On the Use of Shor States for the [7,1,3] Quantum Error Correcting Code
aa r X i v : . [ qu a n t - ph ] N ov On the Use of Shor States for the [[7 , , Quantum Error Correcting Code
Yaakov S. Weinstein, Sidney D. Buchbinder Quantum Information Science Group,
Mitre , 260 Industrial Way West, Eatontown, NJ 07724, USA
We explore the effect of Shor state construction methods on logical state encoding and quan-tum error correction for the [[7,1,3]] Calderbank-Shor-Steane quantum error correction code in anonequiprobable error environment. We determine the optimum number of verification steps to beused in Shor state construction and whether Shor states without verification are usable for practicalquantum computation. These results are compared to the same processes of encoding and error cor-rection where Shor states are not used. We demonstrate that the construction of logical zero stateswith no first order error terms may not require the complete edifice of quantum fault tolerance.With respect to error correction, we show for a particular initial state that error correction usinga single qubit for syndrome measurement yields a similar output state accuracy to error correctionusing Shor states as syndrome qubits. In addition, we demonstrate that error correction with Shorstates has an inherent sensitivity to bit-flip errors. Finally, we suggest that in this type of errorcorrection scenario one should always repeat a syndrome measurement until attaining an all zeroreadout (twice in row).
PACS numbers: 03.67.Pp, 03.67.-a, 03.67.Lx
I. INTRODUCTION
Quantum fault tolerance [1–4] is the framework whichallows for accurate implementation of quantum algo-rithms despite the inevitability of errors during the com-putation. This is done by assuring that an error thatoccurs on one qubit cannot spread to multiple qubits.Application of quantum error correction (QEC) then cor-rects the single qubit error [5–7].However, utilizing the entirety of the fault tolerantframework promises to be an expensive proposition interms of the number of qubits and implemented gates.Thus, it is worth exploring whether it is possible to relaxsome of the strict rules required by the framework. Oneway to do this may be by easing the construction require-ments or simply not using Shor states as syndrome qubitswhen encoding logical computational states and applyingerror correction. In this paper we study the utilizationof Shor states in the encoding of logical zero states andthe application of error correction for the [[7,1,3]] Steanecode [8] with the goal of limiting the number of requiredqubits and implemented gates.A fault tolerant method for encoding a logical compu-tational state in the Steane code is to apply fault toler-ant error correction to any initial state of 7 qubits. Thisrequires construction of proper ancilla syndrome qubitssuch that each ancilla interacts with no more than oneof the 7 data qubits. For the Steane code there are anumber of possible choices for these ancilla includingSteane’s [9] suggestion of using encoded ancilla blocks,and Knill’s [10] method using encoded Bell states andteleportation. In this work we have chosen to utilize four-qubit Shor states [2] for ancilla as they require the leastnumber of qubits and are thus most likely to be experi-mentally accessible. Shor states are simply Greenberger-Horne-Zeilinger (GHZ) states with Hadamard gates ap-plied to each qubit. However, as the Shor states them-selves are constructed in a noisy environment (here the nonequiprobable error environment), verification via par-ity checks is necessary to ensure accurate construction.In this paper, we attempt to determine the number ofShor state verifications necessary to construct logical zerostates or apply error correction with as high a fidelity aspossible. We then ask whether using Shor states withfewer verification steps (thus using fewer ancilla qubitsand requiring fewer gates) will provide sufficient accu-racy to be used in the construction of logical zero statesor the application of error correction. Finally, we explorewhether Shor states are necessary at all in the construc-tion of logical zeros and the application of error correc-tion, or whether sufficient accuracy may be obtained us-ing single qubits for syndrome measurement.The error model used in this paper is a non-equiprobable Pauli operator error model [11] with non-correlated errors. As in [12], this model is a stochasticversion of a biased noise model that can be formulated interms of Hamiltonians coupling the system to an environ-ment. In the model used here, however, the probabilitieswith which the different error types take place is left ar-bitrary: the environment causes qubits to undergo a σ jx error with probability p x , a σ jy error with probability p y ,and a σ jz error with probability p z , where σ ji , i = x, y, z are the Pauli spin operators on qubit j . We assume thatonly qubits taking part in a gate operation will be subjectto error and the error is modeled to occur after (perfect)gate implementation. Qubits not involved in a gate areassumed to be perfectly stored. While this represents anidealization, it is partially justified in that it is generallyassumed that stored qubits are less likely to undergo er-ror than those involved in gates (see for example [13]). Inaddition, in this paper accuracy measures are calculatedonly to second order in the error probabilities p i thus theeffect of ignoring storage errors is likely minimal. Finally,we note that non-equiprobable errors occur in the initial-ization of qubits to the | i state and measurement (in the z or x bases) of all qubits.This paper builds on the previous work of Ref. [14](see also [15]) in which the fault tolerant method of en-coding logical zero states for the [[7,1,3]] code was com-pared to the gate sequence method of encoding to seewhich method led to more accurately encoded zero states.Though the gate sequence method is not fault tolerant(errors can propagate to multiple data qubits) it wasfound that the fidelity of the logical zero states con-structed in this way is comparable to the fidelity of thestates constructed using the fault tolerant method. Ap-plying perfect error correction then revealed that the er-ror probabilities were reduced to at least second order forboth methods (third order for the fault tolerant method),implying the correctability of the errors and suggestingthat either method can be used for practical quantumcomputation. Here we work within the fault tolerantmethod in attempt to determine how best to constructShor states for encoding and error correction. In bothpapers, however, a major goal is to determine whetheraccurate enough protocols can be implemented withoutinvoking the full framework of quantum fault tolerance. II. CONSTRUCTING SHOR STATES
A construction method for the four-qubit Shor statesneeded for the [[7,1,3]] QEC code is shown in Fig. 1.If the construction was done without error, no verifica-tion steps would be needed and the Shor state (withoutthe final Hadamard gates as explained below) would begiven by: | ψ Shor i = √ ( | i + | i ). However, ac-tual implementations of quantum computation will bedone in a noisy environment and thus verifications maybe useful. We simulate construction of Shor states in thenonequiprobable error environment including initializa-tion and measurement errors with different verificationstrategies. We then determine which of the strategiesproduce the highest quality Shor states based on the fi-delity of the constructed Shor states, the fidelity of logi-cal zero states encoded fault tolerantly with the differentShor states used as syndrome qubits, and the fidelity of astate after noisy error correction when the different Shorstates are used as syndrome qubits. The different strate-gies we use are: no verifcation steps, one verification step,and different possible two verification steps. The tenetsof fault tolerance require that at least one verificationstep be applied so as to lower the probability of error tosecond order.To construct the Shor state we start with four qubitsthat we attempt to initialize to the state zero. How-ever, in this work, we assume that initialization itself is anoisy process subject to the same error model as qubitsinvolved in a gate. Thus, the actual state of each initial-ized qubit is ρ i = (1 − p x − p y ) | ih | + ( p x + p y ) | ih | . Wethen apply a Hadamard gate, H , to the first qubit. Thenonequiprobable error environment causes imperfectionsin the gate such that the actual evolution of an attempted FIG. 1: Top: construction of a 4 qubit Shor state. cnot gates are represented by ( • ) on the control qubit and ( ⊕ ) onthe target qubit connected by a vertical line. H represents aHadamard gate. The procedure entails constructing a GHZstate which is verified using ancilla qubits. Hadamard gatesare applied to each qubit to complete Shor state construction.Bottom: fault tolerant bit-flip and phase-flip syndrome mea-surements for the [[7,1,3]] code using Shor states (the Shorstates pictured are assumed to have not had the Hadamardgates applied). To ensure fault tolerance, each Shor state an-cilla qubit must interact with only one data qubit. The errorsyndrome is determined from the parity of the measurementoutcomes of the Shor state ancilla qubits. To achieve fault tol-erance each of the syndrome measurements is repeated twice.Box: a useful equality which allows us to avoid implementingHadamard gates by reversing the control and target of cnot gates. In our context, the cnot gates associated with thephase-flip syndrome measurements are reversed such that theancilla qubits become the control and the data qubits becomethe target, as explained in the text. Hadamard on a single qubit j in the state ρ is: X a =0 ,x,y,z p a σ ja H j ρH † j σ ja , (1)where σ j is the identity matrix, p = 1 − P ℓ = x,y,z p ℓ ,and the terms K ja = √ p a σ ja H j can be regarded as Krausoperators for the Hadamard evolution. The Hadamardis followed by a series of cnot gates. The attempted TABLE I: Relevant fidelity measures for Shor states and encoded logical zeros from different construction methods: Shorstate without verification, Shor state with one verification, Shor state with two verifications, and the accuracy of logical zeroconstruction using single-qubit ancilla for syndrome measurements instead of Shor states. The accuracy measures are thefidelity of the Shor state itself, the fidelity of the seven physical qubits making up the logical zero state, the fidelity of the onequbit of information stored in the seven physical qubits, and the fidelity after perfect error corrction has been applied to theconstructed encoded zero states. no verifications 1 verification 2 verifications 1-Qubit ancillaShor fidelity 1 − p x − p y − p z − p x − p y − p z − p x − p y − p z − p x − p y − p z − p x − p y − p z − p x − p y − p z − p x − p y − p z − p x − p y − p x − p y − p x − p y − p x − p y after QEC 1 − p x − p x p y − p y − p x − p x p y performance of the cnot gate with control qubit j andtarget qubit k , c j not k , in the nonequiprobable error en-vironment on any state ρ actually implements: X a,b =0 ,x,y,z p a p b σ ja σ kb c j not k ρ c j not † k σ ja σ kb , (2)where terms A j,ka,b = √ p a p b σ ja σ kb c j not k can be regardedas the 16 Kraus operators. Note that errors on the twoqubits taking part in the CNOT gate are independentand not correlated. Shor state construction requires three cnot gates, shown in Fig. 1, and thus the final Shor stateis given by ρ Shor − err = ,x,y,z X a,b,c,d,e,f,g A , f,g A , d,e A , b,c K a ρ ⊗ i × ( K a ) † ( A , b,c ) † ( A , d,e ) † ( A , f,g ) † . (3)As explained above, applying the above described gatesequence does not guarantee that the resulting Shorstates are suitable for fault tolerant quantum computa-tion. During syndrome measurement, errors in the Shorstate construction can propagate into the data qubits. Ifonly one Shor state qubit has been compromised by errorthen only one data qubit will be compromised and theerror can be subsequently corrected. However, if multi-ple Shor state qubits are compromised, more than onedata qubit can be compromised and the computationwill fail. Thus, we must test the Shor states to ensurethat multiple qubits have not been compromised by er-ror. This is done utilizing an ancilla qubit, initially inthe state | i , adjoined to the Shor state to measure theparity of random pairs of qubits [2]. Should the test fail(the ancilla qubit measurement yields a | i ), the Shorstate is immediately discarded. Of course, the ancillaqubit initialization and the cnot gate implementationsfor this parity check are themselves performed in thenonequiprobable error environment and thus follow thedynamics described above. We utilize an initial ancillaqubit to measure the parity of qubits 1 and 4. Apply-ing additional verification steps using additional ancillamay, if the cnot s themselves are not too error prone,further ensure the lack of errors in the constructed Shorstates. A second ancilla can recheck the parity of thequbits checked with the first ancilla, or check the par-ity between other Shor state qubits. We have simulated every possible combination for the second parity mea-surement and have found that this choice has little effecton any of our accuracy measures.Our first accuracy measure for the Shor states con-structed with different numbers of verifications is the fi-delity of the constructed Shor state as compared to aperfect Shor state, F = h ψ Shor | ρ Shor − err | ψ Shor i . Thefidelity results for Shor states with zero, one, and twoparity verifications are shown to first order in error prob-ability in the first row of Table I. Note that to first orderthe fidelity for Shor states of two parity verifications isindependent of which qubits are used for the second ver-ification.Comparing the fidelity of the three Shor states we seethat the Shor state with one verification has a higher fi-delity than the Shor state with no verifications unless p z is significantly higher than p x and p y . This demonstratesthat it is usually advisable to perform a verification stepin order to suppress errors that occur during the Shorstate construction. However, the fidelity of the Shor statewith two verification steps is always lower than that ofthe Shor state with one verification step. A second ver-ification step does not give enough benefit to outweighadditional errors that may occur during the verificationprocedure. III. ENCODING WITH SHOR STATES
The Shor state fidelity is a good measure of accuracyfor the Shor state in and of itself. However, our pur-pose for constructing Shor states is to use them to en-code logical zero states and implement fault tolerant er-ror correction. It is possible that different errors in theShor state construction will have more or less of an ef-fect on the accuracy with which these protocols can beperformed. Thus, another way to quantify the qualityof the Shor states is to simulate their utilization in theencoding of logical zero states and in the performance oferror correction and report on the accuracy with whichthese protocols are implemented.We first turn to the construction of logical zero states.To do this in a fault tolerant manner we start with 7qubits all noisily initialized to the state zero. Thoughthis initialization is not perfect we choose to not per-form the first set of (bit-flip) syndrome measurements astheir utility in correcting an initialization error is out-weighed by the noise inherent in applying the necessarysyndrome measurments. Instead, we immediately mea-sure the three phase flip syndromes (each one of the threetwice) with Shor states as the syndrome qubits. To mea-sure phase flip syndromes requires applying a Hadamardgate to each of the seven data qubits before and after thesyndrome measurements. However, we can measure thesyndrome without Hadamard gates if we reverse the rolesof the control and target qubits for the cnot gates, andmeasure the Shor state qubits (noisily) in the x -basis,as explained in [1] and shown in Fig. 1. For the caseof encoding we analyze the scenario where all syndromeresults are zero. Because encoding is done ‘off-line’ onecan choose to utilize only the encoded states with thisoutcome.Encoding in the nonequiprobable error environmentusing Shor states with different numbers of applied ver-ifications will result in logical zero states with differentdegrees of accuracy. We can measure this accuracy in anumber of ways. The first way is simply to look at thefidelity of the seven qubit logical zero state. The accu-racy of this state gives an idea as to how well the entireencoding process was performed. Alternatively, one maylook at the fidelity of only the one qubit of encoded in-formation. This is the only qubit of information that isactually of importance and, if it is protected, the stateof the rest of the system is irrelevant. Measuring thefidelity of this one logical qubit is done by (noiselessly)decoding the constructed logical zero state, tracing outall qubits but the first, and comparing the state of theremaining qubit with the zero state on a single qubit.Both of these fidelity measures have been calculated forlogical zero states constructed using Shor states of zero,one, and two verification parity checks, and are given inTable I.Errors affecting the logical zero state may also be ofvarying degrees of severity. Applying perfect error cor-rection allows us to test the ‘correctability’ of the typesof errors that occur during the encoding. If even perfecterror correction cannot (to first order) correct the errorsin the logical zero state then the encoding method cannotbe used for practical implementations of quantum com-putation. We apply perfect error correction to the statesconstructed using the Shor states with varying numbersof verifications and calculate the fidelity measure of theoutput state. These fidelities are given in Table I, andcorroborate our previous observations that applying oneverifiction to Shor states is optimal. Applying no veri-fication steps to the Shor states leads to lower fidelitiesfor the logical zero states, and applying two verificationsdoes not raise the fidelity. Perfect error correction ap-plied to logical zero states encoded using Shor states withone or two verifications gives unit fidelity up to third or-der. However, perfect error correction applied to logicalzero states encoded using Shor states with no verifica-tions, suppresses errors to second order implying thatthese states may also be useable for practical quantum computation.We compare the above cases of Shor state syndromemeasurement with a logical zero encoding method inwhich a single ancilla qubit is used for each syndromemeasurement. This method does not meet the stan-dards of fault tolerance since an error on the single ancillaqubit, be it an initialization error or an error in one ofthe syndrome measurement cnot gates, can spread tomultiple data qubits. However, using one ancilla qubitremoves the need to construct Shor states thus loweringthe number of gates to be performed. The logical zero fi-delity measures defined above are calculated for the singlequbit syndrome measurement construction method andare shown in Table I. Comparing these fidelity measuresto those calculated for Shor state based encoding, we findthat using single qubit ancilla leads to higher fidelity log-ical zero states. However, upon application of perfect er-ror correction the error probabilities are suppressed onlyto second order, unlike the logical zero states constructedusing Shor states for which the second order error prob-ability terms are also suppressed. IV. QUANTUM ERROR CORRECTION WITHSHOR STATES
We now consider the accuracy with which the differ-ent Shor states can be used as syndrome ancilla qubitsfor quantum error correction. The arbitrary single-qubitinitial state we would like to protect is assumed to havebeen perfectly encoded via the [[7,1,3]] gate encoding se-quence: | ψ i = cos α | L i + e iβ sin α | L i , where | L i and | L i represent the seven qubit logical | i and | i statesrespectively. We assume the environment possibly causesan error such that, before error correction, the system isin a mixed state of no error and all possible single qubiterrors: ρ err = (1 − p x + p y + p z )) | ψ ih ψ | + X i =1 X a = x,y,z p a σ ia | ψ ih ψ | σ ia † . (4)Because there are only single qubit errors in the systemstate, the error can be corrected by perfect applicationof the [[7,1,3]] code.To perform error correction in a fault tolerant manner,Shor states with at least one verification must be used forsyndrome measurements. We apply error correction tothe state ρ err in the nonequiprobable error environmentby implementing the three bit-flip syndrome measure-ments followed by three phase-flip syndrome measure-ments using Shor states with different numbers of veri-fications as the syndrome qubits. Each syndrome mea-surement is repeated twice to account for errors that mayhave occurred during the syndrome measurement itself.We quantify the quality of the error correction via fidelitymeasures comparing the final state after error correctionto the pre-encoded arbitrary state. TABLE II: Fidelity measures for error correction applied to the state ρ err utilizing Shor states with different numbers ofverifications or a single ancilla qubit for syndrome measurement. In the Table a = cos[4 α ] and b = cos[2 β ] sin[2 α ] . In this casethe bit flip syndrome measurements were done first. no verifications 1 verification 2 verifications 1-Qubit ancilla7-Qubit fidelity 1 − p x − p y − p z − p x − p y − p z − p x − p y − p z − p x − p y − p z − ( + a − b ) p x − ( + a − b ) p x − ( + a − b ) p x − ( + a − b ) p x − ( + a − b ) p y − ( − a − b ) p y − ( − a − b ) p y − ( − a − b ) p y − (1 − a ) p z − (1 − a ) p z − (1 − a ) p z − (1 − a ) p z A. Syndrome Measruement Reveals No Error
To read out the syndrome bit the four Shor state qubitsare measured. If the results of the four measurements areof even parity the syndrome bit is a zero. If the resultsare of odd parity the syndrome bit is a one. We first lookat the case where all qubit measurements are zero. Othereven parity measurment results (say 0011 or 0101) givethe same fidelity to first order. The fidelities of the sevendata qubits and the one logical qubit state for this caseare given in Table II.Comparing the seven-qubit fidelities of the QEC pro-cedure utilizing Shor states with different numbers of ver-ifications, we first note that the fidelities for Shor stateswith one and two verifications are identical up to secondorder terms. This fidelity is higher than that attainedby performing QEC using a Shor state with no verifica-tions, again confirming that while performing verificationof the Shor state is important, there is no benefit gainedfrom performing a second verification step. The fidelitiesexhibit little dependence on the initial state of the qubit, α and β only appear in second order fidelity terms. Fur-thermore, regardless of the Shor state used, the p x erroris dominant implying that bit-flips are more harmful tothe error correction procedure than phase flips.Similar trends hold when comparing the single qubit fi-delities except that all of the single qubit fidelities dependstrongly on the initial state. These fidelities are highestwhen α = 0 , π , at which point the first and second order p z terms drop from the fidelity expression, and are lowestwhen α = π . Once again p x is the dominant error term.We note that the presence of first order terms in thefidelity measures indicate that, in this case, noisy QECcannot output a state with no first-order error probabilityterms. Practical quantum error correction in this case isthus reduced to minimizing the coefficients of these firstorder terms.We compare the above QEC performance with that oferror correction done without Shor states, instead using asingle (noisily initialized) ancilla qubit for syndrome mea-surement. While this scheme certainly does not meet thecriteria for fault tolerance, it does allow us to implementQEC with fewer qubits, and the lack of possible errorfrom the construction of the Shor states may, and in factdoes, yield an improved resulting fidelity. The fidelitiesfor this case are shown in the last column of Table II. B. Syndrome Measruement Reveals Error
Above we assumed that errors occur with low proba-bility ( p j ≪
1) and thus the chances of measuring a bit-flip or phase-flip syndrome that is not 000 is extremelysmall. If, however, syndrome measurement does (twicein a row) signify an error a proper recovery operationmust be performed. In such a case we find extremely lowfidelities ( ≈ . c not gate of the sec-ond syndrome bit. If this latter error had occurred therecovery operation would be applied to the wrong qubit,and, thus, the final state would have two errors: the er-ror on qubit 7 which was not corrected and the error onqubit 4 due to the mistaken recovery operation. Becauseboth the gates associated with error correction and thegates applied before error correction are implemented inthe same error environment, there is no a priori rea-son to think that this latter scenario is any less probablethen the presumed error based on the syndrome mea-surement. Therefore, the final state of the system afterthe error correction procedure is a mixed state consistingof a corrected state and states with two errors (not tomention terms from errors that may have occurred dur-ing the final syndrome measurement, recovery operation,etc.) leading to an unacceptably low fidelity.The above suggests a proper procedure to follow uponobtaining a non-zero syndrome measurement (even if thesame syndrome is read out twice in a row). Rather thenaccepting the syndrome measurement and applying therequisite recovery operation, the syndrome measurementshould be redone until the all zero readout is attained(twice in a row). Proceeding based on the non-zero read-out will likely lead to a state with uncorrectable errors. C. Why Bit Flips?
We noted above that σ x errors dominate the loss offidelity. There are a couple of possibilities as to why thismay be so. The first is because the bit-flip syndromemeasurements were implemented first, and thus σ x er-rors that may occur during phase-flip syndrome measure-ments are not corrected. A second possibility is that theuse of (noisy) Shor states may cause the effect of σ x er-rors to be more pronounced. In this section we clarifythis issue by carrying out a series of simulations designedto isolate the cause of increased sensitivity to σ x errors.Our first step is to repeat the above error correctioncalculations implementing the phase-flip syndrome mea-surements first. The fidelities of the resulting states areshown in Table III. Let us first compare the cases wherethe syndrome measurement was done with a single ancillaqubit. In this case the coefficients of the p x and p z termsin the seven qubit fidelity simply switch places while the p y coefficient remains constant. Similarly in the one-qubit fidelity the p y coefficient remains constant whilethe values of the p x and p z terms approximately tradevalues (modulo the contribution of the initial state). Thisalone suggests that the dominance of the p x term in theoriginal simulations was simply because the bit-flip syn-drome measurements were done first. When the phase-flip syndrome measurements are done first p z replaces p x . However, when looking at the QEC simulations thatutilize Shor states for syndrome measurements we do notfind the same trade-off. Instead, though the p z error co-efficients grow and (in most cases) become dominant, wefind much less of a reduction of the p x error coefficients.This suggests that there is something inherent in the useof the (noisy) Shor states that leads to this type of error.To further explore this point we perform two additionalsets of QEC simulations. In the first, we utilize perfectShor states but allow errors (due to the nonequiprob-able error environment) in the error correction process(including syndrome measurement). In the second, weuse Shor states constructed in the nonequiprobable errorenvironment (with one verification) but the error correc-tion itself (including syndrome measurements) is perfect.Both are done with bit-flip syndrome measurements firstand with phase-flip syndrome measurements first. Whenperfect Shor states are used, but the error correction isnoisy, we find that the dominant error depends on whichset of syndrome measurements is done first, if phase cor-rection is done first σ z errors dominate and vice-versa.The other error type is significantly diminished. Whennoisy Shor states are used with perfectly implemented er-ror correction we find that which syndrome is done firstmakes little difference: σ x errors dominate and the fideli-ties do not contain a first order term for σ z errors. Thevarious fidelity measures are displayed in Table IV.Taken together these simulations imply that whennoisy Shor states are utilized for syndrome measurement in the Steane code there is a significant bias towards bit-flip errors. A possible solution is to concatenate into athree-qubit bit-flip QEC code for another level of errorcorrection. This could significantly reduce the sensitivityto bit-flip errors without the resource cost of concatena-tion into another level of the seven-qubit Steane code. V. CONCLUSION
In conclusion, we have calculated quality metrics fordifferent Shor states used as syndrome measurement an-cilla qubits for the [[7,1,3]] CSS QEC code operating ina nonequiprobable error environment. The results sug-gest that while a Shor state constructed in this error en-vironment with one parity check verification is optimalfor suppressing errors in the construction of logical zerostates, Shor states with no checks will also suppress errorprobability terms in the fidelity to second order. In addi-tion, encoding applied without Shor states, instead usingsingle qubit ancilla for syndrome measurement, leads tological zero states with higher fidelity but errors that areless correctable as identified by fidelity after perfect errorcorrection.For error correction applied in a nonequiprobable er-ror environment using the seven qubit Steane code, oursimulations show that not using Shor states leads toa corrected state with higher fidelity than using Shorstates. In addition, we noted that bit-flip errors aredominant whether Shor states are used or not. We firstsuggested that this was due to the fact that the bit-flipsyndrome measurements were done first, meaning thatuncorrected bit-flips may accumulate during phase-flipsyndrome measurements. Simulations switching the or-der of the syndrome measurements demonstrated thatthis is correct when using single qubit ancillae for syn-drome measurement, but does not completely explain theresults of simulations using Shor states. Further simu-lations indicated an inherent sensitivity towards bit-fliperrors when Shor states are used. We suggested that thiscould be overcome by concatenating with a three-qubitQEC code that protects against bit-flip errors. Finally,we suggested that when a non-zero syndrome is detectedimplementing the prescribed recovery operation will leadto a state of unacceptably low fidelity. Rather the syn-drome measurement should be repeated until a zero syn-drome readout is attained.The authors would like to thank G. Gilbert for con-structive comments. This research is supported underMITRE Innovation Program Grant 07MSR205. [1] J. Preskill, Proc. Roy. Soc. Lond. A , 385 (1998).[2] P.W. Shor,
Proceedings of the the 35th Annual Sym-posium on Fundamentals of Computer Science , (IEEEPress, Los Alamitos, CA, 1996).[3] D. Gottesman, Phys. Rev. A , 127 (1998).[4] P. Aleferis, D. Gottesman, and J. Preskill, Quant. Inf. Comput. , 97 (2006).[5] M. Nielsen and I. Chuang, Quantum information andComputation (Cambridge University Press, Cambridge,2000).[6] P.W. Shor, Phys. Rev. A , R2493 (1995).[7] A.R. Calderbank and P.W. Shor, Phys. Rev. A , 1098 TABLE III: Fidelity measures for error correction applied to the state ρ err utilizing Shor states with different numbers ofverifications or a single ancilla qubit for syndrome measurement. In the Table a = cos[4 α ] and b = cos[2 β ] sin[2 α ] . In this casethe phase flip syndrome measurements were done first. no verifications 1 verification 2 verifications 1-Qubit ancilla7-Qubit fidelity 1 − p x − p y − p z − p x − p y − p z − p x − p y − p z − p x − p y − p z − ( − a − b ) p x − ( − a − b ) p x − ( − a − b ) p x − ( + a − b ) p x − ( − a − b ) p y − ( − a − b ) p y − ( − a − b ) p y ( − a − b ) p y − (1 − a ) p z − (1 − a ) p z − (1 − a ) p z − (1 − a ) p z TABLE IV: Fidelity measures for quantum error correction applied to the state ρ err with perfect Shor states and noisy errorcorrection, and noisy Shor states with perfect error correction. Both cases were done with the σ x syndrome measurements firstand the σ z syndrome measurements first. In the Table a = cos[4 α ] and c = cos[2 β ]. bit-flip first phase-flip first7-Qubit fidelity 1 − p x − p y − p z − p x − p y − p z Noisy QEC 1 − ( + ( a − c + ac ) p x − ( + a − b ) p x Perfect Shor States 1-Qubit fidelity − ( − ( a + c − ac ) p y − ( − a − b ) p y − (1 − a ) p z − (1 − a ) p z Perfect QEC 7-Qubit fidelity 1 − p x − p x Noisy Shor States 1-Qubit fidelity 1 − (6 + 2 a − b ) p x − (6 + 6 a ) p x (1996); A.M. Steane, Phys. Rev. Lett. , 793 (1996).[8] A.M. Steane, Proc. Roy. Soc. Lond. A , 2551 (1996).[9] A.M. Steane, Phys. Rev. Lett. , 2252 (1997).[10] E. Knill, Nature , 39 (2005).[11] V. Aggarwal, A.R. Calderbank, G. Gilbert, Y.S. Wein-stein, Quant. Inf. Proc. , 541 (2010).[12] P. Aliferis and J. Preskill, Phys. Rev. A , 052331 (2008).[13] K.M. Svore, B.M. Terhal, D.P. DiVincenzo, Phys. Rev.A , 022317 (2005).[14] Y.S. Weinstein, Phys. Rev. A84