorosome light-harvesting antenna from green sulfur bacteria with over 250 phonon modes coupled to each electronic state. Furthermore, we propose physical setups that can be used to reproduce the quantum dynamics of a standard and multiple-mode Holstein model. The proposed scheme is based on currently available technology of superconducting circuits consist of flux qubits and quantum oscillators.

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Acknowledgements

We acknowledge DTRA Grant No. DTRA1-10-1-0046, AFOSR UCSD Grant No. FA9550-12-1-0046, Department of Energy Award No. DE-SC0008733 and Harvard FAS RC team for Odyssey computer resources. A.J.K. acknowledges the Assistant Secretary of Defense for Research and Engineering under Air Force Contract No. FA8721-05-C-0002. Opinions, interpretations, conclusions and recommendations are those of the author and are not necessarily endorsed by the United States Government.

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Correspondence to Sarah Mostame.

Appendices

Appendix 1: Linear-algebraic bath transformation

Open quantum system approach to the spin-boson model often approximates the environment as a collection of non-interacting harmonic oscillators. This is known as a star-bath model and can be graphically illustrated in a star configuration as shown in Fig. 5a. Linear-algebraic bath transformation [48] converts the star-bath model into a set of weakly coupled multiple parallel chains as shown in Fig. 5b. The multiple-chain bath model has a few primary bath oscillators that are directly coupled to the system (spins/qubits) and the remaining oscillators (secondary bath modes) are coupled to the primary bath modes in a chain. This model employs a simple linear-algebraic approach to reduce the system-bath coupling strength as well as the number of the oscillators that are directly coupled to the system.

Fig. 5
figure 5

a Star-bath model: Non-interacting quantum harmonic oscillators (shown in blue) are directly coupled to a system site (shown in red). b Multiple-chain bath model: A system site is coupled to multiple bath oscillator chains. In this model, the primary modes (shown in blue) are directly coupled to the system and the secondary modes (shown in yellow) are coupled to the primary bath modes in a chain (Color figure online)

To apply the linear-algebraic bath transformation we start by writing the \(H_{\mathrm {el-ph}}+H_{\mathrm {ph}}\) in a compact form [18, 55] and then finding a unitary transformation \({{\varvec{U}}}_n {{\varvec{U}}}_n^\dagger =I\) that satisfies the following conditions [48]:

$$\begin{aligned}&H_{\mathrm {el-ph}}+H_{\mathrm {ph}}= \sum _{n} \begin{pmatrix} L_{n}&\mathbf{b}_{n}^{\dagger } \end{pmatrix} \varvec{\Gamma }_{n} \begin{pmatrix} L_{n} \\ \mathbf{b}_{n} \end{pmatrix} = \sum _{n} \begin{pmatrix} L_n^\dagger&\tilde{\mathbf{b}}_n^\dagger \end{pmatrix} \tilde{\varvec{\Gamma }}_n \begin{pmatrix} L_n \\ \tilde{\mathbf{b}}_n \end{pmatrix} \end{aligned}$$
(7)

with

$$\begin{aligned}&\varvec{\Gamma }_{n}=\begin{pmatrix} \mathbf{0}&{}{\varvec{\kappa }}^{\mathrm t}_{n}\\ {\varvec{\kappa }}_{n}&{}{\varvec{\Omega }}_{n} \end{pmatrix}, \quad \tilde{\varvec{\Gamma }}_n=\begin{pmatrix} \mathbf{0}&{}{\tilde{\varvec{\kappa }}}^{\mathrm t}_n\\ {\tilde{\varvec{\kappa }}}_n&{}\tilde{\varvec{\Omega }}_n \end{pmatrix} =\begin{pmatrix} \mathbf{1}&{} \mathbf{0}^{\mathrm t}\\ \mathbf{0}&{}{{\varvec{U}}}_n^{\dagger } \end{pmatrix} {\varvec{\Gamma }}_{n} \begin{pmatrix} \mathbf{1}&{}\mathbf{0}^{\mathrm t}\\ \mathbf{0}&{}{{\varvec{U}}}_n \end{pmatrix}\, , \end{aligned}$$
(8)

where \(\tilde{\mathbf{b}}_{n}={{\varvec{U}}}_{n}^{\dagger }\,\mathbf{b}_{n}\) with \(\mathbf{b}_n^\dagger \, \, (\mathbf{b}_n)\) being the N-dimensional creation (annihilation) operator vector of the phonons (oscillators). \(L_n\) is an operator that acts on the system, \({\varvec{\kappa }}_n\) is the system-bath coupling strength vector, and \(\varvec{\Omega }_{n}\) is a diagonal matrix, which has the harmonic frequencies as the elements \({\varvec{\Omega }}_{n}=\mathrm{diag}(\omega _{n,1},\ldots ,\omega _{n,N})\). The first column of \({{\varvec{U}}}_{n}\) is \({\varvec{\kappa }}_{n}/\vert \vert {\varvec{\kappa }}_{n} \vert \vert _{2}\) and the other columns are given by the Gram-Schmidt process with random vectors [56]. \(\tilde{\varvec{\Gamma }}_{n}\) is a dense symmetric matrix and \(\tilde{\varvec{\kappa }}_{n}=({\tilde{\kappa }}_{n,1},0,\ldots ,0)^{\mathrm t}\) is the new system-bath coupling strength vector [48].

To complete the multiple-chain transformation, we introduce another unitary transformation \(\tilde{\mathbf{U}}_n=\mathbf{P}_n\mathbf{U}_n\) that follows the following relations

$$\begin{aligned} \tilde{\varvec{\Omega }}_{n}= \tilde{\mathbf{U}}_{n}^{\dagger } \varvec{\Omega }_{n}\tilde{\mathbf{U}}_{n} \quad \mathrm {and} \quad \tilde{\varvec{\kappa }}_{n}= \tilde{\mathbf{U}}_{n}^{\dagger }\varvec{\kappa }_{n} \, . \end{aligned}$$
(9)

The permutation matrix \(\mathbf{P}_n\) is used to rearrange the non-interacting bath oscillators as multiple groups of several interacting oscillators \(\tilde{\mathbf{b}}_{n}=\tilde{\varvec{U}}_{n}^{\dagger }\,\mathbf{b}_{n}\). Note that \(\mathbf{U}_n\) is block diagonal and does not allow the interaction between oscillators from different groups. Now by choosing the l-th subblock \(\mathbf{U}_{n}^{(l)}\) to be \(\mathbf{g}^{(l)}_{n}/\vert \vert \mathbf{g}^{(l)}_{n} \vert \vert _{2}\), we can define the primary modes (the ones that are directly coupled to the system sites) as collective oscillator modes. Here \(\mathbf{g}_{n}\) is the rearranged coupling strength vector

$$\begin{aligned} \mathbf{g}_{n}=\mathbf{P}_{n}^{\dagger }{\varvec{\kappa }}_{n}= \begin{pmatrix} \mathbf{g}^{(1)}_{n}\\ \vdots \\ \mathbf{g}^{(N_{\mathrm{eff}})}_{n} \end{pmatrix}, \end{aligned}$$
(10)

and \(N_{\mathrm{eff}}\) is the number of groups of oscillators. The final step is to tridiagonalize the l-th subblock \(\tilde{\varvec{\Omega }}_{n}^{(l)}\) using the Hessenberg transform [56] via the Householder procedure \(\tilde{\varvec{\Omega }}_{n}^{(l)}={\mathcal T}^{(l)}{\varvec{\Xi }}^{(l)}{{\mathcal {T}}}^{(l)\dagger }\). The diagonal elements of the tridiagonal matrix \({\varvec{\Xi }}^{(l)}\) represent the frequencies of the transformed bath modes and its off-diagonal elements are the coupling strengths between the oscillators in the chain model. \({{\mathcal {T}}}^{(l)}\) is a Hessenberg unitary transform matrix that keeps the primary bath mode unchanged. To adjust the system-bath coupling strengths of primary modes within the experimentally realizable parameter domain, we have also developed a bath mode partitioning scheme [48]. The scheme is called leaping partition (LP), which selects the oscillators far away from each other to form multiple parallel chains. For example, we grouped 253 oscillators of chlorosome into 6 groups as

$$\begin{aligned} \{\omega _{1},\omega _{1+6},\omega _{1+12},\cdots ,\omega _{1+252}\}, \quad \{\omega _{2},\omega _{2+6},\omega _{2+12},\cdots ,\omega _{2+246}\}, \nonumber \\ \{\omega _{3},\omega _{3+6},\omega _{3+12},\cdots ,\omega _{3+246}\}, \quad \{\omega _{4},\omega _{4+6},\omega _{4+12},\cdots ,\omega _{4+246}\}, \nonumber \\ \{\omega _{5},\omega _{5+6},\omega _{5+12},\cdots ,\omega _{5+246}\}, \quad \{\omega _{6},\omega _{6+6},\omega _{6+12},\cdots ,\omega _{6+246}\}, \end{aligned}$$
(11)

where \(\omega _{l}\le \omega _{m}\) if \(l\le m\). See Ref. [48] for more examples on the rearrangement of the oscillators and also the MATLAB code that we have used for the transformation.

Appendix 2: Hierarchically equations of motion approach

In order to estimate the treatable system size for the Holstein model on classical computers we run benchmark calculations with QMaster, which is a high-performance implementation of the hierarchically coupled equations of motion approach (HEOM) [19, 57, 58]. HEOM is based on an open quantum system approach and treats the phonon modes as continuum bath.

The time evolution of the total system, described by the density operator R(t) is given by the Liouville equation

$$\begin{aligned} \frac{\text{ d }}{\text{ d }t}{R}(t)= - \frac{\text{ i }}{\hbar }[{{\mathcal {H}}}(t),{R}(t)]=- \frac{\text{ i }}{\hbar }{{\mathcal {L}}}(t){R}(t). \end{aligned}$$
(12)

At initial time \(t_0=0\) we assume that the density operator \(R(t_0)=\rho (t_0)\otimes \rho _{\mathrm{phon}}(t_0)\) factorizes into the system degrees of freedom, described by the reduced density operator \(\rho (t)\), and vibrational degrees of freedom \(\rho _{\mathrm{phon}}(t)\). The dynamics of the reduced density operator is then obtained by averaging out the vibrational degrees of freedom

$$\begin{aligned} {\rho }(t)=\langle \text{ T }_+\,\exp \Big (-\frac{\text{ i }}{\hbar }\int _0^t \text{ d }s \,{{\mathcal {L}}}(s) \Big )\rangle {\rho }(0) . \end{aligned}$$
(13)

We employ a high-temperature approximation \(\hbar \gamma _m/k_{\mathrm{B}}T<1\) and parameterize the spectral density as a sum over \(N_{\mathrm{peaks}}\) shifted Drude–Lorentz peaks

$$\begin{aligned} J(\omega )=\sum _{k=1}^{N_\mathrm{peaks}}\left( \frac{\nu _k\lambda _k\omega }{\nu _k^2+(\omega +\Omega _k)^2}+\frac{\nu _k\lambda _k\omega }{\nu _k^2+ (\omega -\Omega _k)^2}\right) . \end{aligned}$$
(14)

The time non-local equation can then be cast into a hierarchy of coupled time local equations of motion for a set of auxiliary matrices \(\sigma ^{\vec {n}}\)

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\sigma ^{\vec {n}}(t)= & {} -\frac{\hbox {i}}{\hbar }[{\mathcal {H}}_\mathrm{ex},\sigma ^{\vec {n}}(t)]\nonumber \\&-\sum _{m,k=1,s=\pm 1}^{N,N_{\mathrm{peaks}}} \frac{2}{\beta \hbar ^2}\frac{\lambda _k\nu _k}{{(\gamma _1+\hbox {i}s\Omega _k)}^2-\nu _k^2}V_{m}^{\times }V_{m}^{\times }\sigma ^{\vec {n}}(t) \nonumber \\&-\sum _{m,k=1,s=\pm 1}^{N,N_{\mathrm{peaks}}} n_{m,k,s}(\nu _k+s\,\text{ i }\Omega _k) \sigma ^{\vec {n}}(t) \nonumber \\&+\sum _{m,k=1,s=\pm 1}^{N,N_{\mathrm{peaks}}} \left[ \frac{\hbox {i}}{\hbar }V_m^{\times } \sigma ^{\vec {n}_{m,k,s}^+}(t)+\theta _{m,k,s}\sigma ^{\vec {n}_{m,k,s}^-}(t)\right] , \end{aligned}$$
(15)

Here we define \(\vec {n}=(n_{1,1,+},n_{1,1,-}, \ldots ,n_{1,M,+},n_{1,M,-}, \ldots ,n_{N,M,+},n_{N,M,-}), V_{m}^{\times }\sigma =[a_m^\dag a_m, \sigma ], V_{m}^{\circ }\sigma =[a_m^\dag a_m, \sigma ]_+\) and \(\theta _{m,k,s}=\frac{\hbox {i}}{2}\Big (\frac{2\lambda _k}{k_B T\hbar }V_{m}^{\times } -\hbox {i}\lambda _k(\nu _k+s\,\hbox {i}\Omega _k) V_{m}^{\circ }-\frac{2\lambda _k}{\beta \hbar ^2}\frac{{(\nu _k+\hbox {i}s\Omega _k)}^2}{\gamma _1^2-{(\nu _k+\hbox {i}s\Omega _k)}^2}V_{m}^{\times }\Big ). \) The reduced density matrix is given as \(\rho (t)=\sigma ^{\vec {0}}(t)\). The hierarchy Eq. (15) can be truncated for a sufficiently large hierarchy depth \(\sum _{m,k=1,s=\pm 1}^{N,N_{\mathrm{peaks}}} n_{m,k,s}\ge N_{\mathrm{max}}\), for which convergence is tested by comparing the dynamics for different truncation levels \(N_{\mathrm{max}}\).

Appendix 3: Computational details to estimate the treatable system size

Solving the hierarchy Eq. (15) for a large system size is challenging, and requires a considerable amount of computational resources, both in memory and number of floating point operations per second (FLOPS). The whole set of auxiliary matrices needs to be retained in the CPU memory during the complete propagation of the exciton dynamics, and all entries need to be update for each propagation step. The total number of auxiliary matrices \(N_{\sigma } = (2\,N_{\mathrm{peaks}}\,N + N_{\mathrm{max}})!/(N_\mathrm{max}!(2\,N_{\mathrm{peaks}}\,N)!\) depends on the number of sites N in the Holstein Model (see Eq. (4), main text), the truncation level \(N_{\mathrm{max}}\) and the number of peaks in the spectral density \(N_{\mathrm{peaks}}\). The factor \(2\,N_{\mathrm{peaks}}\) takes into account the shifts of the peaks in the spectral density, Eq. (14) in positive as well as in negative direction along the frequency axis. Thus, for the parameters used to perform the benchmark calculations in Fig. 1 of the main text, we need to propagate up to several millions of auxiliary matrices in log-step, see Table 1.

Table 1 Benchmarks of the treatable system size with respect to the number of sites and number of peaks in the spectral density

We carry out the calculations with the help of a sophisticated algorithm provided by the QMaster package [16]. QMaster is based on massively parallelized vector streaming. The idea behind the algorithm is to efficiently distribute the workload among the available computational units while keeping up a high memory bandwidth. The latter is achieved by a suitable layout of how the auxiliary matrices are stored in the CPU memory. In general QMaster also runs on GPUs and the XeonPhi accelerator. However, todays available GPU memory is limited to 24GB (K80) which is the reason why we run the benchmark calculations on a 64-core AMD Opteron processor with 256 GB memory. QMaster is a single-device implementation, since distributed computation among multiple compute nodes connected by Ethernet is rendered inefficient, due to the large communication overhead [23]. More information about the algorithm as well es a performance analysis are given elsewhere [16].

Table 1 summarizes the technical aspects of the underlying computations to estimate the treatable system size (Fig. 1, main text). The propagation is performed over 10 time steps for a given electronic excitation at \(t_0\). The truncation level is set to \(N_{\mathrm{max}}=3\) which has proven as reasonable value for several light-harvesting complexes [15, 16, 23, 59]. Shown are the results for the maximal treatable system sizes with respect to the number of sites and number of peaks in the spectral density. Once one more peak is added QMaster needs to allocate more than 250 GB of memory, and therefore terminates with a segmentation fault. Thus memory consumption sets the hard limit for the treatable system size. The computation time is a somewhat more soft criteria, since the total computation time depends on the number of propagation steps which strongly depends on the system at hand. For example, calculations for a quadrant of the PSII-supercomplex comprising of 93 sites have been carried out for which the exciton dynamics was propagated over 20,000 time steps with a time increment of 5 fs (100 ps total propagation time) [59]. According to Table 1, the calculations for a non-shifted Drude–Lorentz peak takes about 43 days of total computation time, which allows at least to run simulations to test the convergence of the hierarchy with respect to the truncation level [59].

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Mostame, S., Huh, J., Kreisbeck, C. et al. Emulation of complex open quantum systems using superconducting qubits. Quantum Inf Process 16, 44 (2017). https://doi.org/10.1007/s11128-016-1489-3

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  • Published:

  • DOI: https://doi.org/10.1007/s11128-016-1489-3

Keywords

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