Chapter 11 - Hybrid Methods of Quantum Error Prevention

Introduction

One of the first proposals for combining error prevention methods was due to D.A. Lidar, D. Bacon, and K.B. Whaley [35]. However, a great deal more work has been done since that time to provide methods for combining these seemingly very different methods of error prevention -- quantum error correcting codes (Chapter 7), decoherence-free/noiseless subsystems (Chapter 8), and dynamical decoupling controls (Chapter 9). In this chapter, a few general guiding principles are provided for combining these error prevention methods into hybrid methods of error prevention. Beyond that, a few very specific examples are given which are promising for error prevention and control of quantum systems for quantum information processing.

General Principles for Combining Error Prevention Methods

Perhaps it is not surprising that group theory can be used to describe the error prevention methods and their combinations. In the case of DFS/NS (Chapter 8), it is clear that there is a symmetry involved in the process of identifying and utilizing the method. The stabilizer formalism, leads to the connection between both the classical error correction, where group theory is used, and also the Pauli group. Dynamical decoupling (Chapter 9) can be seen as averaging away errors, but averaging is creating a symmetry as seen in the group-theoretical description of the dynamical decoupling condition.

Dynamical Decoupling for Creating Symmetry

From Section 9.6 we know that dynamical decoupling can produce a Hamiltonian which will commute with an error implying a symmetry. From Chapter 8 we know that a decoherence-free or noiseless subsystem can be utilized if there is a symmetry in the system-bath Hamiltonian. Therefore, one is led to the question, can we create the conditions for a decoherence-free or noiseless subsystem by using dynamical decoupling controls? In other words, can we remove an asymmetry from the Hamiltonian in order to create a symmetry which will enable a DFS encoding to protect against noise? (See Zanardi:99a [41] for an early discussion.)

Notice that this provides a new target for dynamical decoupling. Rather than eliminate a Hamiltonian coupling between the system and bath, we can simply modify the Hamiltonian to be compatible with a certain class of errors. This is true for QECCs as well as DFS. In each case, the elimination of some but not all errors, using dynamical decoupling can yield a set of correctable or avoidable errors.

One way to do this is to directly use the group-theoretical averaging technique presented in Section 9.6.1.

Dynamical Decoupling for Decoupling Logical Qubits

Theorem: Symmetrization with respect to the set of logical operations on a code space, which forms a group $G,$ suffices to completely decouple the dynamics of the encoded sub-space from the bath.

Proof: Symmetrization takes any system-bath Hamiltonian and projects it onto the centralizer of the group generated by the group $G$ of logical operations (i.e., the set of elements that commute with all elements of this group). By irreducibility of the representation of $G$, it follows, from Shur’s Lemma, that the BB-modified system-bath Hamiltonian is proportional to identity on the code space (i.e., the code space dynamics will be decoupled. QED).

The importance of this theorem is two-fold. First, it states that only logical operations are required to remove all the errors using dynamical decoupling controls. One need not apply controls to individual physical qubits. This potentially saves many decoupling controls and means that the time scales involved are those of the logical, not physical, qubit.

Second, one assumes that there exists good and fast logical controls, as are required by the assumptions for quantum computing. Therefore, the required controls are available and need only be tested for time scale constraints. One sometimes encodes in a DFS in order to enable universal quantum computing because not all controls on individual qubits are available in experiment. Note also that this theorem can be used for any logical qubit.

For the original references, consult Byrd/Lidar:02 [39] and Byrd/Lidar:ebb [40] and the references therein.

Quantum Error Correcting Codes and DFS/NS

One way to combine these two methods of error prevention is through concatenation Lidar, Bacon, et al. [35]. Suppose that a set of physical qubits is encoded into a DFS/NS. If the noise protection is imperfect, i.e., the symmetry which gives the protection is not exact, then there will be some "leakage" from the DFS/NS to states outside the protected subsystem. To correct for such leakage errors, one can take the DFS/NS encoded qubits and use those to encode the information redundantly into a quantum error correcting code (QECC). Let $\left|0_{d}\right\rangle \,\!$ be an encoded DFS/NS zero state (a DFS/NS encoded $\left|0\right\rangle \,\!$) state and $\left|1_{d}\right\rangle \,\!$ be the corresponding DFS/NS encoded one state (a DFS/NS encoded $\left|1\right\rangle \,\!$). Then one could use these to encode the information redundantly using a QECC. For example, the three qubit, bit-flip-protected code would be

	${\begin{aligned}\left|0_{L}\right\rangle &=\left|0_{d}\right\rangle \left|0_{d}\right\rangle \left|0_{d}\right\rangle =\left|0_{d}0_{d}0_{d}\right\rangle ,\\\left|1_{L}\right\rangle &=\left|1_{d}\right\rangle \left|1_{d}\right\rangle \left|1_{d}\right\rangle =\left|1_{d}1_{d}1_{d}\right\rangle .\end{aligned}}\,\!$	(11.1)

Of course this is only one example. Any quantum error correcting code and any DFS could possibly be used to encode in these layers. Furthermore, concatenation is still possible for the QECC with essentially no modification.

Operator Quantum Error Correction

An interesting combination of a QECC and a DFS/NS is the one referred to as operator quantum error correction (OPEC). OPEC combines a QECC and DFS/NS in the following way. Suppose we encode in a QECC which has, as its target for correction, a state which is encoded into a DFS/NS. Then, to correct an error, one need only correct the error up to an operation which leaves the DFS/NS invariant. In other words, we only need to implement the error correction operator which takes us into the subsystem, not to the original state. The new state may differ by the old state by an operation which puts the system into another DFS/NS-encoded state.

This situation is describable by using two different systems often referred to as the system and gauge system. These can be denoted by $\left|\psi \right\rangle _{S}\,\!$ and $\left|\phi \right\rangle _{G}\,\!$. Then an error ${\mathcal {E}}\,\!$ and error recovery operation ${\mathcal {R}}\,\!$ results in the following sequence

	$\left|\psi \right\rangle _{S}\left|\phi \right\rangle _{G}{\overset {\mathcal {E}}{\longrightarrow }}\left|\psi ^{\prime }\right\rangle _{S}\left|\phi ^{\prime }\right\rangle _{G}{\overset {\mathcal {R}}{\longrightarrow }}\left|\psi \right\rangle _{S}\left|\phi ^{\prime \prime }\right\rangle _{G},\,\!$	(11.2)

and we say the recovery has been accomplished.

The error correcting code condition, analogous to the Knill-Laflamme condition from QECC Eq.(7.14), is

	$\left\langle i\right|\left\langle k\right|A_{\alpha }^{\dagger }A_{\beta }\left|j\right\rangle \left|l\right\rangle =\delta _{ij}c_{kl\alpha \beta }.\,\!$	(11.3)

Thus we see the final state of the gauge system may be changed by our error and/or recovery operation, but the encoded information can still be retrieved.

Examples of Hybrid Error Prevention

Several examples are given here which show just some of the ways these methods can be combined.

Leakage Elimination Operators

When a set of states is encoded into a logical state for the purposes of error prevention, that state is protected due to the fact that it is encoded. If an operation occurs which takes it out of the protected subspace, that operation is particularly bothersome because the state then loses its error prevention abilities. Such an operator is called a "leakage" operator. Removing those interactions which cause such errors is referred to as "leakage elimination" and is achieved using dynamical decoupling controls.