Jaeger G. Quantum Information: An Overview

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Quantum decoherence and its mitigation

One of the greatest challenges in quantum information science is the develop-

ment of methods for eﬃciently maintaining the uniquely quantum properties

of states during the performance of quantum information-processing tasks. In

particular, quantum information-processing hardware designs are faced with

two competing requirements, that of strong interactions between their inter-

nal components for the performance of controlled quantum gates and that of

isolating qubits from the environment in which the hardware operates.

Quan-

tum decoherence is the loss of coherence within a quantum state behind the

second of these requirements, and can take the form of dephasing or the loss

of state population described by a nonunitary evolution of local system states,

due to unwanted interaction with the environment.

This interaction generally

necessitates the correction of errors it induces, which are highly detrimental

to quantum information-processing tasks.

One of the beneﬁts of information processing with few qubits, say the one

or two qubits of quantum key distribution with photons, is that the coupling

of qubit and environment can be kept small for photons. On the other hand,

eﬀective deterministic quantum computing generally requires many qubits

and the ability to implement conditional quantum gates between diﬀerent,

intersecting sets of qubits. In the case of multi-photon states, the lack of

direct photon–photon coupling renders deterministic quantum computation

ineﬀective, but probabilistic approaches remain viable. In the case of more

strongly interacting particles, their coupling with the environment tends to

be strong as well, leading to increased decoherence. A number of techniques for

avoiding and mitigating decoherence have been developed that are discussed

here, along with principles and methods of quantum error correction that are

vital to the further development of quantum information processing.

See, for example, [429].

A more general theory of decoherence was laid out by Roland Omn`es in [317].

172 10 Quantum decoherence and its mitigation

10.1 Quantum decoherence

Quantum decoherence can result from an environment’s interacting with in-

dividual qubits in such a way that an independent random average phase

shift φ

 (i =0, 1) is given to each basis state, and thus a phase shift rel-

ative to its complement, that is, the environment may induce the following

transformation on these states

|0→e

iφ



|0 , (10.1)

|1→e

iφ



|1 , (10.2)

resulting in an observable relative phase diﬀerence (dephasing) between com-

putational basis states. Decoherence may also transform qubits in such a

way that the population of qubit states is changed in a manner describ-

able by a nonunitary evolution of the statistical operator of each qubit.

Ini-

tially pure states are then transformed to mixed states in which oﬀ-diagonal

terms have simply vanished due to decay (i.e., amplitude damping).

precisely, the decoherence of the state of an ensemble of pure qubit states

|ψ

= a

|0

+ a

|1

may occur in a quantum channel so that



|ψ



→ ρ





∗

−γ(t)

∗

−γ(t)



, (10.3)

where γ(t) is a positive, time-dependent real parameter characterizing cou-

pling to the environment and the statistical operator is kept normalized. For

example, in a d-level system, the dephasing due to coupling with the environ-

ment has the following eﬀect in the case of a generic, possible mixed, initial

qubit state.

→ ρ





I +(1− )ρ

, (10.4)

where  is the strength of depolarization. Such behavior has been carefully

studied experimentally in the important case of the qubit (d = 2), as described

in Chapter 8 [334].

Decoherence can also be viewed as the unitary evolution of a compound

system consisting of the qubit and its environment under which the sys-

tem and environment become entangled. This is just the situation faced by

Schr¨odinger’s cat when it becomes entangled with its environment [365]. Con-

sider two pertinent environmental states |

 and |

 that are not necessar-

ily orthogonal, in particular, that are such that 

 = e

−γ(t)

The pure

See [317] for the presentation of a general theory of decoherence.

See also Sect. 9.6, where the joint evolution of a qubit and environment is de-

scribed. For a detailed treatment of decoherence eﬀects described as quantum

channels and a discussion of the eﬀects on state entanglement, see [465].

The bar notation for the |

 here, as before, serves to distinguish these states

from decomposition operators for CPTP maps used to describe the eﬀects of

quantum channels.

10.2 Decoherence and mixtures 173

time-dependent statistical operator for the total system of qubit and its envi-

ronment can be written

Q+

(t)=|a

Q

| + a

∗

Q

| (10.5)

+ a

∗

Q

| + |a

Q

| .

The reduced statistical operator of the qubit provides its individual descrip-

tion, namely,

(t)=tr

Q+

(t) (10.6)

= 

|ρ

Q+

(t)|

 + 



|ρ

Q+

(t)|



 , (10.7)

where |



 =(|

−e

−γ(t)

)/

√

1 − e

−2γ(t)

is a state eﬀectively orthogo-

nal to |

 at time t. As the qubit and its environment continue to interact

over the decoherence time, an increasingly mixed ensemble of computational-

basis states with no mutual coherence ultimately results, which is the limiting

behavior of the initially pure qubit state described by Eq. 10.3, namely,

(t) → ρ



= |a



|0



+ |a



|1



. (10.8)

Decoherence occurs rapidly in quantum computing systems, which by nature

are generally far more complex than, say, those required to implement quan-

tum key distribution.

Quantum information-processing systems must be engineered and their

states encoded so as to minimize such unwanted interaction with the environ-

ment and to suﬃciently slow the rate of decoherence that all needed quantum

logic operations may be well performed. Random errors accumulate as steps

in a random walk do, that is, with a probability that accumulates linearly in

the number of operations leading to them, say, the number of quantum gates

eﬀected. Three primary methods for mitigating and counteracting the eﬀect of

decoherence in quantum information-processing systems are decoherence-free

subspace methods, quantum error-correction methods, and dynamical decou-

pling methods. The ﬁrst two of these are considered below.

10.2 Decoherence and mixtures

It might be thought that the need to describe quantum systems by statisti-

cal operators arises simply from the inﬂuence of noise on quantum systems,

that is, that all such states could be described by some pure state, being con-

taminated by random external inﬂuences. However, noise is generally local

in character, as in the description above, and is certainly so in cases where

one is dealing with photons separating from each other at the speed of light.

The behavior of such a large number of qubits under decoherence is also described

in Sect. 1.7.

174 10 Quantum decoherence and its mitigation

This severely constrains the ability of noise alone to give rise to mixed states.

Mixed states that can be produced by local noise inﬂuences alone are referred

to as locally contaminated (LC) and those in which local measurements and

classical communication may occur are considered states produced by local

contamination and classical communication (LCCC). This lends further sup-

port to the view that mixed states are fundamental in nature.

Let us consider in some detail the question of which statistical operators

are accessible from pure states under such local processes. For N -partite sys-

tems with qu-d-it subsystems, the initial pure states are described by 2(d

−1)

real parameters, whereas the number of real numbers necessary for describing

a general mixed state of the system is larger, namely d

− 1, as discussed

in Chapter 7. That not all statistical operators are accessible from initially

pure states via noise eﬀects can be seen by noting that the number of real

parameters characterizing state contamination is only linear in the number of

parties, N , so that the number of parameters describing the pure states and

their contamination is less than the number describing the statistical states.

In particular, note that the most general transformation of each of N qu-

d-its |i interacting locally with its environment is

|i|0

→



|j|



, (10.9)

where |



are (not necessarily orthonormal) environmental states speciﬁed

by d

parameters and constrained by d

conditions arising from the orthonor-

mality of the qu-d-it states, which are therefore described by (d

− d

)pa-

rameters. Thus, the number of parameters governing the interaction of all N

qu-d-its with their local environments together with the number of real param-

eters describing the initial pure states is 2(d

−1) + N (d

−d

), whereas the

statistical operators of the qu-d-its may require up to (d

−1) parameters to

be fully described. As a result, it is not possible to account for the appearance

of all statistical operators from initially pure states by local environmental

eﬀects alone for systems composed of more than two qu-d-its [212].

10.3 Decoherence-free subspaces

In the decoherence-free subspace method of decoherence mitigation, one makes

use of a specially chosen type of subspace, a decoherence-free subspace (DFS),

of the full Hilbert space of the information-processing system that by construc-

tion is protected from the inﬂuence of noise from its environment. This sub-

space is spanned by logical-qubit states of several physical qubits [468, 469].

This approach to decoherence mitigation typically assumes that the qubits

under consideration are subject to decoherence of a sort in which the environ-

mental disturbances are identical for all qubits, with each qubit interacting

10.4 Quantum coding, error detection, and correction 175

with the environment in a similar but uncorrelated way, sometimes called col-

lective decoherence. The logical states (sometimes called error-avoiding quan-

tum code states) of decoherence-free subspaces accordingly possess a symme-

try, for example, under permutation of physical components. This symmetry

is associated with the fact that the logical states of DFSs are often maximally

entangled states.

To see how a properly chosen encoding subspace can help mitigate deco-

herence, consider the encoding of a logical qubit into the Bell states |Ψ

,by

the mapping

|0 →|0

 = |Ψ

−

 , (10.10)

|1 →|1

 = |Ψ

 . (10.11)

A dephasing-noise environment such as the ﬁrst one described above in Section

10.1 will have no inﬂuence on encoded states in the subspace spanned by |Ψ

.

It will have the eﬀect

|Ψ

−

→e

iφ



|0e

iφ



|1−e

iφ



|1e

iφ



|0 (10.12)

= e

i(φ

+φ

)



|0|1−|1|0



(10.13)

= e

i(φ

+φ

)

|Ψ

−

 (10.14)

in the case of the Bell state |Ψ

−

, the resulting global phase factor being

unobservable; a similar global phase will clearly also result in the case of

such noise on the Bell state |Ψ

, and therefore on linear combinations of

the two. This sort of noise will accordingly have no computationally rele-

vant eﬀect on the logical states of this decoherence-free subspace, because

|0

→ e

i(φ

+φ

)

|0

and |1

→ e

i(φ

+φ

)

|1

;thelogicalstatesremain

orthogonal despite the noise, as can be seen by taking their inner product.

10.4 Quantum coding, error detection, and correction

Bit errors in classical systems can be corrected, for example, through the use

of a simple repetition code, wherein each bit is encoded into a logical bit

consisting of several duplicates of itself: i → i

= ii...i, i =1, 2. In this

way, provided errors are independent and unlikely, by checking the values

of the encoding bits and maintaining each at the dominant bit value, one

may render single-bit errors incapable of ﬂipping the value of the logical bit,

which is similarly decoded by the receiver using such a majority vote.

generally, linear codes are special subgroups of the full error group consisting

of errors that take codewords to codewords and, as such, are undetectable.

This situation has been well investigated experimentally by the groups of Paul

Kwiat and Aephraim Steinberg; for example, see [264, 302].

The classical case is described in Sect. 4.6.

176 10 Quantum decoherence and its mitigation

A noisy quantum channel can similarly cause the value of a transmitted

bit to ﬂip value with a given probability, p, analogously to the situation in the

classical binary symmetric channel (see Fig. 4.2). A quantum majority vote

procedure can also be carried out. For example, for three qubits, one can use

the encoding

|i →|iii (10.15)

for this purpose. A simple quantum circuit consisting of a C-NOT gate control-

ling each of two ancillae initially in the state |0 suﬃces for the construction

of these codestates. Although an essential component of any quantum error-

correction scheme is the encoding of quantum information, quantum states are

often susceptible to a broader range of errors that may render unrecoverable

superpositions of states encoded in this simple way.

In the quantum context, a majority vote procedure analogous to

the classical one encounters the diﬃculty that one must perform a

measurement on all the “voter” qubits in order to correct errors, a

process that can also destroy quantum coherence. Unlike classical

bit errors, qubit errors are not discrete but rather have a continuous

character just as have qubit states themselves. Thus, a qubit can

enter any of an inﬁnite number of possible states, rather than just

two, as a result of the presence of environmental noise. Because of

this continuous character, the quantum ﬁdelity



P (|ψ),ρ



= ψ|ρ|ψ (10.16)

is a helpful tool to supplement the simple discrete accounting of

errors. It is also important to note that the unitary operations re-

quired by quantum information processing may also be imperfectly

executed. A corrected or processed qubit must have nearly full ﬁ-

delity, F = 1, with its desired state. In fault-tolerant quantum infor-

mation processing, a corresponding amplitude of failure is also often

introduced [258].

The quantum parallelism of quantum computing uses high-visibility multi-

qubit interference to operate eﬃciently, which requires coherence to be main-

tained throughout computation. To avoid the diﬃculties potentially posed

by decoherence, quantum error correction uses entangled states and partial-

state measurements to extract only that information associated with errors

without disturbing vital state coherence. If errors on distinct qubits occur

independently and the probability p of an error to occur on any given qubit

Note that here, because the basis is known and the states involved are orthogonal,

this does not violate the no-cloning theorem. Only bits are actually copied in this

process. Also note that by contrast with quantum data compression, which seeks

to eliminate redundancy, quantum coding introduces it.

10.4 Quantum coding, error detection, and correction 177

is suﬃciently small, it is a good approximation to ignore the possibility of

more than n total errors occurring, because more than n errors will occur

with a very small probability, of the order O(p

n+1

). Therefore, one generally

considers the most common situations, which deal with a limited number of

errors.

Similarly to the logical states of decoherence-free subspaces discussed in

the previous section, quantum error-correcting code (QECC) states are typ-

ically also entangled states. QECCs can be understood as functioning, in

essence, by storing information in the quantum correlations among diﬀer-

ent components of the composite system realizing the code. If a portion of

a system in a code state is inﬂuenced by its environment but remains well

correlated with other portions of the system, then the encoded information

still remains in these correlations and remains salvageable from them through

error recovery procedures.

The continuous nature of quantum errors does not present an insurmount-

able problem because quantum errors can be reduced to a few types of error.

Just as the classical binary symmetric channel produces a number of classical

bit-ﬂip errors e ∈ GF (2) with a given maximum Hamming weight, quantum

Pauli channels are those where single-qubit errors and products thereof pro-

duce at most a given number of errors. The two fundamental types of qubit

error are the bit-ﬂip error and sign-ﬂip error. A bit-ﬂip error on a single qubit

is described by the operation of the Pauli operator σ

on it. Sign-ﬂip errors

(also known as phase-ﬂip errors) are similarly described by the operation of

on the qubit at which they occur, as mentioned in the discussion in Sec-

tion 9.6 on the quantum channels inducing them.

The errors arising from

a number of these basic errors in Pauli channels are thus naturally described

by the Pauli error group, G

,forn qubits:



⊗ e

⊗···⊗e

∈ G, i ∈{1,...,n}



, (10.17)

where G is the single-qubit Pauli group, {±σ

, ±iσ

} (µ =0, 1, 2, 3), which is

Thus, the multiple-qubit error group consisting of such tensor products

is seen to be a subset of the group U (2

) of unitary operators on (C

)

⊗n

An error weight consisting of the sum of the number of bit-ﬂip errors plus

the number of phase-ﬂip errors is attributed to a given error-group element

Both types of qubit error are thus describable by the corresponding gates dis-

cussedinSect.1.4.

See Sect. 9.6 for a discussion of quantum channels inducing the errors comprising

G, the Pauli channels. Note that G is a ﬁnite group of order 16, with a center

consisting of diagonal matrices with a related “eﬀective” error group G

eﬀ

being

G modulo this center, which is a cyclic group of order 4. The eﬀective error group

is thus an Abelian two-group of order 4 that is isomorphic to Z

. The algebra of

the Pauli matrices is such that any of them can be constructed from at most two

of the remaining three.

178 10 Quantum decoherence and its mitigation

e ∈ G

. These errors are correctable, as are linear combinations of them,

because G

is a basis for the space of 2

× 2

matrices.

Consider now a single-qubit state of the generic form

|ψ = a

|0 + a

|1 (10.18)

to be mapped into a linear combination of quantum-code states (logical states)

, |1

 lying in a higher-dimensional Hilbert space of a larger system in-

cluding additional, ancillary qubits. In the DFS example presented in the

previous section, only a single ancilla was used to support this. More robust

logical states for error correction involve a greater number of ancillae, and

hence larger spaces. Speciﬁcally, the state of each qubit Q

of a k-element

quantum register is encoded in a logical state of an element

of a larger

n-element register by a mapping of the form



|0

+ a

|1



|00 ...0 → a



+ a



, (10.19)

so that k logical qubits are present in an integral number n ≥ k of physical

qubits.

Geometrically, QECCs are subspaces such that any error in a small number

of qubits moves the state in a direction perpendicular to them, allowing the

error to be corrected by reference to this change. The (logical) computational

basis {|0

, |1

}

⊗k

is chosen such that errors induced by the environment

also take each basis state to a subspace that preserves the required quantum

coherence and leave the computational register and the environment in a joint

tensor product state. States susceptible to decoherence are initially encoded by

a unitary operation into the corresponding error-correction codespace within

a larger subspace of the full encoding system of original bits plus the ancillae.

The ancillae evolve into mutually orthogonal states dependent on interaction

with the environmental noise, so that error correction is possible based on

the results of measurements of these ancillae, which provide information as

to the speciﬁc errors induced, in a procedure known as error extraction: error

detection is performed, whenever necessary, using a quantum measurement

that projects the computational state onto a superposition of projectors each

derived from a speciﬁc code states by introducing the corresponding error in

it. The original state can be then recovered by a unitary transformation into

the codespace that depends on the result of the detection known as the error

syndrome, which corresponds to a speciﬁc error.

Consider a projector P onto the codespace. To each error syndrome there

will correspond one of a set of orthogonal subspaces. Error correction is car-

ried out via a (trace-preserving) recovery operation R(ρ). In the statistical-

operator description, a proper error-correction recovery process corrects the

error process E(ρ) in the sense that

Note that even in cases of repetition coding this process does not involve the

cloning of qubits because the local physical qubit states obtained by partial trac-

ing out all others will not in general be the same as that of the initial qubit.

10.5 The nine-qubit Shor code 179



E(ρ)



= ρ, (10.20)

where ρ is the statistical operator of the system being corrected. Such an

operation exists for a given error if and only if

†

P = a

P, (10.21)

where the e

are the errors correctable by R(ρ)anda

is a complex scalar.

For a multiple-qubit pure state |Ψ

 lying in the codespace of a nonde-

generate error-correcting code, C,wherebyk qubits are encoded in a n-qubit

space, a quantum codespace is a 2

-dimensional Hilbert subspace such that

Ψ

...e

|Ψ



 =0, (10.22)

where 1 ≤ n ≤ 2K, for any two, possibly identical states |Ψ

, |Ψ



 in the

codespace and any product of Pauli matrices of up to 2n factors in the error

group acting on diﬀerent qubits, K being the number of errors that the code

can correct, where the action of (σ

)

, (σ

)

or (σ

)

on any of the n qubits

is considered to be a single error, the ν

indicating the error locations and the

indicating the multiplicities of the errors. A good error-correcting code will

have a coding rate and an error rate that tend to nonzero values as n becomes

large.

In the face of known types of error, one thus wishes to preserve a 2

dimensional subspace, S, within the full 2

-dimensional space using the code

C, which is then called an [n, k] quantum code.The(n − k) ancillae are

used as primary memory during the error correction process. Let A be a

family of interaction operators e

= µ

|U|χ

, where {|µ

} is a Hilbert-

space basis for the environment the initial state of which is |χ

. The necessary

and suﬃcient conditions for correcting individual errors resulting from G

are

together known as the Knill–Laﬂamme conditions:

0

†

 =0, (10.23)

0

†

 = 1

†

 , (10.24)

[47, 155, 256]. The ﬁrst condition requires that the encoded states remain

orthogonal under any correctable error; the second condition requires that

the lengths and inner product of the encoded states remain equal under any

correctable error. The error behavior in a network of quantum gates can be

represented by a sum of all networks that represent all possible errors, known

as the error expansion of the network [258, 259].

10.5 The nine-qubit Shor code

Let us now consider some speciﬁc error-correction codes used to implement

the above general methods. As mentioned previously, classical bit-ﬂip errors

180 10 Quantum decoherence and its mitigation

have obvious quantum analogues, where the value of a qubit is ﬂipped in the

computational basis. Such bit ﬂips can be detected and corrected in direct

analogy to the manner in which they are detected and corrected for classical

bits, by a majority vote method. However, in particular, the phase-ﬂip error,

described by the action of the Pauli operator σ

, and the bit+phase-ﬂip error

induced by the product of both errors −iσ

,haveno classical counterparts.

Nonetheless, recalling from their discussion in Section 1.4 that the bit-ﬂip and

phase-ﬂip operations interchange roles in the diagonal basis {| , | },one

sees that phase-ﬂip errors can still be corrected if one uses an encoding taking

computational-basis states into diagonal-basis states.

In order to be able to handle both sorts of error at one time, one can, for

example, make use of the following binary linear code and then apply it repet-

itively. Taking the dual of the classical Reed–M¨uller code RM (1, 3), namely

C = {(0, 0, 0), (1, 1, 1)} discussed in Section 4.4, one can use the following two

(three-qubit) GHZ-type entangled states

|⇑=

√



|000 + |111



, (10.25)

|⇓=

√



|000−|111



(10.26)

as logical qubits.

These logical qubits inherit the one-bit-ﬂip-error correction

property of the classical code used in its construction [402]. Using these logical

qubits and an additional repetition coding step, one has

|0

= | ⇑| ⇑| ⇑ , (10.27)

|1

= | ⇓| ⇓| ⇓ . (10.28)

The resulting code is the nine-bit Shor code which makes use of nine physical

qubits to encode each logical qubit [391].

Recall, from the discussion in Section 10.1, that the decoherence process

can be understood as a process in a larger space whereby an environment

described by the pure state |

 becomes entangled with computational basis

state-vectors of one of the qubits, that is,

|0→|a

|0 + |a

|1 , (10.29)

|1→|a

|0 + |a

|1 . (10.30)

Thus, for example, as a result of the entangling of the environment with the

ﬁrst physical qubit of the three-qubit state |⇑, the following evolution of the

overall environment and system state is

An rth order Reed-M¨uller code RM(r, m) consists of the binary strings of length

n =2m associated with the Boolean polynomials of degree at most r.

This code is degenerate in the sense that diﬀerent errors in the total space have

an identical eﬀect in the codespace.