Digital quantum simulation of fermionic models with a superconducting circuit

One of the key applications of quantum information is simulating nature. Fermions are ubiquitous in nature, appearing in condensed matter systems, chemistry and high energy physics. However, universally simulating their interactions is arguably one of the largest challenges, because of the difficulties arising from anticommutativity. Here we use digital methods to construct the required arbitrary interactions, and perform quantum simulation of up to four fermionic modes with a superconducting quantum circuit. We employ in excess of 300 quantum logic gates, and reach fidelities that are consistent with a simple model of uncorrelated errors. The presented approach is in principle scalable to a larger number of modes, and arbitrary spatial dimensions.

The two-, three-and four-mode Trotter step pulse sequences are shown in Supplementary Fig. 1.

B. Initialization
The gate sequences for the initialization of the three-and four-mode simulation are shown in Supplementary Fig. 2. For the two-mode simulation the input state is: We use quantum process tomography to determine the χ matrix. We start by initializing the qubits into the ground state, and prepare input states by applying gates from {I, X/2, Y/2, X} ⊗2 . The process output is reconstructed by applying gates from the same group, essentially obtaining the 16 output density matrices. The χ matrix is then determined using quadratic maximum likelihood estimation, using the MATLAB packages SeDuMi and YALMIP, while constraining it to be Hermitian, trace-preserving, and positive semidefinite; the estimation is overconstrained. Non-idealities in measurement and state preparation are suppressed by performing tomography on a zero-time idle.
The χ matrices for processes ) are determined experimentally, and the matrix of process U 2 U 1 is computed from the experimentally obtained matrices following Ref. [1].
The used quantum circuits are

Supplementary Note 3. RANDOMIZED BENCHMARKING OFẐẐ AND THE TWO-MODE TROTTER STEP
The process fidelity of the exp(−i φ 2 σ z ⊗ σ z ) gate and the two-mode Trotter step are determined using interleaved Clifford-based randomized benchmarking [2][3][4]. This technique is insensitive to measurement and state preparation error, and determines the fidelity properly averaged over all input states, but it restricts the gates to have a unitary which lies within the group of Cliffords. As representative angles we have therefore used φ = π/2, and φ xx = φ yy = φ zz = π/2 for the Trotter step.
The data are shown in Supplementary Fig. 3. We start by measuring the decay in sequence fidelity of sequences of random, two-qubit Cliffords (black symbols). When interleaving we see an extra decrease of sequence fidelity, which can be linked to the process fidelity of the interleaved gate. We find that the exp(−i π 4 σ z ⊗ σ z ) gate and the Trotter step have errors of 0.020 and 0.074, respectively. We note that these values are consistent with estimation by adding individual gate errors (main Letter).

Supplementary Note 4. DIGITAL ERROR
The Trotter expansion introduces digital errors due to discretization. A full analysis of the digital error for the used model can be found in Ref. [5]. For the time-independent model, the two-mode simulation has zero digital error. For the three-and four-mode simulation the full evolution (solid lines), exact digital solution (open symbols connected by dashed lines), and fidelities due to digital error are shown in Supplementary Fig. 4.
For the time-dependent model we find a negligible digital error for two modes, and a significant error for three, see Supplementary Fig. 5. The large error for three modes arises from having to approximate a larger Hamiltonian, as well as using only a single step.
The tunable CZ φ gate works by tuning the frequency of one of the qubits to approach the avoided level crossing of the |ee and |gf states, using an adiabatic trajectory [6]. For large phases we need to closely approach the avoided level crossing, inducing state leakage.
To minimize such leakage we have chosen to increase the length of the CZ φ gate from a typical 40 ns [7] to 55 ns. However, for large phases (> 4.0 rads), see Supplementary Fig. 6a, we still see a considerable amount of leakage, see the Supplementary Fig. 6b. By choosing the leaked state population as a fitness metric, and using Nelder-Mead optimization in a similar approach to Ref. [8] to tune waveform parameters, see Supplementary Fig. 6c-d, we can significantly suppress leakage. We note that this optimization took approximately one minute in real time.

Supplementary Note 6. ASYMMETRIC HUBBARD MODEL
Here, we include the analysis of the fermionic asymmetric Hubbard model for 4 qubits employed in the Letter. Firstly, we present the model in terms of spin operators via the Jordan-Wigner transformation, and describe different limits of the model. Secondly, we analyse the digital quantum simulation in terms of Trotter steps involving the optimized gates (CZ φ ). The asymmetric Hubbard model (AHM) is a variation of the Hubbard model that describes anisotropic fermionic systems. Here, we are going to consider this model for two different fermionic species, that could represent spins, interacting with each other by the Coulomb term, and two lattice sites. The operators for this model have two indices, A ij , where i and j indicate the site position and kind of particle, respectively. Since the fermions might have different masses, we have no reason to assume that the hopping terms will be the same. We can write the Hamiltonian for two sites, x and y, and two kinds of fermions, 1 and 2, as where b † mi and b mi are fermionic creation and annihilation operators of the kind of particle i for the site m. For the main The Jordan-Wigner transformation will be used in our derivation to relate the fermionic and antifermionic operators with tensor products of Pauli matrices, which are operators that we can simulate in the superconducting circuit setup.
This transformation is based on a mapping between fermionic operators and spin-1/2 operators. In this case, the relations are After this mapping, Hamiltonian (1) is rewritten in terms of spin-1/2 operators as where the different interactions can be simulated via digital techniques in terms of single qubit and CZ φ gates.

A. Gate decomposition
We consider the digital quantum simulation of the dynamics of Hamiltonian (3). The Trotter expansion consists of dividing the time t into n time intervals of length t/n, and applying sequentially the evolution operator of each term of the Hamiltonian for each time interval. In this case the evolution operators are associated with the different summands of the Hamiltonian.
In order to describe the digital simulation in terms of Trotter steps involving the optimized gates (CZ φ ), we will first consider the Hamiltonian in terms of exp[−i(φ/2)σ z ⊗ σ z ] interactions. We take into account the relations where R j (θ) = exp(−i θ 2 σ j ) is the rotation along the j coordinate of a qubit. In these expressions the rotations are applied on the two qubits of the product.
The evolution operator associated with Hamiltonian (3) in Note that, in principle, the ordering of the gates inside a Trotter step does not have a sizable effect as far as there are enough Trotter steps. Here, the number of Trotter steps is limited (n approximately ≤ 10) and different orderings will have different results. The different values in the orderings differ in a O(1) constant, while the global digital error depends on the number of Trotter steps n as 1/n (the difference in errors due to different orderings does not depend on n).
If we consider the Trotter error, the fidelity could increase with an optimal ordering where we group terms of the Hamiltonian that commute with each other. Nevertheless, from the experimental point of view, the operators can be rearranged in a more suitable way in order to optimize the number of gates and eliminate global phases. In this sense, we must look for the optimal ordering by considering both aspects.
Here, we simply rearrange the operators in order to optimize the number of gates. If we consider that R j (α) + R j (β) = R j (α + β), then where we use the prime notation in the rotation to distinguish between gates applied on different qubits. This decomposition between even and odd Trotter steps is suitable in order to simplify rotations in x and y, and, therefore, avoid higher number of gates.
The sequence of gates for one odd Trotter step in the digital simulation of the Hubbard model with four qubits is and for one even Trotter step: The gates A i and B j are two-qubit gates in terms of the exp[−i(φ/2)σ z ⊗ σ z ] interactions: A i = exp(i Vi 2 σ z ⊗ σ z t n ) and B j = exp(−i Uj 4 σ z ⊗ σ z t n ). The Z i gates are single qubit rotations: Z i = exp(−i Ui 4 σ z t n ), and X α and Y α are rotations along the x and y axis, respectively.
The exp[−i(φ/2)σ z ⊗ σ z ] interaction can be implemented in small steps with optimized CZ φ gates. The interaction is The quantum circuits for simulating this are shown in the main Letter.

B. Particular case of the model
In order to avoid the gate B x between the first and the fourth qubit, we can consider a particular case of the asymmetric Hubbard model, where U x = 0. In this case, the circuit is the same but without the B x and the Z x gates. That is, for one odd Trotter step: Y π/2 Y −π/2 X π/2 X −π/2 and for one even Trotter step: It is important to note that, for n = 2 Trotter steps, the red gates cancel each other, and we reduce the number of gates that should be applied. For n > 2, the blue gates also cancel each other except in the beginning and in the end of the quantum simulation.

C. Digital quantum simulation of the model
The relation among the values of the parameters in the numerical simulations and the values of the phases in the gates is the following In summary, the fermionic asymmetric Hubbard model with two excitations, one for each kind of fermion, has been analysed and expressed in terms of simulatable spin operators. We have considered the digital quantum simulation in terms of Trotter steps involving the optimized gates (CZ φ ). This is the four-mode system experimentally simulated in the main Letter.