Chapter Quantum Error Correction AQuantum ErrorCorrecting Co de In our study of quantum algorithms wehave found p ersuasive evidence that a quantum computer would have extraordinary p ower But will quantum computers really work Will weever b e able to build and op erate them To do so wemust rise to the challenge of protecting quantum information from errors As wehave already noted in Chapter there are several as p ects to this challenge A quantum computer will inevitably interact with its surroundings resulting in decoherence and hence in the decay of the quan tum information stored in the device Unless we can successfully combat decoherence our computer is sure to fail And even if wewere able to pre vent decoherence by p erfectly isolating the computer from the environment errors would still p ose grave diculties Quantum gates in contrast to clas sical gates are unitary transformations chosen from a continuum of p ossible values Thus quantum gates cannot b e implemented with p erfect accuracy the eects of small imp erfections in the gates will accumulate eventually leading to a serious failure in the computation Any eective strategem to prevent errors in a quantum computer must protect against small unitary errors in a quantum circuit as well as against decoherence In this and the next chapter we will see how clever enco ding of quan tum information can protect against errors in principle This chapter will present the theory of quan tum errorcorrecting co des We will learn that quantum information suitably enco ded can b e dep osited in a quantum mem ory exp osed to the ravages of a noisy environment and recovered without CHAPTER QUANTUM ERROR CORRECTION damage if the noise is not to o severe Then in Chapter we will extend the theory in two imp ortantways We will see that the recovery pro cedure can work eectively even if o ccasional errors o ccur during recovery And we will learn howto process enco ded information so that a quantum computation can b e executed successfully despite the debilitating eects of decoherence and faulty quantum gates Aquantum errorcorrecting co de QECC can b e viewed as a mapping k of k qubits a Hilb ert space of dimension into n qubits a Hilb ert space n of dimension where n k The k qubits are the logical qubits or enco ded qubits that we wish to protect from error The additional n k qubits allow us to store the k logical qubits in a redundant fashion so that the enco ded information is not easily damaged We can b etter understand the concept of a QECC by revisiting an ex ample that was intro duced in Chapter Shors co de with n and k Wecancharacterize the co de by sp ecifying two basis states for the co de sub space we will refer to these basis states as ji the logical zero and ji the logical one They are p ji ji ji p ji jiji each basis state is a qubit cat state rep eated three times As you will recall from the discussion of cat states in Chapter the two basis states can b e distinguished by the qubit observable where x x x i denotes the Pauli matrix acting on the ith qubit we will use the x x notation X X X for this op erator There is an implicit I I I acting on the remaining qubits that is suppressed in this notation The states ji and ji are eigenstates of X X X with eigenvalues and resp ectively But there is no way to distinguish ji from ji to gather any information ab out the value of the logical qubit by observing an y one or two of the qubits in the blo ck of nine In this sense the logical qubit is enco ded nonlocal ly it is written in the nature of the entanglement among the qubits in the blo ck This nonlo cal prop erty of the enco ded information provides protection against noise if we assume that the noise is lo cal that it acts indep endently or nearly so on the dierent qubits in the blo ck Supp ose that an unknown quantum state has b een prepared and enco ded as aji bjiNow an error o ccurs we are to diagnose the error and reverse A QUANTUM ERRORCORRECTING CODE it Howdowe pro ceed Let us supp ose to b egin with that a single bit ip o ccurs acting on one of the rst three qubits Then as discussed in Chapter the lo cation of the bit ip can b e determined by measuring the twoqubit op erators Z Z Z Z The logical basis states ji and ji are eigenstates of these op erators with eigenvalue But ipping any of the three qubits changes these eigenvalues For example if Z Z and Z Z then we infer that the rst qubit has ipp ed relative to the other two Wemay recover from the error by ipping that qubit back It is crucial that our measurement to diagnose the bit ip is a collective measurementontwo qubits at once we learn the value of Z Z butwe must not nd out ab out the separate values of Z and Z for to do so would damage the enco ded state How can such a collective measurement b e p erformed In fact we can carry out collectiv e measurements if wehave a quantum computer that can execute controlledNOT gates We rst intro duce an additional ancilla qubit prepared in the state ji then execute the quantum circuit Figure and nally measure the ancilla qubit If the qubits and are in a state with Z Z either ji ji or ji ji then the ancilla qubit will ip once and the measurement outcome will b e ji But if qubits and are in a state with Z Z either ji ji or ji ji then the ancilla qubit will ip either twice or not at all and the measurement outcome will b e ji Similarly the twoqubit op erators Z Z Z Z Z Z Z Z can b e measured to diagnose bit ip errors in the other two clusters of three qubits A threequbit co de would suce to protect against a single bit ip The reason the qubit clusters are rep eated three times is to protect against CHAPTER QUANTUM ERROR CORRECTION phase errors as well Supp ose now that a phase error j iZ j i o ccurs acting on one of the nine qubits We can diagnose in which cluster the phase error o ccurred by measuring the two sixqubit observables X X X X X X X X X X X X The logical basis states ji and ji are b oth eigenstates with eigenvalue one of these observables A phase error acting on any one of the qubits in a particular cluster will change the value of XXX in that cluster relativeto the other two the lo cation of the change can b e identied by measuring the observables in eq Once the aected cluster is identied we can reverse the error by applying Z to one of the qubits in that cluster X Notice Howdowe measure the sixqubit observable X X X X X p that if its control qubit is initially in the state ji ji and its target is an eigenstate of X that is NOT then a controlledNOT acts according to x p p ji ji jxi ji ji jxi CNOT it acts trivially if the target is the X x state and it ips the control if the target is the X x state To measure a pro duct of X s then we execute the circuit Figure p and then measure the ancilla in the jijibasis We see that a single error acting on any one of the nine qubits in the blo ck will cause no irrevo cable damage But if two bit ips o ccur in a single cluster of three qubits then the enco ded information wil l b e damaged For example if the rst two qubits in a cluster b oth ip we will misdiagnose the error and attempt to recover by ipping the third In all the errors together with our CRITERIA FOR QUANTUM ERROR CORRECTION mistaken recovery attempt apply the op erator X X X to the co de blo ck Since j i and ji are eigenstates of X X X with distinct eigenvalues the eect of two bit ips in a single cluster is a phase error in the enco ded qubit X X X aji bjiajibji The enco ded information will also b e damaged if phase errors o ccur in two dierent clusters Then wewillintro duce a phase error into the third cluster in our misguided attempt at recovery so that altogether Z Z Z will have b een applied which ips the enco ded qubit aji bjiaji bji Z Z Z If the likeliho o d of an error is small enough and if the errors acting on distinct qubits are not strongly correlated then using the ninequbit co de will allow us to preserve our unknown qubit more reliably than if we had not b othered to enco de it at all Supp ose for example that the environment acts on each of the nine qubits indep endently sub jecting it to the dep olar izing channel describ ed in Chapter with error probability pThenabit p and a phase ip with probability p The ip o ccurs with probability probability that b oth o ccur is p We can see that the probabilityofaphase error aecting the logical qubit is b ounded ab ovebyp and the probability of a bit ip error is b ounded ab ovebyp The total error probabilityisno worse than p this is an improvementover the error probability p for an unprotected qubit provided that p Of course in this analysis wehave implicitly assumed that enco ding deco ding error syndrome measurement and recovery are all p erformed aw lessly In Chapter we will examine the more realistic case in whicherrors o ccur during these op erations Criteria for Quantum Error Correction In our discussion of error recovery using the ninequbit co de wehave assumed that each qubit undergo es either a bitip error or a phaseip error or b oth This is not a realistic mo del for the errors and wemust understand howto implementquantum error correction under more general conditions To b egin with consider a single qubit initially in a pure state that in teracts with its environment in an arbitrary manner Weknow from Chapter CHAPTER QUANTUM ERROR CORRECTION that there is no loss or generalitywemay