Measurement-Free Topological Protection Using Dissipative Feedback

Protecting quantum information from decoherence due to environmental noise is vital for fault-tolerant quantum computation. To this end, standard quantum error correction employs parallel projective measurements of individual particles, which makes the system extremely complicated. Here, we propose measurement-free topological protection in two dimensions without any selective addressing of individual particles. We make use of engineered dissipative dynamics and feedback operations to reduce the entropy generated by decoherence in such a way that quantum information is topologically protected. We calculate an error threshold, below which quantum information is protected, without assuming selective addressing, projective measurements, or instantaneous classical processing. All physical operations are local and translationally invariant, and no parallel projective measurement is required, which implies high scalability. Furthermore, since the engineered dissipative dynamics we utilize has been well studied in quantum simulation, the proposed scheme can be a promising route progressing from quantum simulation to fault-tolerant quantum information processing.

Popular Summary

Quantum computers possess an amazing potential to solve certain problems that are intractable with classical computers. However, quantum coherence, which is inevitable for quantum computation, is susceptible to noise and, accordingly, quantum computers can suffer from errors. While the standard quantum error correction technique can be utilized to suppress errors, it is experimentally infeasible given current technology, as it requires parallel measurements of an enormous number of individual particles. To address this problem, we propose a novel way to protect quantum computers from errors, namely, “measurement-free” topological protection.

Measurement-free topological protection does not use projective readouts or selective addressing of individual particles. Instead, we utilize engineered dissipative feedback in a two-dimensional many-body system, which is implemented by local and translationally invariant physical operations. Our goal is to reduce entropy caused by decoherence via dissipative dynamics in a classical system coupled to a quantum system. Our theoretical and numerical analyses show that quantum information is topologically protected, and extremely long coherence times (longer than a day) can be achieved with experimentally feasible physical requirements. The proposed dissipative quantum memory reduces the difficulty in achieving scalability and is experimentally feasible in various physical systems since it does not require any projective measurements of individual particles. Furthermore, the key ingredients utilized in dissipative feedback for topological protection are commonly employed in quantum simulation. Thus, the proposed scheme provides a promising route for progressing from quantum simulation to fault-tolerant quantum computation.

Reliable quantum information storage (or a self-correcting quantum memory) is closely related to the existence of thermally stable topological order, which is an important open problem in physics. Our proposed model of dissipative quantum memory contributes to furthering this field.

Figure 1

MFTP versus standard topological QEC (STQEC). MFTP with discrete-dissipative feedback operations, which are implemented by local and translationally invariant physical operations (left). A STQEC with projective measurements and classical processing (right). Since MFTP does not require any selective addressing, its scalability is evident.

Figure 2

The system and dissipative dynamics for MFTP. (a) The quantum (lower layer) and classical (upper layer) many-body systems for quantum information storage and its feedback controls, respectively. The $Z$ errors occur on the qubits colored by red. The associated vertex stabilizers with the eigenvalue $-1$ are denoted by black squares. The four classical spins adjacent to each vertex interact according to the eigenvalues of the vertex stabilizers. The wrong-sign plaquettes with $b_v=-1$ are shown by gray squares on the upper layer. The double layer is not necessarily required, but different degrees of freedom of the particles located on one layer can be employed. (b) The effect of plaquette interaction. A correct-sign plaquette with $b_v=+1$ favors an even number of $-1$’s, where a chain of $-1$’s is energetically hard to terminate. The wrong-sign plaquette favors an odd number of $-1$’s, where a chain of $-1$’s terminates easily. (c) The competing interactions reveal the location of errors: two wrong-sing plaquettes are connected with a short length.

Figure 3

Cooling dynamics and topological error correction. The location of the $Z$ errors in the quantum system and wrong-sign plaquettes are shown by red and black squares, respectively (top left), where the error probability is $p=0.05$. Equilibrium configurations are shown for $\beta J=0.1$ (top middle), 0.5 (top right), 0.8 (bottom left), and 2.3 (bottom middle), where $-1$ spins are shown by gray squares. An equilibrium configuration at low temperature is put back into the quantum system. The remaining $Z$ errors after the feedback are shown by blue squares (bottom right). Most of the errors are corrected, but some errors, shown by blue squares, remain. These are corrected by subsequent cycles.

Figure 4

Numerical results. The effective decay rate ${\gamma }_{\text{eff}}$ is estimated by numerical simulations for the physical error probabilities $p=0.01,0.02,\dots ,0.05$ and the system size $L=8,10,12,16,20$. (a) The effective decay rate ${\gamma }_{\text{eff}}$ as a function of the physical error probability $p$. From top to bottom $L=8,10,12,16,20$. (b) The effective decay rate ${\gamma }_{\text{eff}}$ as a function of the system size $L$. From bottom to top $p=0.01,0.02,0.03,0.04,0.05$.

Figure 5

Physical implementation of MFTP. (a) The double-layer system for MFTP. Each 2D layer consists of three species of particles (left). The different degrees of freedom of the particles on a single 2D layer can also be utilized (right). (b) The stabilizer pumping scheme for a digital simulation of the cooling process (see Appendix pp2).

Figure 6

One MFTP cycle for the $Z$ error correction. (i) The B qubits and B’ spins are initialized to extract and copy the error syndrome, respectively. (ii) The error syndrome is extracted from the A qubits to the B qubits. The extracted error syndrome is copied to the B’ spins. (iii) By using the copied error syndrome, two-qubit gates, and single-qubit dissipative dynamics, the cooling process under the Hamiltonian $H(\{b_v\})$ is simulated in a digitalized way [see Fig. 5]. (iv) The obtained equilibrium configuration, parts of which reveal the location of the $Z$ errors, is put back into the A qubits to correct the $Z$ errors.

Figure 7

The energy landscape of the ancilla system. At a high temperature, such as $e^{-4\beta J}=10^{-3}$, the system decays to the thermal equilibrium state rapidly. Then the temperature is further reduced to the Nishimori temperature. The relaxation time scales polylogrithmically in the size $L$ of the system at least up to $L=400$, where at most $10^4$ Monte Calro steps are enough for equilibration with unsophisticated schedulings of annealing.