Chapter 8 - Decoherence-Free/Noiseless Subsystems

Introduction

In the last chapter we saw that it is possible, at least in principle, to detect and correct errors in quantum systems. In this chapter a different method for protecting against errors is explored. This method encodes information into quantum states such that the information avoids errors. The information is encoded in such a way that it is invariant under the errors produced by the system-bath Hamiltonian. The initial work involved what are called decoherence-free subspaces and was later generalized to subsystems. (These terms are defined below. Reviews may be found in Whaley/Lidar and Byrd/Wu/Lidar.) Although there are alternative descriptions of decoherence-free subspaces and the subsystem generalization in terms of master equations, in this chapter the Hamiltonian description is used. See Whaley/Lidar and references therein.

General Considerations

In what follows, a general quantum system will be assumed to be coupled non-trivially to a bath such that the entire system-bath Hamiltonian is given by

	$H=H_{S}\otimes I_{B}+I_{S}\otimes H_{B}+H_{I},\,\!$	(8.1)

where $H_{S}\,\!$ acts only on the system, $H_{B}\,\!$ acts only on the bath, and

	$H_{I}=\sum _{\alpha }S_{\alpha }\otimes B_{\alpha },\,\!$	(8.2)

is the interaction Hamiltonian with the $S_{\alpha }\,\!$ acting only on the system and the $B_{\alpha }\,\!$ acting only on the bath. The "error algebra" is denoted ${\mathcal {A}}\,\!$ and is the algebra generated by the set $\{H_{S},S_{\alpha }\}\,\!$. The $S_{\alpha }\,\!$ obviously cause errors because they describe the interaction between the system and the bath. The reason the error algebra contains other terms is that when the system and bath together evolve unitarily, the exponential of the Hamiltonian $H\,\!$ means that products of the $S_{\alpha }\,\!$ with each other and also with $H_{S}\,\!$ will be present in the unitary evolution. In the case that $H_{S}\,\!$ is identically zero, or we can remove it by changing basis (to a rotating frame), the problem simplifies to the consideration only of the algebra of the $S_{\alpha }\,\!$ or the modified $S_{\alpha }\,\!$ ($S_{\alpha }\,\!$ in the rotating frame) need be considered, respectively.

At this point, the objective is to find a set of states which will be immune to the errors which are present. Such states are identified using the error algebra. The way this is done is to put the algebra in a form which is block-diagonal. This type of algebra is said to be "reducible" which means that one may always block-diagonalize it. (See Appendix D.) Suppose that each element of the algebra can be put into the same block-diagonal form using a particular unitary transformation $U_{dns}\,\!$. Then for any element of the error algebra, $A_{\alpha }\,\!$, $U_{dns}A_{\alpha }U_{dns}^{\dagger }\,\!$ is block-diagonal. If the information is stored in states which are acted upon by these blocks, and only these blocks, then the information is protected because the information stays in the states which are defined by these blocks. If the blocks are $1\times 1\,\!$ blocks, i.e., $1\times 1\,\!$ submatrices, which are just numbers, then the states which make up such a system are called decoherence-free "subspaces". If the blocks are larger, then they are called decoherence-free subsystems, or noiseless subsystems.

In the next few sections, the examples will illustrate this construction and how the states are protected.

A Phase-Protected Two-Qubit Decoherence-Free Subspace

One of the simplest example of a decoherence-free subspace is a method for using two physical qubits to encode one logical qubit in such a way that the logical qubit is protected against phase errors which operate on both physical qubits in the same way. This is called a collective phase error and is quite useful as will be shown in Chapter 10.

Let us begin by assuming there is no system Hamiltonian and that the two physical qubits in the system are acted upon by the Hamiltonian

	$H_{cpe}=Z^{(1)}\otimes B+Z^{(2)}\otimes B=(Z^{(1)}+Z^{(2)})\otimes B,\,\!$	(8.3)

where $Z^{(i)}\equiv \sigma _{z}^{(i)}\,\!$ is a phase operator which acts on the $i\,\!$th qubit. This Hamiltonian acts the same on each of the two qubits to produce the same phase error on each. If we now choose our logical states two be

	$\left\vert 0_{L}\right\rangle =\left\vert 01\right\rangle ,\;\;\;\left\vert 1_{L}\right\rangle =\left\vert 10\right\rangle ,\,\!$	(8.4)

then our logical states will be given by

	$\left\vert \psi _{L}\right\rangle =a\left\vert 0_{L}\right\rangle +b\left\vert 1_{L}\right\rangle .\,\!$	(8.5)

If we now suppose that the system and bath are initially uncoupled, $\left\vert \Psi _{SB}\right\rangle =\left\vert \psi _{L}\right\rangle \otimes \left\vert \psi _{B}\right\rangle ,\,\!$ then the states, acted upon by the Hamiltonian $H_{cpe}\,\!$ gives

	$H_{cpe}\left\vert \Psi _{SB}\right\rangle =(0)\left\vert \psi _{L}\right\rangle \otimes B\left\vert \psi _{B}\right\rangle ,\,\!$	(8.6)

which gives a decoupled system and bath. This is clear since the exponential of this Hamiltonian gives the unitary evolution of the state. Thus

	$\exp(-iH_{cpe}t/\hbar )\left\vert \Psi _{SB}\right\rangle =(\left\vert \psi _{L}\right\rangle \otimes U_{B}\left\vert \psi _{B}\right\rangle ),\,\!$	(8.7)

where $U_{B}=\exp(-iBt/\hbar )\,\!$. This Hamiltonian acts as the identity on the logical states of the system. To be even more explicit, when one can trace out the bath degrees of freedom to find that the state of the system remains unchanged by this Hamiltonian. Thus the system has been encoded in such a way that this type of error does not adversely affect the state of the system. This allows for perfect storage.

It is perhaps worth emphasizing that this is a simple model of a particular type of decoherence which would ordinarily lead to collective phase errors on the system states. Such noise has also been called "weak decoherence." However, because the information is encoded, it is protected against these types of errors. In the language of a quantum error correcting codes, this is an infinite distance code since the error does not lead to an error no matter how long, or to what extent the error acts. In the next subsection we will see how this can be extended to collective errors acting on a number of qubits in an arbitrary way, not just codes that will protect against phase errors.

A Decoherence-Free Subspace Using Four Qubits

Consider a Hamiltonian which causes arbitrary collective errors on a collection of four qubits,

	${\begin{aligned}H_{ce4}&=a_{1}(X^{(1)}+X^{(2)}+X^{(3)}+X^{(4)})\otimes B\\&\;\;+a_{2}(Y^{(1)}+Y^{(2)}+Y^{(3)}+Y^{(4)})\otimes B\\&\;\;+a_{3}(Z^{(1)}+Z^{(2)}+Z^{(3)}+Z^{(4)})\otimes B,\end{aligned}}$	(8.8)

where $a_{1},a_{2},a_{3}\,\!$ are arbitrary constants. The standard procedure would be to find irreducible representations of the algebra of errors which is generated by the three collective errors

	${\begin{aligned}S_{x}&=X^{(1)}+X^{(2)}+X^{(3)}+X^{(4)}\\S_{y}&=Y^{(1)}+Y^{(2)}+Y^{(3)}+Y^{(4)}\\S_{z}&=Z^{(1)}+Z^{(2)}+Z^{(3)}+Z^{(4)}.\end{aligned}}$	(8.9)

Here again we are supposing that there is no system Hamiltonian, $H_{S}=0\,\!$.

	$H_{ce4}=\sigma _{z}^{(1)}\otimes B+\sigma _{z}^{(2)}\otimes B=(\sigma _{z}^{(1)}+\sigma _{z}^{(2)})\otimes B,\,\!$	(8.9)

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Quantum Computing on a DNS

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