Fig. 1. Architecture of quantum dynamics simulation using the modified multi-product algorithms with OAA in linearly combined unitary circuits. Multi-product algorithms (yellow boxed) can linearly combine multiple low-order Trotterizations (S1l) to improve the simulation accuracy of quantum dynamics. The LCU circuits (blue boxed) represent the quantum hardware that can implement the multi-product algorithms, and the LCU can be physically realized by a sequence of quantum controlled-unitary operations. There are many failing outcomes in the conventional LCU algorithms, resulting in a low probability of success. We adopt the OAA method (green boxed) to amplify the success probability of the algorithm. This architecture allows enhanced multi-product algorithms for quantum simulation of dynamic evolution, with improved accuracy and a high success probability. By properly choosing the coefficients ci, iteration numbers L(i), and N so that sin2((2N+1)arcsinP)=1, we can amplify the success probability up to unit.

Fig. 2. A programmable LCU quantum simulator in a silicon nanophotonic chip. (a) Quantum circuit for a four-qubit LCU circuit. The top two qubits refer to an ancillary register initialized in |00⟩ state, and the bottom two qubits refer to a data register prepared in |ψ⟩ state. The LCU is enabled by a sequence of quantum controlled-unitaries. {A1,A2,A3,A4} are arbitrary two-qubit unitaries, in which the target Hamiltonian is loaded in. For C(CT), its i-th element in the first column (row) is given by (ci/∑q=1d|cq|)1/2. (b) OAA quantum circuit. W represents the LCU circuit described in (a), and R can flip the amplitude of basis |00⟩ in the ancillary register. The bracketed circuit is repeated for N times to amplify the success probability of the LCU-based quantum algorithms when reading out outcomes by measuring the ancillary register in the |00⟩ basis. (c), (d) Modularized scheme of the LCU quantum simulator. The simulator can implement the LCU and OAA circuits in (a) and (b) and can be fully reprogrammed and reconfigured to implement the quantum algorithms. Inset: photograph of an LCU quantum simulator in a silicon chip. The brackets contain a footprint with one periodic repeating structure.

Fig. 3. Numerical analysis of simulation errors. Simulation error for the standard Trotter formula, multi-product, and modified multi-product algorithms. The simulation error is defined as ‖|ψ0⟩−|ψ′⟩‖, where |ψ0⟩ is the exact state and |ψ′⟩ refers to the output state of quantum algorithms. The curves and shaded regions represent the average value and ±σ of error, respectively, estimated from 1000 randomly generated initial states. The Hamiltonian shown in Eq. (14) that describes the interaction of an electron spin and a nuclear spin is used. The iteration numbers (l) of the Trotter formula, original, and modified multi-product algorithms are 96, {1,2,3,96}, and {4,8,16,32}, respectively. Note that the additional circuit depth and error of OAA have been taken into consideration so that these algorithms require the same circuit depth.

Fig. 4. Emulating quantum dynamics in an LCU quantum simulator. (a) Comparison of standard Trotterization, multi-product, and modified multi-product algorithms, in terms of their simulation accuracy. In the data register, an initial state |01⟩ is input into the simulator, and dynamics for the state of |11⟩ is measured. (b) Simulated error for different algorithms. Error is defined as the discrepancy between the exact solution and computed results. Points denote experimental data, and lines refer to theoretical predictions where algorithmic errors have been included. Both theoretical prediction and experiment show that the modified multi-product algorithm can achieve higher precision than the Trotterization and multi-product algorithms when t≤30. Error bars (±1σ) are estimated from the photon Poissonian statistics.

Fig. 5. Experimental results of modified multi-product algorithms with and without OAA. (a)–(d) Quantum dynamics of four states {|00⟩,|01⟩,|10⟩,|11⟩}, simulated using the modified multi-product algorithm without the OAA. (e)–(h) Quantum dynamics of four states {|00⟩,|01⟩,|10⟩,|11⟩}, simulated using the modified multi-product algorithm with the OAA. (i) Measured success probability for the modified multi-product algorithm with OAA (99.79% for t<30) and without OAA (26.33% for t<30). The success probability of the modified multi-product algorithms thus can be amplified by OAA. Points denote experimental data, and lines refer to their theoretical predictions where algorithmic errors have been included. Error bars (±1σ) are estimated from the photon Poissonian statistics.

Fig. 6. A generalized scheme for implementing LCU circuits. (a) A generalized LCU scheme that can be realized by translating multi-qudit GHZ states |GHZ⟩n+1,d [in (b)] to the LCU circuits. Pi generates the input state |ψ⟩i. (c) An extended 2n-qudit LCU circuit with deep circuit depth. P^i generates the input state |ψ⟩i. In the k-th iteration, when the control qudit state is |j⟩, the unitary operation applied to the i-th qudit is represented as O^i,j(k), or reversely.