Errors and pseudo-thresholds for incoherent and coherent noise
Mauricio Gutiérrez, Conor Smith, Livia Lulushi, Smitha Janardan, Kenneth R. Brown
EErrors and pseudo-thresholds for incoherent and coherent noise
Mauricio Guti´errez , Conor Smith , Livia Lulushi , Smitha Janardan , and Kenneth R. Brown ∗ Schools of Chemistry and Biochemistry; Computational Science and Engineering; and PhysicsGeorgia Institute of Technology, Atlanta, GA 30332-0400 and Department of Physics, Lewis & Clark College, Portland, OR 97219 (Dated: November 5, 2018)We compare the effect of single qubit incoherent and coherent errors on the logical error rate of the Steane[[7,1,3]] quantum error correction code by performing an exact full-density-matrix simulation of an error cor-rection step. We find that the effective 1-qubit process matrix at the logical level reveals the key differencesbetween the error models and provides insight into why the Pauli twirling approximation is a good approxi-mation for incoherent errors and a poor approximation for coherent ones. Approximate channels composedof Clifford operations and Pauli measurement operators that are pessimistic at the physical level result in pes-simistic error rates at the logical level. In addition, we observe that the pseudo-threshold can differ by a factorof five depending on whether the error is calculated using the fidelity or the distance.
I. INTRODUCTION
Error thresholds for fault-tolerant quantum procedures aretypically calculated assuming random, independent Pauli er-rors [1–4]. This error model allows for efficient calculationof thresholds for quantum error correction (QEC) protocols[5–9]. Actual errors in the laboratory often differ from thisapproximation and the question remains if these errors are suf-ficiently small and independent for QEC. There are a numberof methods for measuring the error, from the average fidelity[10, 11], which is easy to measure experimentally [12–15], tothe diamond distance [11, 16], a more challenging measure-ment. For random Pauli errors, there is no substantial differ-ence between using the average fidelity or the diamond dis-tance to measure the gate error. Small incoherent errors canbe well approximated by Pauli errors. In turn, these Pauli er-ror models can accurately reproduce the behavior of quantumerror-correcting (QEC) protocols in the presence of incoher-ent channels [17–20].On the other hand, if the noise process is best described by aunitary operation, there is a large difference between the errormagnitude quantified by the fidelity and the diamond distance[21–23]. One could generate two distinct Pauli channels toapproximate the error. The first one corresponds to the Paulitwirling approximation, an optimistic model that matches theerror in the fidelity. The second one is a pessimistic model thatis constrained to not underestimate a distance-based measure,like the average trace distance or the diamond distance [24,25]. The latter are referred to as honest approximations.The question we address here is how optimistic or pes-simistic are these approximations in practice. We do this inthe context of exact simulation of the errors using the Steane[[7,1,3]] code [26]. We previously used this code to testexpanded efficient error models based on Clifford gates andPauli measurements for incoherent channels [19, 24, 25]. Inorder to understand the difference between incoherent errors, ∗ [email protected] such as amplitude damping, and coherent errors, such as anunwanted rotation, we start with the respective process ma-trices. We then examine the small error limit and performa polynomial expansion of each entry in the process matrixin terms of the error strength parameter and base our analy-sis on the leading order terms. We also construct effective1-qubit process matrices for a QEC step with the respectiveerror channels acting on the physical qubits. This method pro-vides a way to visualize how an error channel is modified byQEC. In real quantum information processing systems, errorsarise from both incoherent and coherent processes [27, 28].However, we focus on purely incoherent and coherent errorsto emphasize their respective distinctive features. We includetwo coherent error models in our analysis, an over-rotationabout the Z and the Hadamard, H , axes. Finally, we quantifythe error magnitude at the physical and logical levels usingvarious metrics.The paper is organized as follows. In Section II we intro-duce our effective 1-qubit process ( χ ) matrix method and givean illustrative example employing the 3-qubit bit-flip code.In Section III we present the physical process matrices andthe effective 1-qubit process for the model error channels andtheir approximations in the low noise limit. We also presentthe error magnitude of each channel for various error metrics.In Section IV we present our level-1 pseudo-threshold valuesfor the Steane [[7,1,3]] code under our model error channels.We compute the pseudo-threshold based on the different errormetrics. Finally, in Section V we conclude and we discussopen questions and future directions. II. ANALYSIS IN THE LIMIT OF LOW ERRORSTRENGTHS
We are interested in understanding why, at the logical error-corrected level, Pauli channels provide exceptionally good ap-proximations to incoherent error channels, but in general apoor one for coherent ones. Our analysis is based on the ef-fective 1-qubit process matrix for the whole circuit, includingthe encoding, occurrence of error, syndrome measurement, er- a r X i v : . [ qu a n t - ph ] M a y ror correction, and decoding. This strategy is motivated by theobservation that in our simulation scheme the final state of thequantum circuit is always completely localized on the logicalcodespace, so that the overall circuit can be compactly repre-sented by a 1-qubit process matrix.For perfect EC, after the stabilizer measurement and cor-rection, it is evident that the final state will live in the logicalcodespace. In general, this will not be the case when the ECis faulty, since errors during the measurement of the stabiliz-ers will cause the logical state to not be projected perfectlyonto the code subspaces. However, in our previous analysis[19], after faulty EC we always perform one round of per-fect EC, to account exclusively for uncorrectable errors. Thishas the effect of completely projecting the final state onto thecodespace. Effective noise channels at the logical EC levelhave been previously employed to study the entanglementin encoded quantum systems [29] and the error-suppressingproperties of the five-qubit code [30].We are particularly interested in the low error rate limit,where it is appropriate to Taylor-expand each entry in the pro-cess matrix in terms of powers of the error rate. This helpsus visualize in a very clear way which terms are more im-portant in determining the relevant characteristics of a givenerror channel. As an example, consider a 1-qubit coherent er-ror channel consisting of a rotation about the X axis by anangle θ : R X ( θ ) = exp( − iθX/ (1)In the normalized Pauli basis √ { I, X, Y, Z } , the processmatrix for this channel is: ( θ/ i sin( θ ) 0 0 − i sin( θ ) 2 sin ( θ/
2) 0 00 0 0 00 0 0 0
In the small error limit ( θ → ), this becomes: − θ / O ( θ ) iθ + O ( θ ) 0 0 − iθ + O ( θ ) θ / O ( θ ) 0 00 0 0 00 0 0 0 The Pauli twirled approximation to this channel is given bythe diagonal entries of its process matrix: ( θ/
2) 0 0 00 2 sin ( θ/
2) 0 00 0 0 00 0 0 0 which corresponds to a channel where the qubit is flippedwith a probability p x = sin ( θ/ : (cid:40) √ − p x I √ p x X (2) For illustrative purposes, imagine the situation where weuse the 3-qubit bit flip code (with stabilizers generated by ZZI and
IZZ ). We perfectly encode our qubit, then 3 in-dependent error instances happen, 1 on each qubit, and finallywe perfectly measure the stabilizer generators, correct, anddecode. If the individual errors correspond to flips with prob-ability p x , the effective channel for the whole circuit is givenby: (cid:40)(cid:112) (1 − p x ) + 3(1 − p x ) p x I (cid:112) − p x ) p x + p x X (3)This channel is still a probabilistic application of an X op-erator. The first Kraus operator corresponds to the situationwhere either no flip or 1 flip occurred. The second Kraus op-erator accounts for the case where 2 or 3 flips occurred, thuscausing a logical X error. If p x = sin ( θ/ , this channel’sreduced process matrix is: (cid:18) − p x ) + 6(1 − p x ) p x
00 2 p x + 6(1 − p x ) p x (cid:19) Here we have only focused on the first 2 rows and columnsof the 1-qubit process matrix. (All the other entries are .)On the other hand, if the 3 independent errors are coherentrotations about the X axis by an angle θ , the effective reducedprocess matrix for the whole circuit is: (cid:18) − p x ) + 6(1 − p x ) p x i p x (1 − p x )] / − i p x (1 − p x )] / p x + 6(1 − p x ) p x (cid:19) Interestingly, for a circuit where the errors are the Paulitwirled approximation to the coherent channels, the effectiveprocess matrix for the whole circuit still matches the diagonalentries perfectly. However, it is completely unable to matchthe off-diagonal entries. In the limit of small error, this be-comes: (cid:18) − (3 / θ + O ( θ ) iθ + O ( θ ) − iθ + O ( θ ) (3 / θ + O ( θ ) (cid:19) At the physical level, the process matrix for the R X ( θ ) channel has diagonal entries proportional to θ and off-diagonal entries proportional to θ . At the logical level withperfect EC, the leading orders get supressed and the effec-tive process matrix now has diagonal entries proportional to θ and off-diagonal ones proportional to θ . In this case, thePauli twirled approximation underestimates the magnitude ofthe error by 1 power of θ both at the physical and logical lev-els. III. EFFECTIVE PROCESS MATRICES FORINCOHERENT AND COHERENT CHANNELS
We have followed the same exact full-density-matrix pro-cedure explained in [19] to compute the final states after errorcorrection with the Steane code. Because of the size of thedensity matrices and the time it takes to cover all the possiblesyndrome branches, we are unable to obtain symbolic expres-sions for the effective 1-qubit process matrices. Instead, weuse quantum process tomography [31–34] to reconstruct thenumerical process matrix for various error strengths and sub-sequently fit each entry to a polynomial to determine the lead-ing order and its coefficient. We test several polynomial fitsand select the one with the smallest total variance. In everycase, the relative variance (variance divided by the value ofthe leading order coefficient) was less than − . Throughoutthe paper, we will refer to the effective 1-qubit process matrixsimply as the process matrix at the logical level.We report the error magnitude for each channel and its ap-proximations using 3 different metrics: the average error rate(average fidelity error), the average trace distance, and the dia-mond distance. Each one of these has a particular importancein our analysis. The average error rate [11] is the measureof choice when experimentally characterizing quantum gatesand it can be efficiently calculated in the laboratory [13, 14].Usually in the literature, the error rate is defined with respectto an ideal gate U . Throughout the paper, the ideal gate U will always be the Identity. There is no loss of generality withthis assumption, since it is equivalent to an interaction picturewhere the error channel E actually corresponds to the discrep-ancy channel between the real operation and the ideal gate.We define the average error rate, r , for a noise process E as: r ( E ) = (cid:104) − F (cid:105) = 1 − (cid:68) (cid:104) ψ |E ( | ψ (cid:105)(cid:104) ψ | ) | ψ (cid:105) (cid:69) , (4)where the average is defined over the space of pure states .Whenever possible, we use the process matrix to analyticallyintegrate over the Bloch sphere surface and obtain the exactexpressions for both the average and the standard deviation.For the cases where the exact symbolic expressions are chal-lenging to compute, we select 150 uniformly distributed stateson the surface of the Bloch sphere and integrate numerically.More specifically, we perform the following procedure:1. For a given initial state, compute the error rate (fidelityerror) at 3 different error strengths.2. Test several polynomial fits, select the one with thesmallest total variance, and store the coefficient of theleading order.3. Repeat this for each one of the 150 initial states.4. Calculate the average and the standard deviation of theset of 150 leading order coefficients.The average trace distance for the error channel E is definedas: (cid:104) D tr (cid:105) = 12 (cid:68) Tr |E ( ρ ) − ρ | (cid:69) , for ρ = | ψ (cid:105)(cid:104) ψ | . (5) In our previous work and in older literature, the fidelity between a purestate, | ψ (cid:105) , and a general state, σ , was defined as (cid:112) (cid:104) ψ | σ | ψ (cid:105) . In this paper,we follow the more recent convention and define the fidelity as (cid:104) ψ | σ | ψ (cid:105) . Once again, the average is calculated analytically wheneverpossible or numerically by following the procedure describedabove. The trace distance is important because in our previouswork [19, 24] we have used it to distinguish between honestand dishonest approximations to error channels.Finally, as a worst-case measure with several useful prop-erties, the diamond distance is the preferred metric in the con-text of fault tolerance [11, 16]. The diamond distance betweenchannels E and F is defined as: D (cid:5) = 12 max ρ || ( E ⊗ I )( ρ ) − ( F ⊗ I )( ρ ) || , (6)where, as mentioned before, the channel F will always betaken as the Identity. For a channel acting on a vector space ofdimension d , the 1-norm is maximized over a vector space ofdimension d . For a linear operator A , the 1-norm is definedas: || A || = Tr (cid:104) √ A † A (cid:105) (7)To find the coefficient of the leading order term in the dia-mond distance, we follow the same procedure described pre-viously: we test several polynomial fits and select the onewith the smallest total variance. To compute the diamond dis-tance we use QETLAB’s diamond norm function [35]. All theother calculations are done with our own python-based soft-ware tools. A. Efficiently simulable approximate channels
In our previous work, we introduced several classicallytractable noise models [24]. These noise models correspondto Kraus channels where each operator is efficiently simula-ble in the stabilizer formalism. The free parameters in ournoise models are the probabilities associated with the Krausoperators. To approximate a target channel, we minimize theHilbert-Schmidt distance [36] between the process matricesof the classically tractable noise model and the target channel.For each target non-Clifford error channel, we study two dif-ferent models: (a) the Pauli channels (PC), which employ onlysingle-qubit Pauli operators, and (b) the expanded channelsor Clifford+measurements channels (CMC), which includeall the single-qubit Clifford operators and the measurement-induced translations. When minimizing the Hilbert-Schmidtdistance between channels, we have the option to perform aconstrained minimization, in which we enforce that for everyinitial pure state its trace distance to the resulting state afterthe target transformation is not greater than its trace distanceto the resulting state after the model transformation. Approx-imate channels that satisfy this condition are referred to ashonest, since they do not underestimate the magnitude of thetarget error. We label unconstrained approximations as “a”and constrained ones as “w”. Finally, we also include in ouranalysis the depolarizing channel (DC), a Pauli channel wherethe probabilities of the X , Y , and Z operators are equal. Theapproximations are summarized in Table I. TABLE I. Summary of the various target and approximate channels.Channel Complete name Honesty constrainedADC amplitude damping –PolC polarization along non-Clifford axis –RZC rotation about the Z axis –RHC rotation about the H axis –PCa Pauli noPCw Pauli yesCMCa Clifford+measurements noCMCw Clifford+measurements yesDC Depolarizing channel no B. Incoherent channels
We define an incoherent channel as a quantum operation E that maps, at least, 1 pure state ρ = | ψ (cid:105)(cid:104) ψ | to a mixedstate σ . As in our previous work [19, 24], we have selected2 representative 1-qubit incoherent channels: the amplitudedamping channel (ADC) and a depolarizing channel about thenon-Clifford π/ axis on the XY plane of the Bloch sphere(Pol π/ C).
1. ADC
Table II presents the process matrices for the ADC and itsapproximations at the physical level and logical level withfaulty EC. Table III describes how the error magnitude forthese channels scales with the damping strength, γ , for themetrics introduced in the previous section. In both tables,the results refer to the behavior in the limit of small damp-ing strength. For the average error rate and the average tracedistance, the standard deviation is also presented. Standarddeviations below − are not reported.There are several interesting trends. At the physical level,the entries of the ADC process matrix are all linear in γ , ex-cept for the χ zz term, which is quadratic. Consequently, theerror magnitude is linear, regardless of the metric used. Atthe logical level, all the linear terms are suppressed, whichconfirms that the Steane code’s correcting procedure is indeedfault tolerant and successful in suppressing single errors. Theterms on the diagonal entries are now proportional to γ , whilethe off-diagonal terms are proportional to γ . At this level, theerror magnitude is quadratic.At the physical level, the PCa matches the diagonal entriesperfectly, since it corresponds to the Pauli twirling approxi-mation (PTA). Its average error rate is also identical to that ofthe ADC, a known property of the PTA. The average trace dis-tance, however, is less than that of the ADC, and therefore weclassify this channel as dishonest. At the logical level, the PCa still approximates the diagonal terms very closely, but not ex-actly, which shows that the off-diagonal terms on the physicalprocess matrix can influence the diagonal terms on the logicalprocess matrix. The error magnitude is practically equivalentto that of the ADC up to second order. In the case of the PCw,the constrained Pauli approximation, at the physical level allthe terms are linear, which guarantees that the error is not un-derestimated. At the logical level, this pessimistic behaviorbecomes even more pronounced, resulting in a very honest,but inaccurate approximation.In contrast to the Pauli channels, the expanded (CMC)channels have access to the off-diagonal entries in the processmatrix. At the physical level, this allows both the constrainedand unconstrained approximations of the expanded channelsto be more accurate. While an advantage at the physical level,the access to the off-diagonal entries becomes unfavorable atthe logical level. The CMCw is a great example: at the logicallevel, it matches the off-diagonal terms almost perfectly, butthis is useless because its approximation to the leading orderin the target channel ( γ ) is less accurate than that achieved bythe PCa, especially on the χ zz entry. In other words, for theADC at the logical level, the critical requirement for a goodapproximation is to be able to match the diagonal entries ac-curately, since these contain the leading orders in the errormagnitude.The constrained approximations at the physical level re-main honest at the logical level, for every error measure used.On the other hand, the unconstrained approximations remaindishonest, with the notable exception of the PCa, which ispractically honest up to second order in γ . Finally, for everychannel and at every level, the average trace distance is con-sistently about twice as the average error rate. The diamonddistance is about 3-4 times larger than the average error rate.
2. Pol π/ C Table IV presents the process matrices for the Pol π/ C andits approximations at the physical level and the logical levelwith faulty EC. Table V describes how the error magnitudescales with p , the depolarizing strength.The trends are very similar to the ones observed on theADC. In this case, at the physical level, all the non-zero en-tries in the process matrices have linear leading orders, whichmeans once again that the error magnitude is linear in the errorstrength. In terms of the average trace distance, at the physi-cal level, the constrained approximations are honest, while theunconstrained ones are dishonest. However, this does not holdfor the average error rate. First, as expected, the average errorrate of the Pol π/ C is equal to the PCa’s. Also, the CMCa re-sults in an honest approximation in terms of the average errorrate.As observed for the ADC and its approximations, at thelogical level, the leading terms in the process matrices be-come quadratic for the diagonal entries, but cubic for the off-
TABLE II. Process matrices for the ADC and its approximations at the physical level and logical level with faulty EC in the low damping limit( γ → ). Only the leading orders are shown.Channel Physical process matrix Effective process matrix at the logical levelADC − O ( γ ) 0 0 γ/ γ/ − iγ/ iγ/ γ/ γ/ γ / − O ( γ ) 0 0 − γ γ i γ − i γ γ − γ γ PCa − O ( γ ) 0 0 00 γ/ γ/ γ / − O ( γ ) 0 0 00 1570 γ γ
00 0 0 491 γ PCw − O ( γ ) 0 0 00 1 . γ . γ
00 0 0 0 . γ − O ( γ ) 0 0 00 7080 γ γ
00 0 0 3020 γ CMCa − O ( γ ) 0 0 3 γ/
80 3 γ/ − i γ/ i γ/ γ/ γ/ γ/ − O ( γ ) 0 0 − . γ γ i . γ − i . γ γ − . γ γ CMCw − O ( γ ) 0 0 γ/ γ/ − iγ/ iγ/ γ/ γ/ γ/ − O ( γ ) 0 0 − γ γ i γ − i γ γ − γ γ DC − O ( γ ) 0 0 00 γ/ γ/ γ/ − O ( γ ) 0 0 00 773 γ . γ
00 0 0 601 γ TABLE III. Behavior of the ADC and its approximations at various levels in low damping limit ( γ → ) for the 3 different error metrics. Onlythe leading orders are shown. Standard deviations smaller than − are not presented.Channel Physical level Logical level with faulty EC (cid:104) − F (cid:105) (cid:104) D tr (cid:105) D (cid:5) (cid:104) − F (cid:105) / (cid:104) D tr (cid:105) / D (cid:5) / ADC (0 . ± . γ (0 . ± . γ γ (0 . ± . γ (0 . ± . γ . γ PCa (0 . ± . γ (0 . ± . γ γ/ . ± . γ (0 . ± . γ . γ PCw (0 . ± . γ (0 . ± . γ . γ (3 . ± . γ (3 . ± . γ . γ CMCa (0 . ± . γ (0 . ± . γ γ/ . ± . γ (0 . ± . γ . γ CMCw (0 . ± . γ (0 . ± . γ γ (1 . ± . γ (1 . ± . γ . γ DC γ/ γ/ γ/ . ± . γ (0 . ± . γ . γ TABLE IV. Process matrices for the Pol π/ C and its approximations at the physical level and logical level with faulty EC in the low noise limit( p → ). Only the leading orders are shown.Channel Physical process matrix Effective process matrix at the logical levelPol π/ C − O ( p ) 0 0 00 (1 + 1 / √ p (1 / √ p
00 (1 / √ p (1 − / √ p
00 0 0 0 − O ( p ) 0 0 − i p p p
00 337 p . p i p p PCa − O ( p ) 0 0 00 (1 + 1 / √ p − / √ p
00 0 0 0 − O ( p ) 0 0 00 5860 p . p
00 0 0 851 p PCw − O ( p ) 0 0 00 (1 + 1 / √ p − / √ p
00 0 0 (1 / √ p − O ( p ) 0 0 00 6350 p . p
00 0 0 2190 p CMCa − O ( p ) 0 0 00 3(3 + 1 / √ p/ / √ p/ / √ p/ / √ p/ − O ( p ) 0 0 − i p p p
00 142 p p i p p CMCw − O ( p ) 0 0 00 (1 + 1 / √ p (1 + 1 / √ p/ / √ p/ / √ p/ − √ p/ − O ( p ) 0 0 − i p p p
00 176 p p i p p DC − O ( p ) 0 0 00 2 p/ p/ p/ − O ( p ) 0 0 00 3090 p p
00 0 0 2400 p TABLE V. Behavior of the Pol π/ C and its approximations at various levels in low noise limit ( p → ) for the 3 different error metrics. Onlythe leading orders are shown. Standard deviations smaller than − are not presented.Channel Physical level Logical level with faulty EC (cid:104) − F (cid:105) (cid:104) D tr (cid:105) D (cid:5) (cid:104) − F (cid:105) / (cid:104) D tr (cid:105) / D (cid:5) / Pol π/ C (0 . ± . p (0 . ± . p p (2 . ± . p (2 . ± . p . p PCa (0 . ± . p (0 . ± . p p (2 . ± . p (2 . ± . p . p PCw (0 . ± . p (0 . ± . p . p (2 . ± . p (3 . ± . p . p CMCa (0 . ± . p (0 . ± . p . p (2 . ± . p (3 . ± . p . p CMCw (0 . ± . p (0 . ± . p . p (3 . ± . p (3 . ± . p . p DC p/ p/ p (1 . ± . p (1 . ± . p . p diagonal ones. This implies that, just like for the ADC, theadvantage of the CMC approximations at the physical levelbecomes a drawback at the logical level, since the diagonalterms are more important than the off-diagonal.There is an interesting difference between the effective pro-cess matrices at the logical level for the ADC and the Pol π/ C.For the former, the zero entries at the physical level remainedzero at the logical level. For the latter, however, this is nottrue. For example, whereas the χ zz entry is zero at the phys-ical level for the Pol π/ C and some of its approximations, atthe logical level there is a term proportional to p . The PCaprovides the most intuitive case to understand where this termis coming from. At the physical level, only X and Y errorsoccur. However, for the Steane code with perfect EC, cer-tain combinations of 1 X and 2 Y errors can result in an un-correctable logical Z error (For example, IIIXY Y I ), whichgive rise to a term in the χ zz entry proportional to the thirdpower of the error strength. On the other hand, if the EC isfaulty, a Y error on the ancillae and another one on the datacan cause a logical Z error, thus turning the error strength pro-portional to p .To summarize, we have found several common features forthe 2 incoherent channels analyzed:1. The expanded (CMC) channels provide more accurateapproximations than the Pauli channels at the physicallevel. However, at the logical level, they become com-pletely eclipsed by the high accuracy of the PCa. Thischaracteristic of the PCa has been observed previously[17–19].2. The high accuracy of the PCa arises, somewhat ironi-cally, from its inability to approximate the off-diagonalterms of the target error process matrix. At the logi-cal EC level, the off-diagonal terms in the process ma-trix are weaker (proportional to the cube of the errorstrength) than the diagonal terms (proportional to thesquare of the error strength), so in the low noise limit,the important contribution to the error at the logicallevel is really made by the diagonal terms.3. At the physical level, the error magnitude for all thechannels is linear in the error strength, whereas at thelogical level, the error magnitude is quadratic in the er-ror strength. This holds for every error metric we havestudied. The average trace distance is consistently abouttwice as the average error rate, while the diamond dis-tance is 3-4 times larger than the average error rate. C. Coherent channels
We define a coherent channel as a quantum operation thatmaps pure states to pure states. They correspond to unitaryrotations about a given axis in the Hilbert space of the system.As model coherent errors, we have selected rotations aboutthe Bloch sphere’s Z and H (Hadamard) axes by an angle θ : RZC = exp( − iθZ/
2) = cos( θ/ I − i sin( θ/ Z (8)RHC = exp( − iθH/
2) = cos( θ/ I − i sin( θ/ H (9)These channels arise if there is an unwanted Hamiltonianduring the gates, for example, from an uncontrolled magneticfield. The angle θ parametrizes the error strength. Just likefor the incoherent channels, we have selected 150 uniformlydistributed states on the Bloch sphere surface to calculate theaverage error rate and the average trace distance.
1. RZC
Table VI presents the process matrices for the RZC andits approximations at the physical level and logical level withfaulty EC. Table VII describes how the error magnitude scaleswith θ , the over-rotation angle, in the limit θ → .The process matrix for the RZC reveals very different char-acteristics from those seen in the incoherent channels. At thephysical level, the process matrix has a quadratic term on itsdiagonal and a linear one on its off-diagonal entries. Sincethe average error rate depends only on the diagonal terms, thisimplies, as seen in Table VII, that there is a considerable mis-match between the average error rate and the distance-basedmetrics. While for both the average trace distance and the dia-mond distance the error magnitude is linear in θ , for the aver-age error rate it is quadratic in θ . This agrees with the scalingof the diamond distance with the average error rate reportedby Kueng et al. [22] and Wallman [21].At the logical level, although not completely coherent any-more, the effective process matrix for the RZC still holdssome features of its unitary nature at the physical level. Inthis case, the diagonal term becomes quartic, while the off-diagonal ones becomes cubic in θ . This implies that the errormagnitude becomes proportional to θ when quantified by theaverage error rate, but proportional to θ when quantified by adistance-based measure.These unique characteristics of the process matrices of theRZC make the approximation by the PCa problematic. Asexpected, at the physical level the PCa matches the diagonalterms and the average error rate exactly, but this means it pre-dicts a quadratic error, when in reality it is linear. In terms ofthe average trace distance and the diamond distance, it under-estimates the real error by one order of magnitude, making itvery dishonest. At the logical level, this extreme dishonesty ismaintained. Once again, in terms of the distance-based mea-sures, the real error is underestimated by one order of mag-nitude ( θ vs. θ ). For the average error rate, something in-teresting occurs: although the error order is the same ( θ ),because the off-diagonal terms in the physical process matrixinfluence the diagonal terms in the effective logical processmatrix, the error magnitude is still severely underestimated. TABLE VI. Process matrices for the RZC and its approximations at physical level and logical level with EC in the limit of small rotation angle( θ → ). Only the leading orders are shown.Channel Physical process matrix Effective process matrix at the logical levelRZC − O ( θ ) 0 0 iθ − iθ θ / − O ( θ ) 0 0 i θ − i θ θ PCa − O ( θ ) 0 0 00 0 0 00 0 0 00 0 0 θ / − O ( θ ) 0 0 00 0 0 00 0 0 00 0 0 206 θ PCw − O ( θ ) 0 0 00 0 0 00 0 0 00 0 0 θ − O ( θ ) 0 0 00 0 0 00 0 0 00 0 0 824 θ CMCa − O ( θ ) 0 0 iθ/
20 0 0 00 0 0 0 − iθ/ θ/ − O ( θ ) 0 0 i . θ − i . θ θ CMCw − O ( θ ) 0 0 iθ/ √
20 0 0 00 0 0 0 − iθ/ √ θ/ √ − O ( θ ) 0 0 i θ − i θ θ DC − O ( θ ) 0 0 00 θ / θ / θ / − O ( θ ) 0 0 00 193 θ . θ
00 0 0 150 θ TABLE VII. Behavior of the RZC and its approximations at various levels in limit of small rotation angle ( θ → ) for the 3 different errormetrics. Only the leading orders are shown. Standard deviations smaller than − are not presented.Channel Physical level Logical level (cid:104) − F (cid:105) (cid:104) D tr (cid:105) D (cid:5) (cid:104) − F (cid:105) / (cid:104) D tr (cid:105) / D (cid:5) / RZC (0 . ± . θ (0 . ± . θ θ/ . ± . θ (0 . ± . θ . θ PCa (0 . ± . θ (0 . ± . θ θ / . ± . θ (0 . ± . θ . θ PCw (0 . ± . θ (0 . ± . θ θ/ . ± . θ (0 . ± . θ . θ CMCa (0 . ± . θ (0 . ± . θ . θ (0 . ± . θ (0 . ± . θ . θ CMCw (0 . ± . θ (0 . ± . θ θ/ . ± . θ (0 . ± . θ . θ DC θ / θ / θ / . ± . θ (0 . ± . θ . θ As can be seen from Table VII, at the logical level, the aver-age error rate for the the RZC is about 37 times larger thanfor the PCa. This highlights a very important limitation of thePCa: its inability to match off-diagonal terms turns it into avery bad approximate channel for coherent operations.In order to not underestimate the error magnitude, at thephysical level the PCw approximation results in a process ma-trix with a linear term on the diagonal. This implies that theerror magnitude is linear for every metric. In terms of theaverage error rate, the error is overestimated by one order ofmagnitude ( θ vs. θ ). Interestingly, in terms of both the av-erage trace distance and the diamond distance, the honestyconstraint is tight: the PCw results in exactly the same errormagnitude as the RZC. At the logical level, however, the PCwbecomes too pessimistic. At this level, the error magnitude isproportional to θ for every error metric.The expanded (CMC) approximations both result in pro-cess matrices at the physical level with linear terms on theoff-diagonal and the diagonal entries. Consequently, the errormagnitude is linear for every error metric. As expected, theCMCa approximation is dishonest when quantified with theaverage trace distance and the diamond distance. However, itis honest by one order of magnitude when quantified with theaverage error rate. The CMCw approximation is honest forevery error metric, and just like the PCw, it saturates the hon-esty bound when quantified by the average trace distance andthe diamond distance. At the logical level, both approximatechannels result in effective process matrices with a quadraticterm on the diagonal and cubic terms on the off-diagonal en-tries, just like for the incoherent channels studied in the pre-vious section. This means that the error magnitude is propor-tional to θ , regardless of the metric employed. Both CMCapproximations overestimate the error magnitude by 2 ordersof θ when employing the average error rate, and 1 order of θ when employing the distance-based metrics.
2. RHC
Table VIII presents the process matrices for the RHC andits approximations at the physical level and logical level withfaulty EC. Table IX describes how the error magnitude scaleswith θ in the limit of θ → .The results are very similar to the RZC. At the physicallevel, the process matrix has off-diagonal quadratic terms andlinear diagonal terms. This implies, once again, that the aver-age error rate is quadratic in θ , while the average trace distanceand the diamond distance are linear in θ .In contrast to the RZC, at the logical level, the effective pro-cess matrix for the RHC is considerably less sparse than at thephysical level. However, the important trends are maintained.The strongest terms are proportional to θ on the diagonal andto θ on the off-diagonal. Once again, because of its lackof access to the off-diagonal entries, the PCa provides an ex- tremely optimistic (and dishonest) approximation both at thephysical and logical levels.The PCw provides an honest approximation at the physicallevel, but this results in an overly pessimistic one at the logicallevel: it predicts the diamond distance to scale like θ , whenin reality it scales like θ . Regarding the expanded (CMC)approximations, at the logical level, they result in effectiveprocess matrices where the strongest terms scale like θ onthe diagonal and θ on the off-diagonal. This means that theyoverestimate the error magnitude by 2 orders of magnitude,when employing the average error rate, and 1 order of mag-nitude, when employing the distance-based metrics. The ex-panded approximate channels (CMCs) show the same trendsas in the RZC case.To summarize, the common features for the 2 coherentchannels analyzed are:1. Just like for the incoherent operations, the expanded(CMC) channels provide more accurate approximationsthen the Pauli channels at the physical level.2. In contrast to incoherent operations, at the logical levelthe PCa results in a very bad approximation. Formally,it is still the most accurate approximation, since its dif-ference from the target channel is proportional to θ (the error magnitude of the target channel), while all theother approximations differ from the target channel bya term proportional to θ (the error magnitude of the ap-proximate channel). However, the PCa underestimatesthe error magnitude of the real error so severely that, forpractical purposes, it is a very bad approximation. Thisholds for every error metric used, but it is more pro-nounced for the average trace distance and the diamonddistance.3. The severe underestimation of the error magnitude bythe PCa is caused by its lack of access to off-diagonalterms in the process matrix. In contrast to incoher-ent channels, coherent rotations have process matriceswhose off-diagonal terms are stronger by 1 order of theerror strength, θ , than the diagonal terms.4. In contrast to incoherent channels, no approximation re-sults in a very accurate description at the logical level.While the PCa severely underestimates the error mag-nitude of the target coherent channel, the rest of theapproximations overestimate it, because of this partic-ular trait of coherent channels of having stronger off-diagonal than diagonal terms. At the logical level, wecannot approximate accurately the behavior of the co-herent channels, at least not with the methods devel-oped in [24]. However, we can use the PCa to provide alower bound and the other approximate channels to pro-vide an upper bound to the error magnitude. The CMCa(unconstrained expanded approximation) provides thetighest upper bound.5. In contrast to incoherent channels, the error magnitudeof coherent channels greatly depends on the metric em-0 TABLE VIII. Process matrices for the RHC and its approximations at physical level and logical level with EC in the limit of small rotationangle ( θ → ). Only the leading orders are shown.Channel Physical process matrix Effective process matrix at the logical levelRHC − O ( θ ) iθ/ √ iθ/ √ − iθ/ √ θ / θ /
40 0 0 0 − iθ/ √ θ / θ / − O ( θ ) i θ O ( θ ) i θ − i θ (1 . × ) θ O ( θ ) O ( θ ) O ( θ ) O ( θ ) O ( θ ) O ( θ ) − i θ O ( θ ) O ( θ ) (1 . × ) θ PCa − O ( θ ) 0 0 00 θ / θ / − O ( θ ) 0 0 00 107 θ O ( θ ) 00 0 0 83 . θ PCw − O ( θ ) 0 0 00 θ/ θ/ θ/ − O ( θ ) 0 0 00 1710 θ θ
00 0 0 1340 θ CMCa − O ( θ ) i . θ i . θ − i . θ . θ − i . θ . θ − O ( θ ) i . θ O ( θ ) i . θ − i . θ θ O ( θ ) O ( θ ) O ( θ ) O ( θ ) (1 . × ) θ O ( θ ) − i . θ O ( θ ) O ( θ ) 106 θ CMCw − O ( θ ) i . θ i . θ − i . θ . θ . θ . θ − i . θ . θ . θ − O ( θ ) i θ O ( θ ) i θ − i θ θ O ( θ ) − . θ O ( θ ) O ( θ ) 31 . θ O ( θ ) − i θ − . θ O ( θ ) 516 θ DC − O ( θ ) 0 0 00 θ / θ / θ / − O ( θ ) 0 0 00 193 θ . θ
00 0 0 150 θ TABLE IX. Behavior of the RHC and its approximations at various levels in limit of small rotation angle ( θ → ) for the 3 different metrics.Only the leading orders are shown. Standard deviations smaller than − are not presented.Channel Physical level Logical level (cid:104) − F (cid:105) (cid:104) D tr (cid:105) D (cid:5) (cid:104) − F (cid:105) / (cid:104) D tr (cid:105) / D (cid:5) / RHC (0 . ± . θ (0 . ± . θ θ/ . ± . θ (0 . ± . θ . θ PCa (0 . ± . θ (0 . ± . θ θ / . ± . θ (0 . ± . θ . θ PCw θ/ θ/ θ/ . ± . θ (1 . ± . θ . θ CMCa (0 . ± . θ (0 . ± . θ . θ (0 . ± . θ (0 . ± . θ . θ CMCw (0 . ± . θ (0 . ± . θ . θ (0 . ± . θ (0 . ± . θ . θ DC θ / θ / θ / . ± . θ (0 . ± . θ . θ D (cid:5) ∝ e / ). However, at the log-ical level, the exponent of the scaling becomes / ( D (cid:5) ∝ e / ). As shown by Kueng et al. [22] and Wall-man [21], a scaling exponent of / is a characteristicsignature of coherent noise. On the other hand, com-pletely Pauli noise has a scaling exponent of . There-fore, at the logical level, the effective noise, althoughnot yet Pauli, becomes less coherent than the physicalnoise. IV. LEVEL-1 PSEUDO-THRESHOLDS
As in [19], the pseudo-threshold is calculated by finding theintersection between the error magnitude curve at the physicallevel with the error magnitude curve at the logical level. Forthe approximate channels, we simulate the real error exactlyat the physical level and use the approximations only at thelogical level. We follow the notation convention of [19] andrefer to these as exact channel / approximate channel . Thismakes the comparisons between the different thresholds esti-mates easier, since they will only depend on the behavior ofthe approximate channels at the logical level. We report thepseudo-thresholds obtained by the different error magnitudemetrics. TABLE X. Level-1 pseudo-thresholds for the ADC and its approxi-mations for the 3 different error magnitude metrics.Channel (cid:104) − F (cid:105) (cid:104) D tr (cid:105) D (cid:5) γ th × γ th × γ th × ADC . ± . . ± . . ADC / PCa . ± . . ± . . ADC / PCw . ± .
86 1 . ± .
84 1 . ADC / CMCa . ± . . ± . . ADC / CMCw . ± . . ± . . ADC / DC . ± . . ± . . Tables X and XI present the level-1 pseudo-thresholds forthe ADC and for Pol π/ C and their corresponding approxima-tions, respectively. The pseudo-thresholds are given in terms
Physical ADCLogical ADCLogical PCaLogical PCwLogical CMCaLogical CMCwLogical DC A v e r a g e e rr o r r a t e −8 −6 −4 −2 γ10 −5 −4 −3 FIG. 1. Average error rate for the ADC and its approximations forvarious damping strengths. The curve for the PCa at the logical levelis located exactly underneath the curve for the ADC at the logicallevel. The level-1 pseudo-thresholds are given by the intersectionbetween the best fits for the physical and the logical error magni-tudes. The lines are just guides to the eye and do not correspond tothe best fits.
Physical ADCLogical ADCLogical PCaLogical PCwLogical CMCaLogical CMCwLogical DC D i a m o n d d i s t a n c e −8 −6 −4 −2 γ10 −5 −4 −3 FIG. 2. Diamond distance for the ADC and its approximations forvarious damping strengths. The curve for the PCa at the logical levelis located exactly underneath the curve for the ADC at the logicallevel. The lines are guides to the eye. of the characteristic error strength: the damping strength ( γ )for the ADC and the error probability p for the Pol π/ C. The2
TABLE XI. Level-1 pseudo-threshold Pol π/ C and its approxima-tions for the 3 different error magnitude metrics.Channel (cid:104) − F (cid:105) (cid:104) D tr (cid:105) D (cid:5) p th × p th × p th × Pol π/ C . ± . . ± . . Pol π/ C / PCa . ± . . ± . . Pol π/ C / PCw . ± .
87 2 . ± .
73 2 . Pol π/ C / CMCa . ± .
91 2 . ± .
83 2 . Pol π/ / CMCw . ± .
77 2 . ± .
69 2 . Pol π/ C / DC .
50 4 .
07 3 . pseudo-threshold values in terms of the average error rate, theaverage trace distance, and the diamond distance can be eas-ily calculated from the expressions on Tables III and V. Fig-ure 1 shows how the average error rate scales with the damp-ing strength, γ , for the ADC and its approximations. Figure2 shows the scaling of the diamond distance. The pseudo-threshold for the ADC and its approximations can be visuallyobtained from these.There are important common characteristics for both in-coherent channels. First, and as observed in [19], the PCaestimates the pseudo-threshold very accurately. Its pseudo-thresholds are practically the same as the exact ones regard-less of the error metric used. Second, since the constrained(“w”) channels remain honest at the logical level, they alwaysprovide lower bounds to the pseudo-threshold. This is notthat useful when approximating incoherent channels, since thePCa is so accurate, but it will be useful for the coherent errors.Finally, the pseudo-threshold does not depend too much on theerror metric. In the case of the ADC, the diamond distancepseudo-threshold ( . × − ) is almost twice as the averageerror rate pseudo-threshold ( . × − ), but they are both inthe same order of magnitude. For the Pol π/ C, the pseudo-thresholds obtained from the different metrics are practicallythe same.
TABLE XII. Level-1 pseudo-thresholds for the RZC and its approx-imations for the 3 different error magnitude metrics.Channel (cid:104) − F (cid:105) (cid:104) D tr (cid:105) D (cid:5) θ th × θ th × θ th × RZC .
92 39 . . RZC / PCa . RZC / PCw .
22 1 . RZC / CMCa .
84 4 . RZC / CMCw .
40 2 . RZC / DC ±
11 147 ±
24 142
Tables XII and XIII present the level-1 pseudo-thresholdfor the RZC and the RHC and their corresponding approxima-tions, respectively. The pseudo-thresholds are given in terms
Physical RZCLogical RZCLogical PCaLogical PCwLogical CMCaLogical CMCwLogical DC A v e r a g e e rr o r r a t e −12 −9 −6 −3 −4 −3 −2 FIG. 3. Average error rate for the RZC and its approximations forvarious over-rotation angles. The curves for the PCw, CMCa, andCMCw do not intersect the physical RZC curve because they allscale quadratically with θ , but the coefficient of the RZC curve isthe smallest. In this case, the PCw and the expanded channels pro-vide a useless lower bound ( ) to the exact pseudo-threshold. Thelines are guides to the eye. Physical RZCLogical RZCLogical PCaLogical PCwLogical CMCaLogical CMCwDC D i a m o n d d i s t a n c e −12 −9 −6 −3 −4 −3 −2 FIG. 4. Diamond distance for the RZC and its approximations forvarious over-rotation angles. The lines are guides to the eye. of the over-rotation angle, θ . Figure 3 shows how the aver-age error rate scales with the over-rotation angle for the RZC.Figure 4 shows the scaling of the diamond distance.3 TABLE XIII. Level-1 pseudo-threshold for the RHC and its approx-imations for different error metrics.Channel (cid:104) − F (cid:105) (cid:104) D tr (cid:105) D (cid:5) θ th × θ th × θ th × RHC . ± . . ± . . RHC / PCa ±
14 181 ±
21 174
RHC / PCw . ± .
12 0 . RHC / CMCa . ± . . RHC / CMCw . ± .
30 0 . RHC / DC ±
11 147 ±
24 142
The level-1 pseudo-threshold trends are very different forcoherent channels. The PCa gives an overly optimistic esti-mate of the pseudo-threshold. The PCw and the expandedchannels give an overly pessimistic pseudo-threshold. Thisholds for every error metric used. In contrast to the inco-herent case, none of our approximations predicts the pseudo-threshold accurately, although we can bound the pseudo-threhsold. The PCa will always give an upper bound, whilethe PCw and expanded channels will always give a lowerbound. Notice, however, that this lower bound is not usefulat all if we employ the average error rate as our metric. Inthis case, the pseudo-threshold given by the PCw and the ex-panded approximations is exactly 0. When the error is quan-tified by the average trace distance and the diamond distance,the lower bound is tighter.For coherent channels, the level-1 pseudo-threshold quan-tified by the diamond distance is 1 order of magnitude higherthan the one quantified by the average error rate. This is quiteunexpected, since the assumption is that the threshold valuegiven by the diamond distance would actually be lower thanthe one given by the average error rate, because the diamonddistance is a worst-case measure. Intuitively, this is a conse-quence of the average error rate being very small ( . θ )and the diamond distance being considerably larger ( θ/ ) atthe physical level. At the logical level, the diamond distanceis still larger than the average error rate, but by a smalleramount. This can be observed on Figure 5. The appreciatehow the different scalings cause a huge discrepancy betweenthe two pseudo-thresholds, it is convenient to normalize theerror magnitude at the logical level by the error magnitude atthe physical level. This is shown in Figure 6. Since we arenormalizing by the respective error magnitude at the physi-cal level, both curves at this level become equal to 1 and thecurves at the logical level become quadratic in θ . V. CONCLUSIONS
We have computed the physical process ( χ ) matrix and theeffective 1-qubit process ( χ ) matrix at the first level of ECwith the Steane [[7,1,3]] code for different incoherent and co-herent error models in the limit of low noise. For incoher- Physical RZC (1 - F)Logical RZC (1 - F)Physical RZC (D)Logical RZC (D) E rr o r m a g n i t u d e −9 −6 −3 θ / 2π10 −4 −3 −2 FIG. 5. Error magnitude for the RZC quantified with two differenterror metrics: the average error rate and the diamond distance. Theintersection of the two blue curves gives the level-1 pseudo-thresholdbased on the average error rate. The intersection of the two cyancurves gives the pseudo-threshold based on the diamond distance. Inthis case, the θ th obtained from the diamond distance is about 1 orderof magnitude higher than the one obtained from the average errorrate, as can be seen on Table XII. Logical RZC (1 - F)Logical RZC (D) N o r m a li z e d e rr o r m a g n i t u d e −3 −1 θ / 2π10 −4 −3 −2 FIG. 6. Normalized error magnitude for the RZC. The normaliza-tion factor is the respective error magnitude for each method at thephysical level. ent errors, at the logical level, the off-diagonal terms decayfaster than the diagonal terms, which explains the high accu-4racy of the PCa for incoherent channels. On the other hand,for coherent errors, the off-diagonal terms, stronger than thediagonal ones at the physical level, remain stronger at the log-ical level. This implies that the PCa approximation (and anystochastic approximation that matches the average fidelity atthe physical level) to coherent channels will unavoidably un-derestimate the error magnitude at the logical level. On theother hand, a stochastic channel that matches (or does not un-derestimate) a distance-based measure at the physical levelwill result in an approximation that is too pessimistic at thelogical level. These trends provide bounds on the pseudo-threshold of coherent channels, but these are not very tight.We have also observed that for coherent channels, the level-1pseudo-threshold depends strongly on the error measure em-ployed. However, distance-based measures result in consider-ably higher pseudo-thresholds than the average fidelity.As several authors have shown [22, 37], a characteristic fea-ture of stochastic noise is that the diamond distance scales lin-early with the average error rate. In contrast, for coherentnoise the diamond distance scales like the square root of theaverage error rate. As we can see from the effective processmatrices at the logical level, if the physical noise is unitary,at the first level of concatenation, the effective noise has θ off-diagonals terms and θ diagonal terms. This means thatdiamond distance scales like (cid:104) − F (cid:105) / . The scaling of this exponent as a function of the level of concatenation or thecode distance remains an open question.We have examined channels that are either coherent or in-coherent while in reality most channels contain both aspects[37]. For the unitary error models considered, the errors canbe completely removed by dynamic-decoupling techniques[38–40]. In general, open-loop control techniques can be usedto transform error channels that have a larger coherent char-acter into errors that have less coherent character [41, 42]. Atthe gate level the addition of Pauli twirling gates [43, 44] canalso reduce the coherent noise. Given the negative effect ofcoherent errors on the pseudothreshold, we expect these co-herent noise reducing methods will be essential for achievinglogical qubits that outperform physical qubits. ACKNOWLEDGMENTS
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