Realizing Universal Non-Markovian Noise Suppression

Abstract

Non-Markovian noise, arising from environmental memory effects, is the most general and challenging form of noise in quantum computing, and is typically difficult to characterize and suppress. Here, we analyze and experimentally demonstrate a non-Markovian noise suppression scheme inspired by quantum purification protocols. We theoretically prove that, even without noise calibration and assumptions on specific noise models, the scheme can exponentially reduce non-Markovian error rates with respect to the ancillary system size. We implement the protocol using nuclear spins, demonstrating that non-Markovian noise can be suppressed for both unitary operations and non-unitary channels. The observed fidelities and process tomography show close agreement with theoretical predictions, confirming the practicality and effectiveness of the scheme.

Figures (17)

• Figure 1: An intuitive figure that shows the difference between Markovian and non-Markovian noises, where "E" and "M" stand for environment and memory, respectively. (a) Markovian noise: the interactions between system and environment at different time points are independent (memoryless). (b) Non-Markovian noise: the interactions at different time points come from the same environment, which keeps a memory. It can feed information from the past back into the system, leading to history-dependent noises.
• Figure 2: Non-Markovian noise suppression protocol, where the yellow circles and lines represent the Pauli twirling and the pink ones represent the effective Pauli noises after Pauli twirling. The summation represents averaging the Pauli operators under a specific probability distribution. (a) The noisy circuit with non-Markovian noise, represented with two explosion signs and a demon, and the Pauli twirling. (b) Pauli twirling turns a non-Markovian noise into two classically correlated Pauli noises at two time points, with $\rho=U_1 \rho_0 U_1^\dagger$ and $U=U_2$. (c) Purification circuit for non-Markovian noise. (d) The simplified circuit. The pink dashed box denotes the entire non-Markovian noise after purification; compared with (b), it affects the ideal gate $U$ more weakly.
• Figure 3: (a) Molecular structure of the five-qubit NMR sample, comprising three $^{19}$F and two $^{1}$H spins as qubits. The dashed lines indicate the strong couplings, with the corresponding coupling strengths (in Hz) labeled alongside. (b) Experimental circuit for non-Markovian noise suppression in the case of unitary gates. The yellow circles $P_{i,j}$, $P_{i',j'}$ and the depolarizing operations $\mathcal{D}$ denote uniformly random Pauli operations, used for Pauli twirling and resetting the ancillary register to the maximally mixed state, respectively. The non-Markovian noise is generated by the joint Hamiltonian evolution governed by the Hamiltonian $H=\vec{\omega}_1 \cdot \vec{\sigma}_1+\vec{\omega}_2 \cdot \vec{\sigma}_2 +J\sigma_{1z}\sigma_{2z}$, where $\vec{\sigma}_1$ and $\vec{\sigma}_2$ are Pauli vectors for the upper and lower qubits, respectively. Here $\vec{\omega}_1=(\omega_{1x},\omega_{1y},\omega_{1z})=(2.0,1.3,1.0)=-\vec{\omega}_2$ and $J=1$. (c) Fidelity results for the unitary operation case. For each evolution time $t \in [0,0.2]$, 10 configurations of random Pauli operators are sampled, and the experiment is repeated once for each configuration. Blue and green circles represent the experimental fidelities with and without the noise suppression protocol, respectively, while solid lines denote the simulated results. At $t=0.2$ (red star), the expectation values obtained from five random configurations are shown on the right. The bars denote the simulated expectation values, while the gray solid circles and error bars indicate the corresponding experimental results. The Pauli gate combinations used in the experiment are illustrated above each group.
• Figure 4: (a) Experimental circuit for the non-unitary channel $\mathcal{C}$. Shown here is the segment that replaces the part enclosed by the red dashed box in the unitary operation case of Fig. \ref{['nMfig1:exp']}(b). The left circuit shows the target evolution, while the right depicts the experimental implementation, with $\tau=0.35$. (b) Fidelity results for the non-unitary channel case. For evolution times $t \in [0,0.2]$, 10 random Pauli configurations are sampled, with one experimental run performed for each configuration. Experimental fidelities with and without the suppression protocol are shown as blue and green circles, respectively, while the solid lines represent numerical simulations. At $t=0.2$ (red star), the expectation values obtained from five random configurations are shown on the right. The bars denote the simulated expectation values, while the gray solid circles and error bars indicate the corresponding experimental measurements. The Pauli gate combinations used in the experiment are illustrated above each group. (c–d) Process matrices of the non-unitary channel. Gray solid bars correspond to the ideal partial SWAP channel, while black solid bars and colored bars with dashed outlines represent the simulated and experimental results at $t=0.2$, respectively. In (c), green bars correspond to the result with the noise suppression protocol, and in (d), red bars correspond to the result without it.
• Figure 5: The generalized version of our scheme. The non-Markovian noise consists of $n{+}1$ time points, separated by $n$ target quantum gates $U_i$. The joint probability distributions of Pauli errors on the main and ancillary registers are assumed to be identical.

Theorems & Definitions (2)

• proof
• proof