Universal quantum simulation of single-qubit nonunitary operators using duality quantum algorithm

Quantum information processing enhances human’s power to simulate nature in quantum level and solve complex problem efficiently. During the process, a series of operators is performed to evolve the system or undertake a computing task. In recent year, research interest in non-Hermitian quantum systems, dissipative-quantum systems and new quantum algorithms has greatly increased, which nonunitary operators take an important role in. In this work, we utilize the linear combination of unitaries technique for nonunitary dynamics on a single qubit to give explicit decompositions of the necessary unitaries, and simulate arbitrary time-dependent single-qubit nonunitary operator F(t) using duality quantum algorithm. We find that the successful probability is not only decided by F(t) and the initial state, but also is inversely proportional to the dimensions of the used ancillary Hilbert subspace. In a general case, the simulation can be achieved in both eight- and six-dimensional Hilbert spaces. In phase matching conditions, F(t) can be simulated by only two qubits. We illustrate our method by simulating typical non-Hermitian systems and single-qubit measurements. Our method can be extended to high-dimensional case, such as Abrams–Lloyd’s two-qubit gate. By discussing the practicability, we expect applications and experimental implementations in the near future.

Special cases. If one UE-parameter f k (t) in the main text is equal to zero, F(t) can be expressed by two UE-terms in some special cases.
(a) In the case of f 0 (t) = 0, In fact, it is accordance with the phase matching condition I [see Eq. (S2)] if we set f 0 (t) = 0 · e iθ 3 . From Eq. (S3) to (S5), it is not difficult to calculate the parameters and matrices in the main text that a 0 , a 1 , V 0 and V 1 are equal to c 2 , c 1 , U 2 and U 1 , respectively.

3/6 Explicit expressions of UEs in the section of 'Illustration' PT-symmetric Hamiltonians.
A time-independent PT-symmetric two-level Hamiltonian has a general form of satisfying [PT, H PT ] = 0, where P = σ 1 is the parity operator and T is the time-reversal operator having the effect that i → −i.
The eigenvalue of H PT are ε ± = rcosθ ± s 2 − r 2 sin 2 θ , and we set the difference of them as ω is either real or imaginary depending on the system is in exact-PT or PT-broken phase, respectively. We now expand the time-evolution operator e −i(t/h)H PT as the form of Eq.
(2) in the main text. The four UE-parameters are and Terms in the brackets are always real, while terms outside the brackets are the phases. the phase angles θ 0 and θ 3 of f 0 and f 3 satisfy phase matching condition I in Eq. (S2).

Anti-PT-symmetric Hamiltonians.
A Hamiltonian is anti-PT-symmetric if {PT, H APT } = 0. In fact, the general form of the Hamiltonian is equal to the imaginary unit i times H PT . For a time-independent two-state system, The eigenvalues are ε ± = i rcosθ ± √ s 2 − r 2 sin 2 θ , and we set the difference of them as λ = 2 r 2 sin 2 θ − s 2 = 2i s 2 − r 2 sin 2 θ = iω, where ω is that in Eq. (S23). The unitary expansion of the time-evolution operator e −i(t/h)H APT can be expanded as Eq.
(2) in the main text, and the four UE-parameters are and f 3 (t) = i r sin θ iλ e i λt 2h − e −i λt 2h e th r cos θ .
Phases of the UE-parameters are outside the brackets, and they meets the phase matching condition II in Eq. (S6). Therefore, a time-independent anti-PT -symmetric two-level system can be simulated by two qubits.

P-pseudo-Hermitian Hamiltonians.
A P-pseudo-Hermitian Hamiltonian, say H PPH , satisfies PH † PPH P = H PPH . For a time-independent two-level system, of which the eigenvalues are ε ± = rcosθ ± us − r 2 sin 2 θ . The difference of the two eigenvalues is The time-evolution operator e −i(t/h)H PPH can be expanded as the form of Eq.
(2) in the main text with the four UE-parameters as follow and Comparing with phase matching conditions (see Sec. 2), the phases of the four UE-parameters meet both the condition I and IV in Eq. (S2) and (S16).
(i) In an arbitrary phase, the time-evolution operator e −i(t/h)H APPH has the form of Eq.
(2) in the main text, where the four UE-parameters are equal to f 0 (t) = 1 2 e i λt 2h + e −i λt 2h e th r cos θ ,
Noticing that λ is either real or imagine, the terms in the brackets are real. The phases of the four UE-parameters are 0, 0, 0 and π/2, so none of the phase matching conditions are met. Therefore, three qubits are necessary to simulate this system in a general case by either the six-or eight-dimension protocols.
(ii) In some special cases (see Sec. 2.5), the system can be simulated by two qubits. For example, when the system is in the anti-PT-symmetric phase, i.e. s = u, one of the UE-parameters f 2 (t) of the time-evolution operator e −i(t/h)H APPH becomes zero. As discussed in the first section above, it meets both the phase matching condition II [Eq. (S6)] and IV [Eq. (S16)]. So the system can be simulated by two qubits in this anti-PT-symmetric phase.
Single-qubit measurements: σ 3 -measurement. As a simple case, we illustrate how to simulate a single-qubit σ 3 -measurement. Assuming |0 and |1 are the two eigen states of σ 3 , the effect of this measurement is to apply either of the two nonunitary matrices with probabilities e 0 |ψ e or e 1 |ψ e , respectively. We now apply a U = H 2 , C 0−σ 0 , C 1−σ 3 and another H 2 as quantum circuit in Fig.(6) in the main text. Now, the system evolves to a state |0 a M 0 |ψ e + |1 a M 1 |ψ e .
Finally, a measurement is performed on the ancillary qubit. If an output |k a is obtained, the work qubit will evolve to |k e with a probability of e k |ψ e (k = 0, 1). Although the ancillary qubit is annihilated, the work qubit is kept for further use.

The Abrams-Lloyd's gate
The explicit forms of the UE-terms and parameters of N = 4