Physical Implementability of Linear Maps and Its Application in Error Mitigation

Abstract

Completely positive and trace-preserving maps characterize physically implementable quantum operations. On the other hand, general linear maps, such as positive but not completely positive maps, which can not be physically implemented, are fundamental ingredients in quantum information, both in theoretical and practical perspectives. This raises the question of how well one can simulate or approximate the action of a general linear map by physically implementable operations. In this work, we introduce a systematic framework to resolve this task using the quasiprobability decomposition technique. We decompose a target linear map into a linear combination of physically implementable operations and introduce the physical implementability measure as the least amount of negative portion that the quasiprobability must pertain, which directly quantifies the cost of simulating a given map using physically implementable quantum operations. We show this measure is efficiently computable by semidefinite programs and prove several properties of this measure, such as faithfulness, additivity, and unitary invariance. We derive lower and upper bounds in terms of the Choi operator's trace norm and obtain analytic expressions for several linear maps of practical interests. Furthermore, we endow this measure with an operational meaning within the quantum error mitigation scenario: it establishes the lower bound of the sampling cost achievable via the quasiprobability decomposition technique. In particular, for parallel quantum noises, we show that global error mitigation has no advantage over local error mitigation.

NISQ: Error Correction, Mitigation, and Noise Simulation

“…The first is that N is invertible, and N −1 is CPTP. For an invertible N , the inverse N −1 is unique, and is Hermitian preserving (HP) and trace preserving (TP) [11]. If the dimensions of input and output space are the same, the channel has a CPTP inverse iff the channel is an unitary channel [12,13].…”

“…For non-invertible channels, this construction Eq. (11) does not satisfy the condition of generalized inverse (N • N + •N = N when the dimension of the nilpotent Jordan block is greater than one), and we will call N + the quasiinverse.…”

“…Recall that this is true iff N i is a unitary channel. Then in principle one can insert an additional gate implementing N −1 i after each U i to fully invert the noise effect [11]. In reality, the experimentally obtained noise models are Ñi .…”

Information recoverability of noisy quantum states

“…Both practically relevant and theoretically interesting, this framework combines an efficient way to manipulate quantum states and general quantum operations. Note that similar protocols are employed in predicting properties of quantum states [14,22] and mitigating quantum error [23][24][25][26][27]. In particular, we suppose that multiple copies of a noisy state N (ρ) are available.…”

“…We analytically obtain the values of this measure along with the concrete retrieving protocols for generalized amplitude damping channels and depolarizing channels. Our retrieving costs set ultimate limits on the cost required for shadow retrieving, and the corresponding protocols outperform existing probabilistic error cancellation (PEC) methods [23][24][25][26][27]. Specifically, we employ our method in tasks of estimating the ground state energies of several molecules with VQE under the assumption that the noise model of the entire circuit is known.…”