Abstract
In digital quantum simulation of fermionic models with qubits, nonlocal maps for encoding are often encountered. Such maps require linear or logarithmic overhead in circuit depth which could render the simulation useless, for a given decoherence time. Here we show how one can use a cavity–QED system to perform digital quantum simulation of fermionic models. In particular, we show that highly nonlocal Jordan–Wigner or Bravyi–Kitaev transformations can be efficiently implemented through a hardware approach. The key idea is using ancilla cavity modes, which are dispersively coupled to a qubit string, to collectively manipulate and measure qubit states. Our scheme reduces the circuit depth in each Trotter step of the Jordan–Wigner encoding by a factor of N^{2}, comparing to the scheme for a device with only local connectivity, where N is the number of orbitals for a generic twobody Hamiltonian. Additional analysis for the Fermi–Hubbard model on an N × N square lattice results in a similar reduction. We also discuss a detailed implementation of our scheme with superconducting qubits and cavities.
Introduction
Quantum computers are widely touted as a new frontier for simulating quantum systems.^{1,2} The simulation of quantum chemistry,^{3,4,5,6,7} strongly correlated fermionic systems,^{8,9,10,11,12} and lattice gauge theories,^{13,14} are among the crucial applications.^{15} However, apart from ultracold fermionic atoms, all quantum simulation platforms are based on bosonic/spin degree of freedom. Therefore, one has to encode the fermionic problem into simulationfriendly spin models.
In the literature, there are a number of methods for doing so and we will focus on the methods that require implementing a nonlocal map, e.g., Jordan–Wigner (JW) or Bravyi–Kitaev (BK) mappings.^{16} Our approach relies on the use of a cavity–QED system to achieve the nonlocal coupling directly. This is in contrast to other ideas for improving the nonlocality of the fermionspin mapping, such as direct simplification of the quantum circuit^{17} or using gate teleportation^{18} to lower the cost of the Jordan–Wigner and Bravyi–Kitaev schemes. Another alternative to the approach taken here is to introduce additional qubits to achieve improved locality of the spinrepresentations of fermonic operators.^{5,19,20} Lastly, we mention a recently introduced technique for quantum simulation using plane waves rather than typical electronic structure basis sets composed of quasilocal Gaussian orbitals.^{21} The approach taken there has been shown to achieve linear circuit depth for a certain class of electronic systems. We do not pursue subspace encodings and consider arbitary electronic systems with a focus on approaches that directly implement the nonlocal maps rather than circumventing them.
Here, we present a hardwareefficient scheme to perform digital fermionic simulations on a physical system made of spins. Our approach makes use of cavity–QED physics,^{22,23,24,25} where one or several ancilla cavity modes are used to encode, simulate the Hamiltonian and measure the desired observables. The selective nonlocal coupling of ancillae to a qubit string allows for implementation of JW and BK mappings in one shot and reduces the simulation time. More specifically, in exponentiating each term of the Hamiltonian, our scheme reduces the circuit depth of both JW and BK to O(1) operations. This improvement reduces the simulation time, and therefore, mitigates the decoherence effects.
We then present an experimental implementation of our scheme in a circuitQED platform,^{26,27,28,29,30,31,32,33,34,35,36,37,38} where experimental progress on fermionic and quantum chemistry simulation has been recently achieved.^{4,7} In particular, we use dispersive coupling of microwave cavity photons to superconducting qubits^{30,38} to generate nonlocal string operations nonperturbatively. This digital approach offers better scaling in the collective gate time than a previous analog scheme where multispin interactions are generated perturbatively,^{39} resulting in an exponential decrease with the number of Pauli operators to be implemented. Moreover, experimental advances have been achieved in probing inhomogeneity in resonate frequencies in the context of both superconducting qubitarray and resonatorlattice,^{40,41} and hence pave the way for the realization of collective manybody gates. Therefore, our scheme is preferable for implementing large strings, and it also remedies the disadvantage of circuitQED architecture, i.e. low connectivity, compared to ion trap architectures.^{42}
Furthermore, we compare our scheme to conventional local schemes for various fermionic models, such as Fermi–Hubbard model and generic Coulomb Hamiltonian. In these comparisons, we introduce a parallelization scheme, which further improves the simulation. Specifically, by parametrically coupling multiple cavity modes, we further decrease the circuit depth for each Trotter step by an additional factor of N. This results in an overall O(N^{2}) reduction for Jordan–Wigner and Bravyi–Kitaev transformation in the cases of a Fermi–Hubbard model on an NbyN lattice and a quantum chemistry problem with N orbitals, implemented on a device with local connectivity.
Results
Fermionic encoding with the nonlocal cavity–QED interaction
Coulomb Hamiltonian and Fermionic encoding
We consider a generic electronic model with hopping and twobody Coulomb interaction. The form of the Hamiltonian is given by
Here, κ_{ ij } is the hopping matrix and V_{ ijkl } represents the interaction matrix. The indices i, j, k and l can label orbitals either in realspace or the reciprocalspace and can also absorb spin indices.
In order to simulate fermions with qubits, the simplest scheme is the Jordan–Wigner transformation:
The index j can be used to label sites in any dimension. For example, the string in 2D can be chosen as a ‘selfavoiding snake’ as illustrated by the red string in Fig. 1. In addition to the JW transformation, the Bravyi–Kitaev transformation^{16} also requires strings of Pauli operators although the form is more complicated (see Supplemental Information VI). The length of Pauli strings are on average logarithmically shorter than JW using the Bravyi–Kitaev transformation. In order to implement the time evolution with such string operators, we will consider using the cavityassisted conditional string operation in the following sections.
Cavity–QED interaction and controlledstring operation
We consider the quantum nondemolition (QND) interaction^{43} of a cavity–QED system in the dispersive regime:
where χ is the dispersive interaction strength.
We prepare the cavity photon state in the restricted subspace n_{ a } = 0, 1. For circuitQED implementation, the cavity nonlinearity introduced by the qubits are large enough, such that the cavity itself can be operated as a qubit. To collectively manipulate a qubit string, we simply apply the dispersive interaction for a period of τ. The time evolution operator is expressed as
Here, we used the property that photon and spin operators commute, and the Paulimatrix property \(\left( {\sigma _j^z} \right)^2 = 1\). If we choose the operation time to be τ = π/(2χ), we end up with
The additional phase factor (−i)^{N} depends on the length of the string and can be canceled by applying an additional phase gate on the ancilla cavity, and we call the resulting evolution operator \(C_{\overline Z }\), i.e., a conditional\(\overline Z\) string operator, controlled by the cavity photon state: (1) If n_{ a } = 0, no operation is performed; (2) If n_{ a } = 1, a string operator \(\overline Z = \mathop {\prod}\nolimits_j {\kern 1pt} \sigma _j^z\) is applied. Such a cavitycontrolled string operation has also been proposed to manipulate and engineer the topological ground state of the toriccode model.^{23,44,45}
Exponentiation of the string operators, time evolution, and phase estimation
In order to perform digital quantum simulation of a Fermionic Hamiltonian H, one needs to perform Trotter evolution with small time steps,^{2} i.e., e^{−iHΔt}. After breaking the Hamiltonian down to subterms \(H = \mathop {\sum}\nolimits_q {\kern 1pt} h_q\), one exponentiates each of these subterms as \({\rm e}^{  {\rm i}h_q{\rm{\Delta }}t}\). The subterm h_{ q } is composed of a qubit string operator. For example, a hopping term in Eq. (1) is represented by qubit operators under JW encoding as h_{ ij } = \(\kappa _{ij}\left( {\sigma _i^ + \sigma _j^  + {\mathrm{H}}{\mathrm{.c}}{\mathrm{.}}} \right)\mathop {\prod}\nolimits_{k \in {\rm{string}}} {\kern 1pt} \sigma _k^z\). This can be split into two pieces \(h_{ij}^{(1)} = \frac{1}{2}\kappa _{ij}\sigma _i^x\sigma _j^x\mathop {\prod}\nolimits_{k \in {\rm{string}}} {\kern 1pt} \sigma _k^z\) and \(h_{ij}^{(2)} = \frac{1}{2}\kappa _{ij}\sigma _i^y\sigma _j^y\mathop {\prod}\nolimits_{k \in {\mathrm{string}}} {\kern 1pt} \sigma _k^z\), and will be exponentiated separately. The conventional approach realizes the exponentiation of these string terms by a CNOT ladder (a sequence of nearestneighbor CNOTs) illustrated in Fig. 2a (upper panel, see Supplemental Information I for details). Here, we present a hardwareefficient quantum circuit which uses the cavitycontrolled string operation Eq. (5) as shown in Fig. 2a (lower panel). The essence is to collect the global parity information into the cavity ancilla with a single \(C_{\overline Z }\) gate and another \(C_{\overline Z }\) gate to erase the parity information after the rotation of the ancilla along xaxis by an angle 2Δt. Note that this circuit reduces the number of gates and circuit depth by a factor of N (N being the length of the string) due to its nonlocal and highly parallel feature, and hence greatly reduces the operation time.
To derive the properties of the circuit, we start with the conditional string operation \(C_{\overline Z }\), and the rotation of the ancilla
where X_{a} is the PauliX operator of the ancilla photon state. The three successive gates \(C_{\overline Z }R_x(2{\mathrm{\Delta }}t)C_{\overline Z }\) can be expressed as
where we have used the property \(\overline Z ^2 = {\mathbb{1}}_q\). The final expression represents a conditional evolution with the nonlocal manybody Hamiltonian \(H_{{\rm{string}}} = \overline Z = \mathop {\prod}\nolimits_{j \in {\rm{string}}} {\kern 1pt} \sigma _j^z\), controlled by the ancilla photon state \(\left \pm \right\rangle _a\).
In general, arbitrary manybody interactions along the string can be exponentiated, by choosing the proper singlequbit rotations in the beginning and end of the circuit (see Fig. 2a). In Fig. 2b,c, we show explicitly the circuits to implement the exponentiation of the hopping subterm \(h_{ij}^{(1)} = \frac{1}{2}\kappa _{ij}\sigma _i^x\sigma _j^x\mathop {\prod}\nolimits_{k \in {\rm{string}}} {\kern 1pt} \sigma _k^z\) and the interaction subterm \(h_{ijkl}^{(1)} = \frac{1}{4}V_{ijkl}\sigma _i^x\sigma _j^x\sigma _k^x\sigma _l^x\mathop {\prod}\nolimits_{m \in {\rm{string}}} {\kern 1pt} \sigma _m^z\) coming from the Coulomb interaction term in Eq. (1), both under JW encoding. Here, we have used Hadamard gates to turn certain σ^{z} operators into σ^{x} with the identity \({\mathrm{H}}_j\sigma _j^z{\mathrm{H}}_j = \sigma _j^x\). On the other hand, a typical term in the Bravy–Kitaev encoding may involve all types of Pauli operators, e.g., \(\sigma _1^y\sigma _2^x\sigma _3^y\sigma _5^z\). This qubit string can be exponentiated with the circuit in Fig. 2d, where the combined Hadamards and phase gates (S and S^{†}) realized with a single pulse turn the σ^{z} operators into σ^{y}.
If one starts the ancilla in the \(\left + \right\rangle _a\) (\(\left  \right\rangle _a\)) state, one only gets forward (backward) evolution after n Trotter steps, e^{−inΔtH} (e^{inΔtH}), as suggested by Eq. (7). However, if one starts with the ancilla in state \(\left 0 \right\rangle _a = \frac{1}{{\sqrt 2 }}\left( {\left + \right\rangle _a + \left  \right\rangle _a} \right)\), one gets a conditional evolution CU = \({\rm e}^{  {\rm i}Ht}\left + \right\rangle \left\langle + \right_a + {\rm e}^{{\rm i}Ht}\left  \right\rangle \left\langle  \right_a\), where t = nΔt. This property can be applied to quantum phase estimation^{46,47} for extracting energy spectrum and state preparation (see Supplemental Information VIII for details). Note, after the state preparation, one can extract fermionic correlation function such as \(C_{ij} = \left\langle {\psi \left {c_i^\dagger c_j} \right\psi } \right\rangle = \left\langle {\psi \left {\sigma _i^ + \sigma _j^  \mathop {\prod}\nolimits_k {\kern 1pt} \sigma _k^z} \right\psi } \right\rangle\) with conditional string operations. For example, the circuit shown in Fig. 2e implements the xxpart of the correlator, i.e. \(\left\langle {\psi \left {\sigma _i^x\sigma _j^x\mathop {\prod}\nolimits_k {\kern 1pt} \sigma _k^z} \right\psi } \right\rangle\), where setting ϕ = 0 (ϕ = π/2) in the phase gate gives the real (imaginary) part. The measurement of dynamical correlator is discussed in Supplemental Information VII.
Parallelizations with multiple ancillary cavity modes
Another advantage of the cavity–QED approach is that one can further parallelize the exponentiation of all the mutually commuting subterms h_{ ij } using multiple cavity ancillae. This can be realized with multiple cavities or different modes in the same cavity as discussed further in the next section. Parallelization is trivial if the string operators to be exponentiated do not overlap with each other. It is also possible to exponentiate multiple overlapping strings in parallel, namely \(\mathop {\prod}\nolimits_\nu {\kern 1pt} {\rm e}^{{\rm i}\kappa {\rm{\Delta }}t\overline {\cal S} _\nu }\), where ν labels different strings. A concrete example is exponentiating hopping terms between two neighboring rows in parallel which appears in the Hubbard model (illustrated in Fig. 2f). The detailed derivation can be found in section “Methods”.
Implementation with circuit–QED architecture
In this section, we focus on the experimental implementation of the QND interactions of Eq. (3). We also discuss implementation of parallelization with multiple ancilla modes in the same cavity either by higher level contribution or alternatively by periodical modulation of the flux couplers.
Realization with circuit QED
We consider a collection of multilevel superconducting qudits inductively coupled to a single or multiple transmissionline cavities or 3D cavities as shown in Fig. 3a. The simplest case with one cavity mode can be described by a generalized Tavis–Cummings model:^{48}
Here, a is the annihlation operator for the cavity mode with frequency ω, \(\left l \right\rangle _j\) represents the lth level of the jth qudit with corresponding energy \(\epsilon _l\), and g_{ll′} = \(g\left\langle {l\left \phi \rightl^{\prime}} \right\rangle \equiv g\phi _{ll^{\prime}}\) is proportional to the inductive coupling strength g and the phase matrix element (ϕ being the superconducting phase operator). The strength g can be made uniform even in the presence of nonuniform mode function with the fluxtunable inductive coupler,^{49} as shown in Fig. 3a.
In the dispersive regime, namely
(N represents the total number of coupled qudits and Δ_{ll′} the detuning), one can adiabatically eliminate the direct inductive coupling V between qudits and the cavity. The effective Hamiltonian after a Schrieffer–Wolff transformation^{48,50,51} up to secondorder is given by
Apart from H_{0}, the terms app earing in secondorder perturbation have three types: (1) The energy shift of level l is given by: \(\chi _l = \mathop {\sum}\nolimits_{l^{\prime} \ne l} {\kern 1pt} \chi _{ll^{\prime}} = \mathop {\sum}\nolimits_{l^{\prime} \ne l} {\kern 1pt} g_{ll^{\prime}}^2\left( {\frac{1}{{{\mathrm{\Delta }}_{ll^{\prime}}}}  \frac{1}{{{\mathrm{\Delta }}_{l^{\prime}l}}}} \right)\), summed over the contributions χ_{ll′} from virtual transitions to all other levels l′, where the first term is AC Stark and the second term is Bloch–Siegert shift, in the absence of rotatingwave approximation; (2) the Lamb shift \(\kappa _l = \mathop {\sum}\nolimits_{l^{\prime} \ne l} {\kern 1pt} \frac{{g_{ll^{\prime}}^2}}{{{\mathrm{\Delta }}_{ll^{\prime}}}}\) which only renormalizes the qudit energy level: \(\epsilon _l\) → \(\epsilon _l\) + κ_{ l }; (3) the flip–flop interactions between any two qudits mediated by virtual photons with strength \(\mu _{ll^{\prime}} = \mathop {\sum}\nolimits_{l^{\prime \prime} \ne l,l^{\prime}} \frac{{g_{ll^{\prime \prime}}g_{l^{\prime \prime}l^{\prime}}}}{2}\left( {\frac{1}{{{\mathrm{\Delta }}_{ll^{\prime}}}}  \frac{1}{{{\mathrm{\Delta }}_{l^{\prime \prime}l}}} + \frac{1}{{{\mathrm{\Delta }}_{l^{\prime}l^{\prime \prime}}}}  \frac{1}{{{\mathrm{\Delta }}_{l^{\prime \prime}l^{\prime}}}}} \right)\), which we need to cancel out to avoid the induced crosstalk errors in our manybody gates. One can choose specific superconducting circuits, such as fluxonium^{38,48,52,53} focused here (alternatively flux qubit^{54} or protected 0π qubit^{55,56}). In particular, we consider the situation that phase matrix elements obtain selectionrule property^{38,53,57} at large ratio of Josephson and charging energy E_{J}/E_{C} (e.g. E_{J} = 20 GHz, with fixed E_{C} = 0.5 GHz from now on): ϕ_{01} = ϕ_{12} = ϕ_{03} = 0 as shown in Fig. 3c. In the case of fluxonium, this is due to the feature that the ground and excited states are persistentcurrent states with different winding numbers m, which can be seen from their wavefunctions being trapped in different wells of the Josephson potential −E_{J} cos ϕ and have negligible overlap (Fig. 3b). Therefore, the contribution from χ_{01} (as well as any other interwell virtual transition) is nearly zero (<10^{−5} at E_{J} = 20 GHz). A QND interaction \(H_{{\mathrm{QND}}} = \mathop {\sum}\nolimits_j {\kern 1pt} \chi a^\dagger a\sigma _j^z\) arises in secondorder perturbation with strength χ = ∑_{ l }(χ_{0l} − χ_{1l})/2j, while the nonzero contributions are from intrawell virtual transitions to higher levels, such as χ_{02} and χ_{13}, which has recently been experimentally observed (see ref. ^{38}). On the other hand, the singleexcitation flipflop term \(_j\left 0 \right\rangle \left\langle 1 \right_{j^{\prime}}\) disappears (μ_{01} = 0) due to the forbidden interwell transitions (g_{01} = g_{12} = g_{03} = 0, etc.), and the lowestlevel contribution is from \(_j\left 0 \right\rangle \left\langle 2 \right_{j^{\prime}}\). During the simulation process, we only occupy levels 0 and 1 which act as the qubit degree of freedom, therefore the flip–flop process does not play any role and hence will not introduce the unwanted crosstalk error in the manybody \(C\overline Z\) gate. When we need to implement singlequbit Hadamard (H) and phase (S) gates to get PauliX and Y (Fig. 2a), we can go to the smallE_{J}/E_{C} regime (e.g. E_{J} = 4 GHz) by quasiadiabatically tuning the flux into the junction loop. In this regime, 0–1 transition can be implemented indirectly via a Raman process (0 → 2 → 1) utilizing the lowlying Λstructure,^{57} as shown in Fig. 3b,c. A direct transition is also possible since the 0–1 matrix element is sizable and can be accessed by the classical drive. Alternatively one can stay constantly at an intermediate parameter regime (such as E_{J} = 10 GHz) so that selection rules hold while the suppressed but still nonvanishing 0–1 transition is enabled by enhancing the power of the classical drive.
Note that due to the condition of dispersive regime Eq. (9), the QND interaction strength χ has to decrease when the number of coupled qubits N increases due to resonance enhancement. According to the constraint \(g{\mathrm{/\Delta }} \ll 1{\mathrm{/}}\sqrt N\) \(\left( {{\mathrm{\Delta }} \equiv {\mathrm{Min}}\left {{\mathrm{\Delta }}_{ij}} \right} \right)\), one can fix g and increase the detuning magnitude Δ and get the asymptotic scaling \(\chi = g \cdot (g{\mathrm{/\Delta }}) \ll g^2{\mathrm{/}}\sqrt N\). This scaling is exponentially better than a previous scheme where multispin interactions are generated perturbatively^{39} with exponential decreasing interaction strength with the length of the string, i.e., O(g^{N}/Δ^{N−1}).
For small N [i.e. O(10)], it is possible to remedy the insignificant decay of maximum interaction strength due to resonance enhancement by varying the parameters (external flux or E_{J}) of individual fluxoniums such that frequency of different qudits (\(\epsilon _{l,j}\)) are detuned. The QND interaction strength χ will not decrease significantly because it contains contributions from multiple levels χ_{0l} and χ_{1l}. One can then avoid the asymptotic \(1{\mathrm{/}}\sqrt N\) scaling by modular construction of multiple cavities with N ~ O(10) qubits together connected with quantum teleportation as discussed in Supplemental Information IX. Alternatively, instead of obtaining the QND interaction perturbatively as the above scheme, it is in principle possible to directly engineer the QND (crossKerr) interaction such as utilizing nonlinear coupling with Josephson junctions.^{30}
Although we focus on fluxonium qubits here, one can generate QND interaction in more general cases for other qubits such as transmons. In those cases, one can detune the qubit frequency to avoid unwanted flip–flop interactions [for N ~ O(10)], or using a balance cavity mode as discussed further in Supplemental Information III.
Coupling to multiple ancillary modes with parametric coupler
In order to gain further parallelizability and shorten the time complexity, one can couple the qubits to multiple ancillary cavity modes as mentioned in the previous section, which certainly poses additional experimental challenges. One first needs to selectively address the qubits on different strings with a certain cavity mode which is usually distributed extensively and touches all the qubits. Second, one needs to couple the qubits dispersively to cavity modes with different frequencies. These two challenges can be solved by one trick, i.e., parametrically modulating the coupling of the qubits to the transmissionline cavity. One option is to periodically modulate the flux in the inductive coupler shown above in Fig. 3b (see e.g. refs. ^{58,59}) with multiple tones, i.e. \(g_j[{\mathrm{\Phi }}(t)]\) = \(\mathop {\sum}\nolimits_\nu {\kern 1pt} \tilde g_{\nu ,j}{\rm{cos}}(f_\nu t)\), where j labels the qubit and f_{ ν } represents the modulating frequencies, with f_{0} = 0 (static coupling). The scheme is illustrated in Fig. 3d.
The multitone modulation technique is mature in microwaveengineering and turns out to be a valuable computational resource. The weight \(\tilde g_{\nu ,j}^\prime\) and driving tones f_{ ν } are controllable. We choose f_{ ν } such that the qubit frequency \(\epsilon\) is upconverted to a frequency close to but still offresonant with the sideband ancillary tones (f_{ ν }). In this case, they are dispersively coupled by the QND interaction H_{QND} = \(\mathop {\sum}\nolimits_\nu {\kern 1pt} \mathop {\sum}\nolimits_j {\kern 1pt} \tilde \chi _{\nu ,j}a_\nu ^\dagger a_\nu \sigma _j^z\) with strength \(\tilde \chi _{\nu ,j} = \left( {\tilde \chi _{02}^{\nu ,j}  \tilde \chi _{13}^{\nu ,j}} \right){\mathrm{/}}2\), where \(\tilde \chi _{ll^{\prime}}^{\nu ,j} = \tilde g_{\nu ,j}^2{\mathrm{/}}(\epsilon _l  \epsilon _{l^{\prime}}  \omega _\nu + f_\nu )\). Note that f_{ ν } can decrease the detuning to make the interaction sizable. We choose \(\tilde g_{\nu ,j}\) such that each qubit is only coupled to the tones of the selected strings, as illustrated in Fig. 3d with multiple colors. As we see, the inductive couplings of qubits 4 and 5 are constant such that the qubits are only dispersively coupled to the fundamental mode a_{0}, while the couplings of qubits 1 and 8 are modulated by three tones and hence connect the qubits to four cavity modes, etc. It is clear that the number of cavity modes one can upconvert (or downconvert) to is limited since the upconverted detuning has to be made different to avoid crosstalking between different ancillae modes, but one should be able to couple 10–20 modes. To couple more ancillae, the solution is again teleportationbased modular architecture discussed in Supplemental Information IX. As we will discuss in the following section, for a Fermi–Hubbard model on a N × N square lattice in real space, the number of modes one needs to couple to is N. Therefore, for a 100qubit system which can be realized in the near future for a shortcircuit algorithm still requiring no quantum error correction, it is possible to realize our parallelization scheme.
Time complexity
In the previous sections, we focused on how to exponentiate a single term h_{ p } in the system Hamiltonian H = \(\mathop {\sum}\nolimits_p {\kern 1pt} h_p\). In the following, we compare the time complexity (circuit depth) of our cavity–QED approach with the conventional approach of a single Trotter step e^{−iHΔt}.
Fermi–Hubbard model
As the first example, we consider the spinful 2D Fermi–Hubbard model in realspace and on an N × N square lattice. We use qubits on two sublattices to encode fermions with different spin s = ↓ (purple) or s = ↑ (yellow) as shown in Fig. 4. The spinful Fermi–Hubbard model is a restricted form of Eq. (1) given by
where j → (n_{ x }, n_{ y }) is a twocomponent label for the 2D sublattice. The first and second terms represent hoppings and onsite Hubbard interaction, respectively. The types of terms and their corresponding time complexity is listed below (for more details see Supplemental Information V).
(1 and 2) Onsite Hubbard interaction and Horizontal hopping: translates to ZZ interaction and 2local flip–flop interaction without string in the qubit representation, both of which have O(1) circuitdepth. (3) Vertical hopping (even and odd): typically contains a “snakeshape” JW string (Fig. 4) and hence dominates the time complexity.
With one transmissionline cavity coupled to each pair of rows, one can parallelize the vertical hopping terms (see Supplemental Information V for details). For the vertical hopping between the same pair of rows, one can exponentiate these terms in series, resulting in the Trotter step circuit depth (time complexity) O(N). With the multimode scheme shown in Figs. 2f and 3d, one can exponentiate these terms and reduce the depth to O(1). In contrast, the conventional approach needs O(N^{2}) due to the linear overhead of implementing the CNOT ladder in Table 1.
The generic Coulomb Hamiltonian
For the generic Coulomb Hamiltonian described in Eq. (1), which is the relevant model for quantum chemistry or strongly correlated electronic materials simulated in reciprocal space, the indices i, j, k, and l are typically not neighbors. The type of terms that dominate the computational resource is the fourlocal interaction term \(V_{ijkl}c_i^\dagger c_j^\dagger c_kc_l\), which requires a sequence of O(N^{4}) unitary transformations for a system with N orbitals (i, j, k, l = 1, 2, …, N) in a single Trotter evolution step due to all possible choices of the four fermion indices. Taking into account the JW string, which has length of O(N), the Trotter step circuit depth of the conventional approach becomes O(N^{5}).^{60}
For our cavity–QED approach, we list the circuit depth for the two approaches. (1) Series: O(N^{4}), due to the reduction of the linear overhead of the Jordan–Wigner string. (2) Parallel: O(N^{3}), assuming N ancilla cavity modes. The remaining O(N^{3}) terms cannot be exponentiated in parallel because they do not commute with each other (e.g. when the first index i coincide, but the remaining three indices j, k, and l are all different). However, note that for an actual quantum chemistry Hamiltonian, although the total number of terms scales as O(N^{4}), a large number of integrals vanish between distant orbitals or due to symmetry. The number of noncommuting terms also scales as O(N^{3}) though similarly sparse. This can be seen from the example molecules discussed in Table 1 (operator information collected from refs. ^{6,7}), which has typically only O(N) to less than O(N^{2}) noncommuting terms (equivalent to the minimum number of commuting groups listed in the table). Therefore, there is a huge potential for parallelization in practice.
Summary of the comparison between cavity–QED and conventional approaches
Here, we summarize and compare the various properties of the cavity–QED scheme versus the conventional scheme, as shown in Table 2.
In order to compare both schemes, we first compare their gate time. With the stateoftheart technology, the secondorder QND interaction strength between qubits and cavity with the form \(\chi \mathop {\sum}\nolimits_j {\kern 1pt} a^\dagger a\sigma _j^z\), can typically reach about 50–100 MHz,^{30} corresponding to gate time of 20–40 ns. On the other hand, the conventional approach needs nearestneighbor CNOT gates between qubits, coming from the secondorder ZZ interaction, \(\frac{{4g^{\prime 2}}}{\eta }\mathop {\sum}\nolimits_{i,j} {\kern 1pt} \sigma _i^z\sigma _j^z\) (e.g. due to the thirdlevel contribution in the context of transmon qubits,^{61} where η is the nonlinearity of the transmon). The typical strength of the ZZ interaction is around 50 MHz,^{32} corresponding to a gate time of 40 ns. Since both types of interactions are of perturbative nature (up to second order), the gate time in both cases are of the same order of magnitude. The relevant parameters are summarized in Table 2. We also include the asymptotic prefactor \(\sqrt N\) (reduces to \(\sqrt {{\mathrm{log}}{\kern 1pt} N}\) with the Bravyi–Kitaev encoding) of the cavity–QED gate time due to the dispersive regime condition Eq. (9), which can be remedied by the modular architecture connecting multiple cavities (Supplemental Information IX). The average number of strings (cavity ancilla modes) a single qubit touches simultaneously is of O(10), so one does not need to worry about crosstalk between the ancillae due to frequency crowding in these cases either.
We emphasize that having a scheme with a shorter operation time in each Trotter step enables more evolution steps within the coherence time of the system, and hence increases the precision of the algorithms, such as phase estimation. Besides the cavity–QED scheme presented in this paper, there are some other schemes which can reduce the overhead due to the nonlocal string operator, such as refs. ^{17,18}. We compare our scheme with theirs in Supplemental Information X.
Another significant advantage of our scheme over the conventional scheme is the gate fidelity, in particular, the fidelity due to the control pulses. In the conventional scheme, in order to implement N CNOTs in the CNOT ladder, one has to send N control pulses. Assuming the fidelity is F for each pulse, the overall fidelity due to imperfect pulse becomes F^{N} as shown in Table 2. On the other hand, in the case of our manybody gate, one can actually just use a single control pulse with error F′ to detune the cavity frequency. In this case, the overall fidelity due to imperfect pulse is just F′, which does not have an exponential decay. Therefore, our collective manybody gate has a significant advantage in terms of quantum control and pulse fidelity.
Numerical simulation in the presence of decoherence
In this section, we numerically simulate and compare different approaches with two simple but representative experiments: (1) a 2D spinful Fermi–Hubbard model on a 2 × 2 lattice (simulated by 8 qubits). (2) A quantum chemistry problem, i.e., the outer shell electrons of a BeH_{2} molecule (simulated by 6 qubits), which has been simulated with superconducting qubits in a recent experiment.^{7}
The simulation takes into account decoherence of qubits and cavity, represented by the jump operators \(l_j = \tilde l_j\sqrt {{\mathrm{\Gamma }}_j}\), where Γ_{ j } is the corresponding decay rate and \(\tilde l_j\) the normalized operator. The types of jump operators of our numerical simulation is listed in the caption of Fig. 5, along with the realistic estimation of experimental parameters chosen according to ref. ^{31}.
In particular, we simulate the Kitaev phase estimation protocol (see Supplemental Information VIII) for both systems and for the Fermi–Hubbard model also the measurement of spectral function A(ω) = −2Im[G(ω)], where G(ω) is extracted from the Fourier transform of the dynamical correlators including \(\left\langle {\psi \left {{\kern 1pt} c_i(t)c_j^\dagger (0)} \right\psi } \right\rangle\) (see Supplemental Information VII). Since both measurement protocols involve time evolution U(t), the dissipation of the system will affect the measurement result, as shown in Fig. 5. We compare four different situations: the ideal situation without dissipation, the conventional approach, and the cavity–QED approach in series and in parallel, respectively. Since each approach needs different operation time per Trotter step, the effects of dissipation are different.
For the Fermi–Hubbard model, we use JW encoding in all cases and three transmission line cavities are needed to couple each pair of rows (four rows in total) in parallel. For the BeH_{2} molecule, we use the modified BK encoding discussed in ref. ^{7}. With this encoding, there are a total of 164 terms, which can be divided into eight groups, where all the terms in the same group commute with each other, as shown in Table 1. In this case, one can reduce the circuit depth to eight by exponentiating all the terms in the same group in parallel with multiple ancilla modes in the same central cavity. This would require about 20 tones in the flux modulation using the trick in Fig. 3d. On the other hand, the series cavity–QED approach will exponentiate all the terms sequentially with a single cavity ancilla.
Regarding to the phase estimation protocol in Fig. 5a,c, the cavity ancilla expectation \(\left\langle {Z_{\rm a}(t)} \right\rangle\) (PauliZ) oscillates in time in the ideal case, i.e. \(\left\langle {Z_{\rm a}(t)} \right\rangle = {\rm{cos}}(E_{\rm g}t)\), where E_{g} is the groundstate energy of the prepared eigenstate. Nevertheless, in the presence of decoherence, the signal decays significantly in time, while the peaks in frequencyspace signal \(\left\langle {Z_{\rm a}(\omega )} \right\rangle\) also shrinks due to dissipation. For the Fermi–Hubbard model in (a), we prepare the ground state in the beginning, and one can see that E_{g} (shown by the dashed line) can be clearly resolved in the biggest peak in \(\left\langle {Z_{\rm a}(\omega )} \right\rangle\) in the blue and purple curves (ideal dissipationless case). The purple curve has a Fourier transform over the period 0 ≤ t ≤ 1000, namely 10 times long as the others, and hence has much better resolution. With dissipation, the signal dies out in a short time. While this peak still has the correct position for the cavity–QED parallel approach (red dashed), it shifts slightly for the series approach (green dashed) and becomes obscured in the conventional approach local (light blue dashed). For the phase estimation in BeH_{2} molecule in (c), we see that the parallel cavity–QED approach (red dashed) approximates the dissipationless signal (blue) with almost the same resolution of the groundstate energy while the height of the peak is reduced. The series cavity–QED approach (green dashed) has significant broadening in the resolution, while the conventional local approach has all the peaks being smeared out and is hence hard to tell the actual energy.
For the spectral function measurement in panel (b) for Fermi–Hubbard model, we prepare the initial state as the ground state. The two biggest peaks correspond to the hole (left) and particle (right) resonance, and the distance is approximately U, namely the Mott gap. We can see that the dissipation effect leads to the shrinking and asymmetry of the two peaks. The shrinking is proportional to the operation time of different approaches. The asymmetry is due to the fact that the qubit has much larger loss rate than absorption rate as listed in the figure caption. Due to our encoding of 0 (1) electron as spin up (down) of the qubit, the qubit loss induces loss of holes but not particles. Therefore, the hole peak (left) shrinks more than the particle peak. In practice, one could choose two different ways of encoding and average the signal to get rid of this asymmetry.
Conclusion and discussion
In this article, we have shown that, in the context of cavity/circuitQED architecture, the use of the common cavity modes greatly simplifies the nonlocal stringlike encoding needed for fermionic simulation, such as Jordan–Wigner and Bravyi–Kitaev transforms. In particular, we are able to get rid of a polynomial overhead, i.e., N^{2} of the Trotterstep circuit depth in the conventional local approach, which reduces the time complexity of the simulation for a given precision and in turn reduces the decoherence effects. The nonlocal quantum control and parallelization of multiple ancillacontrolled processes developed in this paper may have profound applications in many others areas, such as quantum information processing, lattice gauge theory simulation and measurement of entanglement spectrum in quantum manybody systems.^{62}
Methods
Derivation of parallelizations with multiple ancillae
Here, we show the detailed derivation of multiancilae parallelization mentioned above. We use conditional string\(\overline Z\) operations with multiple cavity ancilla modes, namely
where each ancilla mode a_{ ν } is dedicated to a particular string ν. This collective gate can be realized by dispersively coupling qubits simultaneously to multiple modes resulting in the QND interaction
As explained below, by proper conditional rotations, we can achieve a generic conditional string\(\bar {\cal S}\) in different Paulibases, i.e. \(C_{\bar {\cal S}_\nu }\), where the \(\overline Z _\nu\) string in Eq. (12) is replaced by \(\bar {\cal S}_\nu\). We consider the case where all the strings commute with each other, i.e. \(\left[ {\bar {\cal S}_\nu ,\bar {\cal S}_{\nu ^{\prime}}} \right] = 0\). Thus the conditionalstring also commutes, i.e. \(\left[ {C_{\bar {\cal S}_\nu },C_{\bar {\cal S}_{\nu ^{\prime}}}} \right] = 0\). Therefore, following the derivation in Eq. (7), we can reach the identity
where \(R_x^\nu\) and \(X_{{\rm a},\nu }\) is the xaxis rotation and PauliX operator of the ancilla mode ν. If all the ancillae are initiated at \(\left + \right\rangle _\nu\), the exponentiation of multiple strings is achieved in parallel, i.e. \(\mathop {\prod}\nolimits_\nu {\kern 1pt} {\rm e}^{{\rm i}\kappa {\rm{\Delta }}t\bar {\cal S}_\nu }\).
Now we consider how to convert the conditional\(\overline Z\) into conditional\(\bar {\cal S}\). We illustrate the idea with example shown in Fig. 2f]. This involves turning the head and tail of each string into PauliX operators. To achieve this, we split the \(C_{\bar {\cal S}_\nu }\) operator into two parts applied sequentially (order is arbitrary): the main \(C_{\overline Z _\nu ^1}\) string and the \(C_{\overline X _\nu ^2}\) part in the ends as shown in the green box in Fig. 2f. To achieve \(C_{\overline X _\nu ^2}\), we just need to sandwich the \(C_{\overline Z _\nu ^2}\) operators with Hadamards H_{ j } performed on the qubits in parallel. The application of all the \(C_{\overline Z _\nu }\) gates are performed in parallel with multimode QND interaction \(H_{{\rm{QND}}}^\prime\) Eq. (13). Therefore, the overall circuit depth of parallelizing N such hopping terms is of O(1). The generalization to arbitrary type is shown in Supplemental Information II.
Data availability
The data sets generated during and analyzed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
We thank Vladimir Manucharyan for the suggestions of the scheme using fluxonium qubits and providing experimental details and parameters. We thank Ignacio Cirac for pointing out the scaling of dispersive interaction. We also thank Peter Zoller, Jens Koch, and Eran Ginossar for helpful discussions. G.Z. and M.H. were supported by AROMURI, NSFPFC at the JQI, YIPONR, and the Sloan Foundation. The work by G.Z. was performed in part at Aspen Center for Physics, which is supported by National Science Foundation grant PHY1607611. J.D.W. acknowledges startup funds from Dartmouth College. This work was performed under the auspices of the U.S. DOE contract No. DEAC52 06NA25396 through the LDRD program at LANL.
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Zhu, G., Subaşı, Y., Whitfield, J.D. et al. Hardwareefficient fermionic simulation with a cavity–QED system. npj Quantum Inf 4, 16 (2018). https://doi.org/10.1038/s4153401800653
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