Quantum state preparation and nonunitary evolution with diagonal operators

Realizing nonunitary transformations on unitary-gate-based quantum devices is critically important for simulating a variety of physical problems, including open quantum systems and subnormalized quantum states. We present a dilation-based algorithm to simulate nonunitary operations using probabilistic quantum computing with only one ancilla qubit. We utilize the singular-value decomposition (SVD) to decompose any general quantum operator into a product of two unitary operators and a diagonal nonunitary operator, which we show can be implemented by a diagonal unitary operator in a one-qubit dilated space. While dilation techniques increase the number of qubits in the calculation, and thus the gate complexity, our algorithm limits the operations required in the dilated space to a diagonal unitary operator, which has known circuit decompositions. We use this algorithm to prepare random subnormalized two-level states on a quantum device with high fidelity. Furthermore, we present the accurate nonunitary dynamics of two-level open quantum systems in a dephasing channel and an amplitude-damping channel computed on a quantum device. The algorithm presented will be most useful for implementing general nonunitary operations when the SVD can be readily computed, which is the case for most operators in the noisy intermediate-scale quantum computing era.

Figure 1

Circuit for implementing a nonunitary diagonal operator $\widehat{\mathrm{\Sigma }}$ on $|\psi \rangle$ using only unitary gates with a one-qubit dilation. The final gate on each rail is the measurement gate, $H$ is the Hadamard gate, and the multiqubit gate is the diagonal operator in Eq. (1). The algorithm is successful when the ancilla is measured in state $|0\rangle$.

Figure 2

Circuit for implementing the nonunitary operator $\widehat{M}$ on $|\psi \rangle$ using only unitary gates with a one-qubit dilation. The final gate on each rail is the measurement gate, and all other gates are defined in the text. The only operator acting over the dilated space is the diagonal operator, and the linear combinations are performed using the Hadamard gates.

Figure 3

Real and imaginary parts of the off-diagonal element (coherence) of a two-level state in a $Z\otimes Z$ dephasing channel. The results from the quantum device (dots) are accurate for all times in the simulation, where the exact solution (lines) is generated classically. The results from the quantum computer are generated from density-matrix tomography.

Figure 4

The trajectory of a two-level system undergoing dephasing with a one-qubit environment is shown on a Bloch sphere with exact results represented by the dotted line and results from IBMQ Lagos represented by dots. The trajectory is in the $z=0$ plane, represented by the dark gray circle. One period is shown (about 60 ps), where the initial state, $\frac{1}{\sqrt{2}}\left(\right|0\rangle +|1\rangle )$, is the first red dot on the right. The system explores mixed states when it is in the interior of the sphere.

Figure 5

Time dynamics of a zero-temperature two-level system in an amplitude-damping channel. The results from the IBMQ Lagos device (dots) are in good agreement with the exact results (solid lines). Results were generated from a single run using the device maximum of 32 000 samples (shots) with error mitigation.