Error Suppression and Error Correction in Adiabatic Quantum Computation: Techniques and Challenges

Adiabatic quantum computation (AQC) has been lauded for its inherent robustness to control imperfections and relaxation effects. A considerable body of previous work, however, has shown AQC to be acutely sensitive to noise that causes excitations from the adiabatically evolving ground state. In this paper, we develop techniques to mitigate such noise, and then we point out and analyze some obstacles to further progress. First, we examine two known techniques that leverage quantum error-detecting codes to suppress noise and show that they are intimately related and may be analyzed within the same formalism. Next, we analyze the effectiveness of such error-suppression techniques in AQC, identify critical constraints on their performance, and conclude that large-scale, fault-tolerant AQC will require error correction, not merely suppression. Finally, we study the consequences of encoding AQC in quantum stabilizer codes and discover that generic AQC problem Hamiltonians rapidly convert physical errors into uncorrectable logical errors. We present several techniques to remedy this problem, but all of them require unphysical resources, suggesting that the adiabatic model of quantum computation may be fundamentally incompatible with stabilizer quantum error correction.

Popular Summary

The adiabatic variant of quantum computing (AQC) promises computation speedups comparable to standard “circuit-model” quantum computing as well as intrinsic robustness against certain kinds of noise. However, a useful quantum computer will have to be fault tolerant—robust to all forms of noise. While fault tolerance is theoretically attainable in the circuit model, nobody yet knows how to achieve it in AQC. Fault tolerance relies heavily on error correction, which has not been integrated naturally with AQC so far.

In this work, we take a step toward fault tolerance by formulating error correction in the natural control language of AQC (slowly varying Hamiltonians, rather than instantaneous “gates”). We discover and resolve a fundamental incompatibility between error correction and AQC, and we unite all the known techniques for error suppression in AQC within a unified theory.

We use our new framework for adiabatic error correction to demonstrate major obstacles to achieving fault-tolerant AQC. In fact, adiabatic error correction may be too effective: While suppressing unwanted errors, it also eliminates our ability to perform the desired computation. Actually performing AQC while error correction is active requires many-body Hamiltonians that do not appear in nature.

Any future work on fault-tolerant AQC will have to address this fundamental problem: Error-corrected logical computation in the AQC model, an essential component of fault-tolerant AQC, requires unphysical resources. In the absence of a new error-correction paradigm that surmounts this problem, we suspect that fault-tolerant AQC may be unachievable.

Figure 1

Logarithmic plot of the failure probabilities as a function of the total evolution time $T$ for a two-logical-qubit AQC defined by the Hamiltonian $H(t)=(1-t/T)(\sum _i{X}_i)+(t/T)(Z_1+Z_2+Z_1{Z}_2)$ under the influence of Hamiltonian noise $H_{\eta }(t)=\sum _i{\eta }_i(t)Z_i$. In these numerical simulations, ${\eta }_i$ is a classical stochastic process of $1/f$ type with spectrum $S(\omega )={10}^{-3}/\omega$, and the success probabilities are averaged over 1000 instances of this noise process for each evolution time $T$. Four simulations are performed: (i) the unencoded system (thick black line), (ii) the system encoded into four physical qubits using the [[4, 2, 2]] quantum code [15] (dashed red line), (iii) the encoded system with EGP applied at strength $\Omega =1$ (thin blue line), and (iv) the encoded system with DD applied at frequency $\Omega /2\pi$ (black crosses). The initial performance increase is due to reduction of diabatic errors, while later performance degradation is due to the accumulation of errors. Both EGP and DD are capable of suppressing this type of noise equally well, as shown by the approximately equal success probabilities of cases (iii) and (iv). Encoding without error suppression [case (ii)] performs especially poorly because twice as many qubits are exposed to noise as in the unencoded case and no measures are taken to suppress it.

Figure 2

The filter functions for periodic dynamical decoupling (solid line) and continuous amplitude driving (dashed line). The filter minima can be made to overlap at first order in the Magnus expansion.