Increasing sensing resolution with error correction
IIncreasing sensing resolution with error correction
G. Arrad, Y. Vinkler, D. Aharonov, and A. Retzker Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Givat Ram, Israel (Dated: November 9, 2018)The signal to noise ratio of quantum sensing protocols scales with the square root of the coherence time. Thus,increasing this time is a key goal in the field. Dynamical decoupling has proven to be efficient in prolongingthe coherence times for the benefit of quantum sensing. However, dynamical decoupling can only push thesensitivity up to a certain limit. In this work we present a new approach to increasing the coherence time furtherthrough error correction which can improve the efficiency of quantum sensing beyond the fundamental limits ofcurrent state of the art methods.
Quantum sensing and metrology[1] are key goals of quan-tum technologies. Impressive achievements have been madein both in recent years. The frequency uncertainty ofatomic clocks has decreased dramatically[2, 3], the signal tonoise ratio of magnetic field measurements has considerablyincreased[4–8] and the contrast of spin imaging has improved[10, 11]. Since the sensitivity of quantum sensing scales as √ T , where T is the coherence time, a great deal of effort hasbeen devoted to designing and realizing protocols to increasethis time while also maintaining the sensing signal. However,the state of the art protocols, which are based on dynamicaldecoupling(DD), can only tackle low frequency noise sincethe DD control has to be faster than the correlation time of thenoise. Thus DD can only increase the coherence time up to acertain limit. This limit can be overcome by the use of errorcorrection(EC), which need not be faster than the noise corre-lation time, only its effect, which is much slower. Therefore,the utilization of EC for quantum sensing objectives could in-crease the signal to noise ratio substantially, and thus enhancethe sensitivity of field measurement and the contrast of imag-ing.EC tackles high frequency noise by using redundant qubits.Following Shor’s work[12], various protocols have been pro-posed, including Stean’s code[13] and various fault tolerantmethods[14]. Recently, several EC protocols have been real-ized [15, 16]. Here we introduce the concept of using EC forquantum sensing and present a set of protocols that achievethis goal. The basic idea is shown in fig. 1. An EC schemeis composed of a code sub-space {| ψ , ψ , ... ψ N (cid:105)} , in whichall the pertinent information is found; i.e., the sensing sig-nal should work inside the code (e.g. H s = g | ψ i (cid:105)(cid:104) ψ j | + h . c . ,where throughout this article g is the signal). The code issusceptible to errors, which map it to orthogonal subspaces.Correction of the errors is done by means of projective mea-surements of the various subspaces, and applying a correctionsequence.There are two main differences between EC for quantumcomputing and EC for sensing. While a logic operation can berealized by fast arbitrary pulses that rotate between the codestates, the sensing signal for most scenarios is weak, continu-ous, and very specific. This difference is the basis for the maincomplication in the schemes presented here. However, thesensing mechanism can also potentially benefit from the use Figure 1. The basic mechanism of the combination of EC with sens-ing. Errors map the code space to orthogonal spaces, and is followedby EC sequence, composed of a detection and then a correction, thatbrings the state back to the code. of fully protected qubits, which are neither sensitive to noisenor to the measured fields. Thus, we will make use of one ormore qubits which are assumed to be ”good”, in contrary tothese that are sensitive to the signal. These ’good’ qubits couldbe produced by clock states or robust nuclear spins. There isno analog for this in the case of quantum computing since inthat case we use, by design, the most robust qubits available.These two characteristics thus define the main features of thesensing codes.
Improving dynamical decoupling with error correction —
DD fails in two main scenarios. The first is when DD is notfast enough to overcome the correlation time of the noise andthe second is when DD suffers from noise in the control. Weaddress these issues below. We start by presenting a few phys-ically relevant sensing models in which the effects of errorscan be ameliorated by an appropriate error-correction proto-col.Classical Drive Noise — The first model we consider iscomprised of a single Two Level System (TLS), composedof the basis states | ↑(cid:105) and | ↓(cid:105) , which are separated by anenergy gap ω , and are driven by an external drive Ω at fre-quency ω . The sensing signal g is coupled to the TLS and a r X i v : . [ qu a n t - ph ] O c t the system is described by the Hamiltonian H = (cid:104) ω + f ( t ) (cid:105) σ z + ( Ω + δ Ω ) σ x cos ( ω t ) + g σ z cos ( Ω t ) . Here f ( t ) and δ Ω represent the external and the controlnoise and σ i are the Pauli operators. This Hamiltonian rep-resents the main magnetometry scenario which has been re-alized in NV centers[4, 7], ions[8] and atoms[9]. By trans-ferring first to the interaction picture with respect to ω σ z / Ω σ x , as-suming that f ( t ) (cid:28) Ω (cid:28) ω and taking advantage of therotating-wave-approximation, we are left with the Hamilto-nian: H I = g σ z + δ Ω σ x . We designate the eigenstates of σ z as | ↓(cid:105) and | ↑(cid:105) , andwe assume that we have a good qubit {| (cid:105) , | (cid:105)} , with a knownenergy gap ν , that can be controlled at will. The code isdefined as: {| ↓ (cid:105) , | ↑ (cid:105)} , (1)the signal g inflicts a phase shift between the two states whichmay be detected and the error δ Ω represents a bit-flip oper-ation on the first qubit, taking the system into the orthogonalsubspace {| ↑ (cid:105) , | ↓ (cid:105)} , and thus a projective measurementcould correct the error. However, the noise is only correctableat a short time scale with respect to 1 / g since the sensingtends to rotate the noise, hence generating an effective generalnoise that cannot be corrected by the code. We have checkedthat numerically (fig. 2) and have seen that indeed for a highEC repetition rate, the error is correctable.Noise in all directions — In case of general noise , the sys-tem Hamiltonian is described by: H = f z σ z + f x σ x + f y σ y + Ω σ x + g σ z cos Ω t , and in the interaction picture with respect to the drive, in thelimit when Ω is much faster than the noise, it is approximatedby: H = f x ( t ) σ x + g σ z , and this noise can be dealt with by the previous code.This method can be incorporated in a pulsed DD scheme aswell. Suppose we have a system with the following Hamilto-nian H sense = g σ z cos ω t + f z ( t ) σ z + f x ( t ) σ x , where we estimate g (cid:39) g ,i.e. g = g + ∆ g , and we aim toevaluate the correction ∆ g . The f i ( t ) represents the noise,where f x ( t ) is assumed to be fast and thus cannot be dealtwith by DD. In order to correct this, we can work with theprevious code: {| ↑ (cid:105) , | ↓ (cid:105)} . The f x ( t ) σ x term can be dealtwith by EC as was the case in the previous scheme, and the σ z term can be dealt with by pulsed DD the following way.The sensing signal will induce a phase shift between thetwo states. Assuming that the measurement is repeated ev-ery time τ which is followed by a π pulse switching the Figure 2. (Color online) The fidelity of a state as a function oftime, for different time differences between EC operations. Hereeach point represents an average over N = π / g . In each run either no EC procedure wasapplied (green), an EC procedure was applied each time interval of g ∆ T = . g ∆ T = . ( − g / , g / ) . In-set: The probability of measuring the initial state of the system, asa function of time. Here the black line represents the case with nonoise, and the different lines show the case where no EC was made(green), or where an EC was made at intervals of g ∆ T = . g ∆ T = . ( − g , g ) . population of the two code states, the following state is real-ized: | ↑ (cid:105) + e − i (cid:82) t gcos ( ω t ) dt | ↓ (cid:105) → e − i g ω sin ( ω t ) | ↑ (cid:105) + | ↓ (cid:105) . The | ↓ (cid:105) continue acquiring the phase φ = (cid:82) ττ gcos ( ω t ) dt until the next pulse. Suppose we choose τ = πω , andrepeat the same procedure n = t / τ times. Note that: ∑ nk = (cid:82) ( n + ) τ n τ gcos ( ω t ) dt = n g ω = t g π we gain the phase ina linear process: | ψ t (cid:105) = e − i g π t | ↑ (cid:105) + e i g π t | ↓ (cid:105) , This procedure illustrates the pulsed version of the combina-tion of DD and error correction. In case that a noise in the y direction exists as well, part of it will be corrected by errorcorrection and part by the DD.Measuring interaction — The next model we present iscomposed of two identical two-level systems with energy gaps ω , which are coupled by the sensing signal g , as describedby the Hamiltonian H = ∑ j (cid:104) ω + f j ( t ) (cid:105) σ zj + g σ x σ x , (2)where f j ( t ) represents noise in the same direction as the en-ergy gap, and we have arbitrarily taken the signal to be g / | ↓ , ↑ (cid:105) and | ↑ , ↓ (cid:105) and the second one spanned by the states | ↓ , ↓ (cid:105) and | ↑ , ↑ (cid:105) . Going into the interaction picture withrespect to ω (cid:0) σ z + σ z (cid:1) / H I = ∑ j f j ( t ) σ zj + g (cid:0) σ + σ − + h . c . (cid:1) . (3)Working in that subspace, we note that the states ( | ↓ , ↑ (cid:105) ± | ↑ , ↓ (cid:105) ) / √ ( σ + σ − + h . c . ) with eigenvalues ( − ) and (+ ) , respectively. We again use the existence of a ’good’qubit {| (cid:105) , | (cid:105)} with energy gap ν in order to define the codestates in an identical manner to the ones defined in Eq. (1): (cid:26) | ↓ , ↑ , (cid:105) − | ↑ , ↓ , (cid:105)√ , | ↓ , ↑ , (cid:105) + | ↑ , ↓ , (cid:105)√ (cid:27) . (4)The error ( E j = σ zj ) maps the code to an orthogonal subspace,allowing for a correction by projective measurement. Prolonging T — In order to deal with decay errors, i.e. T errors, we need a more elaborate EC scheme. One canuse traditional codes, which assume that an error can occur inany of the qubits. However, in order to utilize these codes forsensing, extremely sophisticated protocols are required. Wepropose much simpler codes that use ’good’ qubits. By con-structing these codes we develop the main ideas which couldalso be used as building blocks for combining sensing withtraditional codes. Note that dealing with T noise is not thesame as dealing with general errors. (The generality of T errors is discussed in [18] )Sensing becomes more challenging for T noise since aspecially tailored protocol should be used to distinguish thesignal from the noise. Specifically, the code states must differfrom each other by the state of at least two of the qubits. Themost obvious way to do this is with a Raman transition be-tween two codes states via a state which is outside the code, autility state. As the utility state is not inside the code, errors onit will not be correctable. Moreover, some errors can directlyconnect the code and the utility state. (see [18])The following protocols manage to overcome the noise andenlarge coherence times because two factors come into play:1) The small population of the utility state. 2) The specialcharacteristics of the dissipation noise. The main problemwith this protocol is explained below.In this error model the first qubit has a limited T time; inother words, it is susceptible to dissipation, whereas the othertwo are ’good’ qubits. The proposed code is: | A (cid:105) = | ↓ + ↑ (cid:105)| (cid:105) ; | C (cid:105) = | ↓ − ↑ (cid:105)| (cid:105) which is fully correctable; i.e., both bit flips and phase flipscan be corrected. The sensing signal, however, is a phase flipitself and thus a scheme which distinguishes between the noiseand the signal is needed. Let’s look at the following proce-dure. By opening a gap between these two states and the util-ity state | B (cid:105) = | ↓ − ↑ (cid:105)| (cid:105) , a Raman transition mediating Figure 3. The level structure of the basic model to overcome generalnoise. g is the sensing signal while g is the external driving. the sensing signal between the code states, can be designed.The level structure is shown in fig. 3. The Hamiltonian is: H s = g | B (cid:105)(cid:104) A | + g | B (cid:105)(cid:104) C | + h . c . + ∆ | B (cid:105)(cid:104) B | . In this case theutility state will always be occupied by the small amplitude ε = Ω δ . Suppose we arrive, at time t, at that state: (cid:16) α | − (cid:105)| (cid:105) + β | + (cid:105)| (cid:105) + e it ∆ ε | − (cid:105)| (cid:105) (cid:17) | n = (cid:105) , (5)The emission of a photon will result in: (cid:2) α | (cid:105)| (cid:105) + ( β + e it ∆ ε ) | (cid:105)| (cid:105) (cid:3) | n = (cid:105) + [ − α | (cid:105)| (cid:105) + ...... + ( β − e it ∆ ε ] | (cid:105)| (cid:105) ) | n = (cid:105) , Measuring S z S z , i.e. sepa-rating according to different n populations, and correcting,we get: α | + (cid:105)| (cid:105) + ( β ± e it ∆ ε ) | − (cid:105)| (cid:105) [18] .This shows that (for ∆ T (cid:29) β undergoes a randomwalk and thus correction to the signal would only result insecond order terms, giving us a longer coherence time. Theuncertainty, however, stays the same as the original uncer-tainty δ g = √ T (up to a small numerical factor) since: δ g Raman = δ Ω √ T = √ T , where the first equation is due tothe Raman transition’s slow effective rotation frequency of ω e f f = g Ω δ , and in the second we substituted: √ T = δ Ω √ T which we derived from the condition that the variance of thenoise random-walk reaches unity 1 = ε √ N = Ω δ (cid:113) T T . Working in the strong noise regime —
A possible appli-cation for the above scheme can be found in systems with a T · g (cid:28) P = cos ( g · t ) and T decay time,is shown here [17] to depend on time as: δ g = (cid:112) e t / T − cos ( gt ) √ nT t (cid:113) sin ( gt ) , (6)where T is the total experiment time and n the number of si-multaneously running systems. The optimal time is achievedwhen sin ( gt ) (cid:39) δ g ∝ √ T Tn [17]. But, inthe strong noise regime, T · g (cid:28) ( gt ) (cid:39) ( gt ) . Substituting intoeq.[6] differentiating and solving for best timing, we arive at t max = T giving us: δ g strong ∝ g · T − / , which is worseby a factor of T g [18] .One way to prevent this is by making a high detuning.In such interferometry our signal becomes g e f f = g + ∆ (where ∆ is the detuning); thus we can get to the middleof the cosine even for signal g, and enhance the sensitiv-ity. Still, this is not always simple to achieve, for exam-ple when measuring the strength of a weak drive, and inthose cases we can use the above-mentioned scheme for EC.This EC will give us the sensitivity eq.[6], where we change g → ω e f f = g Ω δ and T → T and now, using T = δ Ω T ,we get (cid:0) ω e f f T (cid:1) = ( gT δ Ω ) (cid:29) ( ω e f f t ) (cid:39) δ g EC ∝ √ T . Checking the relative accuracy we get: δ g EC δ g strong ∝ T g (cid:28) Spin spin interactions —
In this section we devise a schemefor measurement of the interaction strength between a dis-sipative TLS and a stable one. The dipole-dipole interac-tion will induce flip flops between the two. Here we use thecode: ( | ↑ (cid:105) + | ↓ (cid:105) ) / √ ( | ↓ (cid:105) + | ↑ (cid:105) ) / √ , wherethe first qubit is the sensing qubit and the other two are goodqubits. This is a fully correctable code, as the error maps thecode to orthogonal subspaces. Moreover, the flip flop inter-action H = g (cid:0) σ + σ − + h . c (cid:1) couples the two code states di-rectly and thus the sensing protocol does not use a utility statewhich lies outside the code. However, note that in order touse this we need to be able to apply non local interactions inthe correction sequences [18]. This code could be useful inmeasuring spin - spin interactions between ions or measuringdistances between an NV center and a nucleus. Measuring the sideband interaction term —
The sidebandinteraction is the main building block for quantum informa-tion processing with trapped ions. As the strength of the side-band interaction is proportional to the Rabi frequency, pre-cise measurement of this term is analogous to measurementof laser, microwave or rf fields.Since the sideband interaction, H = η Ω ( σ − a + + h . c ) , where σ − is the raising spin operator and a is thephonon distraction operator, creates flip flops, the previouscode could be used when one of the good qubits is re-placed by a phonon: ( | ↑(cid:105)| n = (cid:105)| (cid:105) + | ↓(cid:105)| n = (cid:105)| (cid:105) ) / √ ( | ↓(cid:105)| n = (cid:105)| (cid:105) + | ↑(cid:105)| n = (cid:105)| (cid:105) ) / √ . The precision of themeasurement of this protocol is limited by the coherence timeof the phonon. [18]. This procedure could be used to measureRabi frequencies and the Lamb-Dicke parameter.
Molmer Sorensen coupling measurement —
A naturalquestion to explore is the utilization of MS gates for EC. Asthe MS coupling term scales as η g Ω δ measurement, of thisterm provides additional information on top of the sideband;namely the detuning.The code states are | A (cid:105) = |↑ (cid:105) + |↓ (cid:105)√ | vib (cid:105) ; | C (cid:105) = |↓ (cid:105) + |↑ (cid:105)√ | vib (cid:105) and the ancillary state is | B (cid:105) = |↓ (cid:105) + |↓ (cid:105)√ | vib (cid:105) . The red side-band interaction is Figure 4. Red sideband interaction. The typical behavior of the redsideband scheme in the strong noise limit. The red line is the refer-ence sin ( Ω t ). The blue line, which follows it almost exactly, is thesimulated signal using EC with T τ · = − and Ω δ = − . Theyellow line, which forms a rather crude sine signal, is the simulatedsignal using EC with T τ · = − and Ω δ = · − . The fast de-caying green line is the same simulation without EC and Ω δ = − .The inset represents the energy level diagram of the scheme. applied on both the first and the second ions, resulting in aRaman transition via the third state. The code is correctable;however, as the utility state is not correctable under phaseflip the protocol is not perfect. We simulated this protocol tovalidate that a considerable gain exists, see fig.4 and [18] . Utilizing super-radiance —
One way to measure the dis-tance between two emitters that are closer than a wavelengthapart is to measure the energy gap between the super radiantstate and the irradiant state. Although the irradiant state hasa longer lifetime and thus can be measured with higher accu-racy, the supperradiant state has short lifetime and thus lowaccuracy. The following procedure shows that by using dissi-pation we can still measure the energy gap with high accuracy.The code is composed of the following two states: | A (cid:105) = | (cid:105) , | B (cid:105) = | (cid:105) + | (cid:105)√ , with the Hamiltonian H = g (cid:0) σ z + σ z (cid:1) + g (cid:0) σ + σ − + h . c (cid:1) yielding an energygap of ∆ E = g + ω G which can be measured by Ramseyinterferometry.It is evident that the code is not generally correctable. How-ever, since superradiance changes the error model, the codebecomes correctable. The T error changes to superradiancedecay; namely, a coherent decay of both the first and the sec-ond qubits, as described by the operator O = (cid:0) σ − + σ − (cid:1) a † .The third qubit is chosen to be a good qubit. Assuming wehave the state | ψ (cid:105) = a | A (cid:105) + b | B (cid:105) before the decay, we getafterward | ψ (cid:105) = cos ( θ )[ a | (cid:105) + b | (cid:105) + | (cid:105)√ ] ⊗ | ph (cid:105) + sin ( θ )[ a | (cid:105) + | (cid:105)√ + b √ | (cid:105)√ ] ⊗ | ph (cid:105) This error is correctable by measuring (cid:0) s z s z s z (cid:1) and then cor-recting. Conclusions and perspectives —
We have proposed and an-alyzed the use of EC for increasing the signal to noise ratio ofvarious sensing protocols. Due to the very specific character-istics of the sensing signals and the noise model, special ECprotocols were designed. We have shown that this is a pow-erful method that could have considerable implications forquantum technologies goals and on precision measurements.
Acknowledgments.–
This work was supported by EU Inte-grating Project DIADEMS. We thank Roee Ozeri, and NadavKatz for useful discussions. [1] V. Giovannetti, S. Lloyd and L. Maccone, Nature Photonics ,222 (2011).[2] P. O. Schmidt et al., Spectroscopy Using Quantum Logic Sci-ence , 749 (2005).[3] T. Rosenband et al., Frequency Ratio of Al+ and Hg+ Single-Ion Optical Clocks; Metrology at the 17th Decimal Place Sci-ence 319, 1808 (2008).[4] G. Balasubramanian et, al., Nature , 648 (2008). [5] K. Jenesen et, al., Room Temperature Atomic ensembles forquantum memory and magnetometry. Laser Spectroscopy -Proceedings of the XIX International Conference (2010).[6] S. Hong et al., MRS BULLETIN , 155 (2013).[7] J. R. Maze et al,. Nature , 644 (2008).[8] S. Kotler et al., Nature , 61 (2011).[9] W. Wasilewski et al., Phys. Rev. Lett. , 133601 (2010)[10] Staudacher et al., Science
561 (2013)[11] P. London, Phys. Rev. Lett. , 067601 (2013)[12] Peter W. Shor, Phys. Rev. A , R2493?R2496 (1995).[13] Steane, Andrew, Proc. Roy. Soc. Lond. A ,1954 (1996).[14] Daniel Gottesman, http://arxiv.org/abs/0904.2557[15] P. Schindler, et al., Science , 6033 (2011)[16] G. Waldherr et al., arXiv:1309.6424[17] Huelga et al., Phys. Rev. Lett. , 3865 (1997).[18] The Supplementary materials for this paper can be found on -- - - - - - - - - - - - . The methods for the T Raman schemeappear in (sec 1.1), strong noise (1.2), spin spin(1.3), SorensonMolmer simulation (1.4), red sideband (1.5). Nv centers are sec4. sec 3 discuss ’generality of T errors’. sec 2 discuss multi-level systems and dissipative spin spin interactions. SUPPLEMENTARY MATERIAL
In the following sections, we provide a detailed introduction to the methods used to derive the results presented above, anddiscuss some of the more complex, less intuitive notions applied in this article. The following topics are presented:1.1)
The Three qubits Raman transition scheme for EC
EC in the strong noise limit
Spin spin interactionsscheme
Simulation of the Sorenson-Molmer scheme
Red side band schemes Multiple level systems for EC and sensing
A general application scheme
Flip-flops where both spins dissipate The generality of T noise Proving the generality statements Methods for NV center schemes T errors Here we provide a detailed explanation of the EC mechanism presented in the ’prolonging T ’ section above, for the Ra-man transition between the two three-qubits states | A (cid:105) = | (cid:105) + | (cid:105)√ | (cid:105) and | C (cid:105) = | (cid:105)−| (cid:105)√ | (cid:105) through the intermediate state | B (cid:105) = | (cid:105)−| (cid:105)√ | (cid:105) . The transition is carried out by lasers shone on the qubits, under the assumption that transition to the | (cid:105) + | (cid:105)√ | (cid:105) state has been suppressed by the creation of a large energy gap from this state.The key parameters are the two Rabi frequencies g , Ω and the detuning of the two lasers (which are identical) δ . Note thatthe population of the intermediate state ( | B (cid:105) ) is proportional to ε = Ω δ (cid:28) (cid:16) β | + (cid:105)| (cid:105) + α | − (cid:105)| (cid:105) + e it δ ε | − (cid:105)| (cid:105) (cid:17) | n = (cid:105) (7)A photon is emitted, resulting in: (cid:0) α | (cid:105)| (cid:105) + ( β + e it δ ε ) | (cid:105)| (cid:105) (cid:1) | n = (cid:105) + (cid:0) − α | (cid:105)| (cid:105) + ( β − e it δ ε ) | (cid:105)| (cid:105) (cid:1) | n = (cid:105) , where e it δ is the fast rotating phase of the middle Raman state, and we assume that the emission of the photon is fast relative tothe sensing Hamiltonian.To correct this we measure the spin correlation operator S z S z : If the first two qubits turn out to be in the same state,we make the correction: | (cid:105)| (cid:105) → | + (cid:105)| (cid:105) , | (cid:105)| (cid:105) → | − (cid:105)| (cid:105) , otherwise we make: | (cid:105)| (cid:105) → | + (cid:105)| (cid:105) , | (cid:105)| (cid:105) → −| − (cid:105)| (cid:105) and thus we get: ( β + e it δ ε ) | + (cid:105)| (cid:105) + α | − (cid:105)| (cid:105) or ( β − e it δ ε ) | + (cid:105)| (cid:105) + α | − (cid:105)| (cid:105) Because the times between subsequent decays varies, and since (cid:104) δ t (cid:105) (cid:39) δ T (cid:29) e it δ is essentially random, resultingin a random walk of β in 2D, i.e.: (cid:104) β (cid:105) = (cid:104)| β ( t ) |(cid:105) = m ε where m is the number of measurement circles until time t,and ε is the size of one step. The factor 2 comes from the dimensionality of the random walk.From this we get the STD (standard deviation) of σ β = √ m ε √ . Note that the random walk also occurs when no dissipation ismeasured, so we have m = tT . Now we define T ∗ , the (new) decoherence time with EC, as the time after which β can nowlonger be separated from the induced noise, that is σ β (cid:39) = (cid:113) T ∗ T ε or T ∗ = T / ε . As seen above, due to theRaman transition rotation frequency ω e f f = g Ω δ , the new accuracy is δ g = Ω δ δ ω e f f ∝ ε (cid:112) T ∗ = εε √ T = √ T , up to a factor of the order of unity. In a frequency measurement, we can achieve the sensitivity of [17]: δ g = (cid:112) e t / T − cos ( gt ) √ nT t (cid:113) sin ( gt ) (8)as function of the measurement time. To find the optimal time we look for a time such that sin ( gt ) (cid:39) δ g ∝ √ T Tn as can be seen in [17], where T is the total experiment time and n is the number of, simultaneouslyrunning, separate systems, but, in the strong noise regime, T · g (cid:28) gt (cid:28) ( gt ) (cid:39) ( gt ) . By substituting in [8] we get δ g = (cid:113) e t / T − + ( gt ) √ nT t ( gt ) , (9)differentiating according to t and solving gives us t max > T . Since we cannot prolong the measurement for so long, we willchoose the longest time possible ,i.e. t = T giving us: δ g strong (cid:39) √ e − √ nT g (cid:16) T / (cid:17) ∝ T − / This is not good since it increases the error by factor gT (cid:29) . If we use the three-qubits-scheme for error correction described above in this system, we reduce the frequency to the new ω e f f = Ω g δ = g ε , but we can prolong the decoherence, getting T ∗ (cid:29) T . Specifically we have T ∗ ε ∝ T , and thus can choose T ∗ ω e f f = g ε T ∗ = gT / ε (cid:29) δ g = δ Ω · δ g Ω δ (cid:39) δ Ω (cid:114) e t / T ∗ − cos (cid:16) g Ω δ t (cid:17) √ nT t (cid:114) sin (cid:16) g Ω δ t (cid:17) where we replaced the old T of the system to the T ∗ of the corrected system, and plugged in the Raman transition frequencyinstead of the simple system’s g.Now, as implied above, we can choose t so that sin (cid:16) g Ω δ T (cid:17) = sin ( gT δ Ω ) = . Noting that δ g = (cid:16) d f ( g , x ) dg (cid:17) − δ f ( g , x ) (since we assume δ x = δ g cor = √ e δ Ω √ T T n = √ T T n , just as in [17]. Checking the relative accuracy we get: δ g cor δ g strong ∝ T g (cid:28) We describe in detail error corrections when measuring filp-flop type interaction. The code we need to correct is: | A (cid:105) = ( | ↑ (cid:105) + | ↓ (cid:105) ) / √ | B (cid:105) = ( | ↓ (cid:105) + | ↑ (cid:105) ) / √ , where the first qubit is subject to T noise (i.e. dissipation),and the signal flip-flops between this qubit and a second ’good’ qubit, that is: H s = Ω ( σ + σ − + h . c . ) where σ are the Paulioperators. The third qubit is an auxiliary ’good’ qubit. The Hamiltonian in the code space, is: H = Ω ( | B (cid:105)(cid:104) A | + | A (cid:105)(cid:104) B | ) which induces rotation between the two code states.Suppose we reach the following state: | ψ (cid:105) = a | ↑ (cid:105) + | ↓ (cid:105)√ + b | ↓ (cid:105) + | ↑ (cid:105)√ , and then a dissipation takes place, with amplitude w (where w (cid:28) | ψ (cid:105) = w √ ( a | ↓ (cid:105) + b | ↓ (cid:105) ) ⊗ | ph (cid:105) + (cid:32) a | ↓ (cid:105) + | ↓ (cid:105)√ + (cid:112) ( − w ) √ [ a | ↑ (cid:105) + b | ↑ (cid:105) ] (cid:33) ⊗ | ph (cid:105) . Now we measure the operator S z S z ( S iz is the spin, in the z direction, of the i-th qubit), thus casting the code into the state (upto normalization): | ψ (cid:105) = ( a | ↓ (cid:105) + b | ↓ (cid:105) ) ⊗ | ph (cid:105) or | ψ (cid:105) = (cid:32) a | ↓ (cid:105) + | ↓ (cid:105)√ + (cid:112) ( − w ) √ [ a | ↑ (cid:105) + b | ↑ (cid:105) ] (cid:33) ⊗ | ph (cid:105) . In the first case, the error can be corrected by the pulses | ↓ (cid:105) → |↑ (cid:105) + |↓ (cid:105)√ and | ↓ (cid:105) → |↑ (cid:105) + |↓ (cid:105)√ . In the second caseanother measurement is needed; specifically measuring the operator S z will transform the state into either: | ψ (cid:105) = ( a | ↓ (cid:105) + | ↓ (cid:105) ) ⊗ | ph (cid:105) or | ψ (cid:105) = ( a | ↑ (cid:105) + b | ↑ (cid:105) ) ⊗ | ph (cid:105) , and in both cases completing the correction should be straight forward. Another possibility for EC while sensing flip-flops is to use Ramsey interferometry, for example in the following scheme. Forthe code we use three qubits. The first qubit state is denoted by | u (cid:105) ; | d (cid:105) , this qubit is undergoing dissipation. The second qubit,whose state is denoted by | ↑(cid:105) ; | ↓(cid:105) is a ’good’ qubit. Flip-flops occur between the first and the second qubits. The third qubit is a’good’ auxiliary qubit whose state is denoted by | (cid:105) ; | (cid:105) . The Hamiltonian will take the form of H s = Ω ( σ − σ + + h . c . ) + ν ( σ z ) The code will be: {| A (cid:105) = | u ↓(cid:105) + | d ↑(cid:105)√ ⊗ | (cid:105) , {| C (cid:105) = | u ↓(cid:105)−| d ↑(cid:105)√ ⊗ | (cid:105)} and we get the sensing H s = ( ν + Ω ) | A (cid:105)(cid:104) A | which allowsfor Ramsey interferometry for measuring Ω . For the measurement, apply a field ˆ B = B σ z · cos ( ω · t ) where ω = Ω + ν + δ iswell-known. Note that this field works only on the third qubit. Apply ˆ B for a π / | A (cid:105) , then wait while makingerror corrections (every τ seconds) for time t, then apply ˆ B for another π / | C (cid:105) will give the expected cosine signal.EC procedure: assume we start in the state | ψ (cid:105) = a | A (cid:105) + b | C (cid:105) . A dissipation occurs on the first bit, taking us to: | ψ (cid:105) = sin ( θ ) a | d ↓ (cid:105) + b | d ↓ (cid:105)√ ⊗ | ph (cid:105) +[ cos ( θ ) a | u ↓ (cid:105) + b | u ↓ (cid:105)√ + a | d ↑ (cid:105) − b | d ↑ (cid:105)√ ] ⊗ | ph (cid:105) . By measuring the (local) operators S z and S z we can separate the system to | ψ − , − (cid:105) = a | d ↓ (cid:105) + b | d ↓ (cid:105)√ | ψ , − (cid:105) = a | u ↓ (cid:105) + b | u ↓ (cid:105)√ | ψ − , (cid:105) = a | d ↑ (cid:105) − b | d ↑ (cid:105)√ , where | ψ − , (cid:105) fits the measurement results of < S z > = − / , < S z > = / a | d ↑ (cid:105)− b | d ↑ (cid:105)√ ↔ a | u ↓ (cid:105)− b | u ↓ (cid:105)√ occurring during the dissipation process, whichare second order in the noise parameter, giving us T ∗ ∝ τ T · T . The Sorenson Molmer scheme we proposed is not as simple as it might appear, since it involves a Raman transition througha state which is outside the code. Here, unlike the original scheme, the EC sequence do not induce an error every time (this iswhy this scheme works better). Nevertheless, every time an emission of a photon is measured, a small error is induced throughthe correction sequence; in essence this error occurs since we send the population of the intermediate state to 0.As the intermediate state is a fast oscillating state, whose phase relative to the other states depends on initial conditions,we assumed that each such correction would effectively send the system to the beginning of the last oscillation cycle of theintermediate state, causing a 1 / δ delay (this happens on average every T , resulting in the very small change of T / δ whichcan be accounted for exactly).Also we assumed that every time we decay, and correct that decay, we effectively lose half the time since last EC sequence,as this is the average time in the lower state in which the sensing Hamiltonian does not operate. In addition, we note that theeigenvectors of the Hamiltonian are of the form g | A (cid:105) − Ω | C (cid:105) and g | A (cid:105) + Ω | C (cid:105) + ε | B (cid:105) , which are populated in the Ramantransition process, and ε ( g | A (cid:105) + Ω | C (cid:105) ) + | B (cid:105) . This latter state should only have a population of order ε , and any excessivepopulation means that the population is trapped in the intermediate | B (cid:105) state and causes degradation of the signal. For thisreason we had to assume that the EC does not cause a population transfer to this last state. All these assumptions were validatedby simulation. A typical output is visualized in fig. 4 in the paper, showing that EC vastly increases the accuracy.Furthermore, in fig. 5, the dependency of the system’s T ∗ on different parameters is shown, as calculated from the resultsof the simulation. The fit is rather crude and bears only qualitative resemblance to the expected T ∗ = . · T ε − Figure 5. Sorenson Molmer simulationThe graph depicts the dephasing time with error correction ( T ) of the Sorenson Molmer simulation system, estimated from the simulation asa function of τ T and of ε = g δ (in the inset). Here T is the original decay rate of the system (i.e. without EC). The illustrated curves aredescribed by the formula T = . T ε a qualitative agreement can be seen. Note that although τ does not appear in the final formula, theformula is only valid in the τ T << can be attributed to four factors: 1) given our limited resources, each point on the graph shows the average over the results ofonly a few runs, rather than few thousands, as ideally it should. 2) The actual results depend to a great extent on the number oftimes in which the emission of a photon was actually detected 3) T ∗ were only estimated up to a factor of order unity ratherthan calculated exactly 4) higher order contributions were ignored when calculating the expected result. Dissipation takes place on the first bit. by setting ε = √ g + g δ we have the typical state of | ψ (cid:105) = a | A (cid:105) + c | C (cid:105) + e i φ B ε | B (cid:105) and,in the fast ECS limit, dissipation bring us to: | ψ (cid:105) = a ( − τ Γ ) |↑ (cid:105) + |↓ (cid:105)√ | vib (cid:105) + c |↓ (cid:105) +( − τ Γ ) |↑ (cid:105)√ | vib (cid:105) + e i φ B ε | ↓ (cid:105) + | ↓ (cid:105)√ | vib (cid:105) + τ Γ ( a | ↓ (cid:105) + b | ↓ (cid:105)√ | vib (cid:105) ) (10)It should be possible to design a measurement that will separate the state, depending on the measurement outcome, in to: | ψ (cid:105) = a | ↓ (cid:105)| vib (cid:105) + c | ↓ (cid:105)| vib (cid:105) + e i φ B ε | ↓ (cid:105)| vib (cid:105)√ | ψ (cid:105) = a | ↑ (cid:105)| vib (cid:105) + c | u ↑ (cid:105)| vib (cid:105) + e i φ B ε | ↓ (cid:105)| vib (cid:105)√ | ψ (cid:105) = a | ↓ (cid:105)| vib (cid:105) − c | ↓ (cid:105)| vib (cid:105)√ τ / Γ τε which is third order. In other words, we achieved enlarged T and enhanced precision, as was also validated by the simulationpresented in the previous section.0
2. MULTI-LEVEL SYSTEMS, SENSING , AND EC
In this paper we mainly assumed that we were dealing with physical qubits; i.e. each distinct qubit is also a distinct atom,spinor or, ion (and so forth). The main implication of this assumption is some loss of generality in the possible relation ofthe sensing Hamiltonian to the error model: for example, when we say (under ’prolonging T ’) that the most obvious wayto connect two states which are more than one step apart is a Raman transition, we are obviously referring to some natural,physical, partition of the qubits.If we have two states say, | (cid:105) and | (cid:105) connected by some field, we can always denote | dddd (cid:105) = | (cid:105) and | uuuu (cid:105) = | (cid:105) and now the same field connects these two seemingly very different states. Of course, such notions aresuperficial and only cause confusion because they have no real implications, since the errors will now also connect very differentstates. In other words, when we measure ’distances’ between code states we should really measure them relative to the possibleerrors. As we noted above, all qubits that are sensitive to the measured field must be also sensitive to errors (otherwise we willuse only ’good’ qubits and need no EC). This notion is always true, and this is why our ’physicality’ assumption is a very goodone.Nevertheless, when describing multilevel systems in terms of ’good’ and ’noisy’ qubits, a simple system might still need tobe described by very complicated qubits structures. Thus for some systems, the obvious error models might differ from the oneswe refer to in this article. ( changing the errors is interchangeable with changing the sensing Hamiltonian as regards EC becausethe orthogonality relations between the two is all that matters).In practice, finding systems with error models which are correctable is not easy, as the added complexity of the systems tendsto cause more evolved errors rather then simply different ones. Still, the possibility exists, and two examples are given below. One way to achieve better performance from our EC models is by using systems that decay to states other than the groundstates. On the down side, such systems will undergo decay from all their subsequent states rather than only from the ”up” stateof the qubits. On the up side, each state will decay into a different orthogonal state, enabling additional freedom in separatingand correcting the errors.We use atomic states in the example. Denote | (cid:105) = | J = m = (cid:105) , | (cid:105) = | J = m = (cid:105) and | u (cid:105) = | J = m = (cid:105) , | d (cid:105) = | J = m = (cid:105) . The decay is thus either | (cid:105) → | u (cid:105) ⊗ | phA (cid:105) or | (cid:105) → | d (cid:105) ⊗ | phB (cid:105) , whereall the different photons populations contribute to orthogonal states of the environment; i.e. the decays resulting in differentphotons are not coherent.We will use a code made up of the two-atom states: | (cid:105) + | (cid:105)√ , | (cid:105) + | (cid:105)√ . The sensing Hamiltonian will be H s = g | (cid:105) + | (cid:105)√ (cid:104) | + (cid:104) |√ + h . c . which can be achieved by means of magnetic field in the ˆ x direction, acting on each atom separately. Note that the decay on different atoms is non-coherent in other words there aredifferent phase-shifts for different atoms as well as for different decays.Starting with: | ψ (cid:105) = a | (cid:105) + | (cid:105)√ + b | (cid:105) + | (cid:105)√ , A decay brings us to | ψ (cid:48) (cid:105) = w (cid:18) a | (cid:105) + | (cid:105)√ + b | (cid:105) + | (cid:105)√ (cid:19) + A a | u (cid:105) + b | u (cid:105)√ ⊗ | phA (cid:105) + B b | d (cid:105) + a | d (cid:105)√ ⊗ | phB (cid:105) + C a | u (cid:105) + b | u (cid:105)√ ⊗ | phC (cid:105) + D a | d (cid:105) + | d (cid:105)√ ⊗ | phD (cid:105) where A , B , C , D are the amplitudes to emit the 4 respective possible distinct photons. Measuring each atom’s J value will eithercorrect the errors or bring us (w.l.o.g.), up to normalization, to:1 A a | u (cid:105) + b | u (cid:105)√ ⊗ | phA (cid:105) + B b | d (cid:105) + a | d (cid:105)√ ⊗ | phB (cid:105) . Now measuring the J z state of the first atom (that is, its m value); i.e. measuring in the {| u (cid:105) ; | d (cid:105)} basis, we get (w.l.o.g.): ( a | u (cid:105) + b | u (cid:105) ) ⊗ | phA (cid:105) where now the photon number has no significance, and can be ignored. Now make | u (cid:105) → | (cid:105) + | (cid:105)√ and | u (cid:105) → | (cid:105) + | (cid:105)√ andthe error is fully corrected. Note that this method, in fact, enables the measurement of generic magnetic field. This can be done by using multiple level systems. We use two prob atoms in the J=1 and J=0 states, and one good qubit insome other states {| u (cid:105) ; | d (cid:105)} . Denote | (cid:105) = | J = m = (cid:105) , | (cid:105) = | J = m = (cid:105) and | ! (cid:105) = | J = m = (cid:105) . We assume flip flopsof the two qubits (i.e. H s = g | (cid:105)(cid:104) | + h . c . ), both of which may undergo decay in the form: | (cid:105) → | ! (cid:105) ⊗ | phA (cid:105) ; | (cid:105) → | ! (cid:105) ⊗ | phB (cid:105) , where the two modes of decay are non coherent, and the decays of different atoms are also non-coherent. We use the three-atomcode: | u (cid:105) + | d (cid:105)√ , | u (cid:105) + | d (cid:105)√ . Starting with : | ψ (cid:105) = a | u (cid:105) + | d (cid:105)√ + b | u (cid:105) + | d (cid:105)√ , a decay brings us to | ψ (cid:48) (cid:105) = w (cid:18) a | u (cid:105) + | d (cid:105)√ + b | u (cid:105) + | d (cid:105)√ (cid:19) + A (cid:18) a | !0 u (cid:105) + | !0 d (cid:105)√ ⊗ | phA (cid:105) + B | !1 u (cid:105) + | !1 d (cid:105)√ ⊗ | phB (cid:105) (cid:19) + C (cid:18) a | u (cid:105) + | d (cid:105)√ ⊗ | phC (cid:105) + D | u (cid:105) + | d (cid:105)√ ⊗ | phD (cid:105) (cid:19) where A , B , C , D are the amplitudes to emit the 4 respective possible distinct photons. Measuring the J state of each atoms, weeither correct the error or reach (w.l.o.g.) as the state: (cid:18) A | !0 u (cid:105) + | !0 d (cid:105)√ ⊗ | phA (cid:105) + B | !1 u (cid:105) + | !1 d (cid:105)√ ⊗ | phB (cid:105) (cid:19) , now, measuring the M state of the second atom brings us to: ( a | !0 u (cid:105) + b | !0 d (cid:105) ) or ( a | !1 d (cid:105) + b | !1 u (cid:105) ) where we have left out the photons, as they are now unimportant. Correcting from here is straight forward.2 T DECAY ERROR VERSUS GENERAL ERROR
In the language of error correction it is customary to refer to two kinds of errors; phase-flip (i.e. errors proportional to the σ z Pauli operator) and bit-flip (i.e. σ x errors). This is because these two errors are simple to comprehend, and span all possibleerrors, in the sense that if a bit-flip and a phase-flip can be corrected any general error can be corrected. These kind of codes are’fully correctable’ and can be corrected for ’general errors’.Specifically this means that in the former case we can also correct errors induced by decay. Sometimes the converse is alsoheld to be true, but it is not. Below we demonstrate, by means of an example, that correcting decay errors does not require thepower to correct any error. We also present and explain why one might, naively, believe the converse.For clarity, we define errors induced by decay (also refereed to as T errors). These errors are caused by coupling the systemto the environment via the term a † i σ − + h . c , where a i is the annihilation operator of the i-th mode of a of the environment. Thiserror, when one traces-out the state of the cavity (i.e. the environment), causes decoherence. The rate of the error is defined bythe typical time scale, denoted T , after which a decay is likely to occur. In what follows we will suppose, for simplicity, thatthe photon mode was originally found in the | n = (cid:105) state. T errors are not general errors To prove this consider the following eight qubit system, which can be corrected for bit-flip and T errors on each qubit, butnot for the effects of phase-flip errors. Denote | + (cid:105) = | (cid:105) + | (cid:105)√ and |−(cid:105) = | (cid:105)−| (cid:105)√ , we shell define our code states, andour initial state, to be: | A (cid:105) = | + (cid:105)| + (cid:105) ; | B (cid:105) = |−(cid:105)|−(cid:105)| ψ (cid:105) = a | A (cid:105) + b | B (cid:105) Bit -flip error , occurring with amplitude ε (cid:28) | ψ (cid:48) (cid:105) = ( − ε ) | ψ (cid:105) + ε (cid:18) a | (cid:105) + | (cid:105)√ | + (cid:105) + b | (cid:105) − | (cid:105)√ |−(cid:105) (cid:19) + ... + ε (cid:18) a | + (cid:105) | (cid:105) + | (cid:105)√ + b |−(cid:105) | (cid:105) − | (cid:105)√ (cid:19) + higher order Note that while we assume all qubits might have errors, we also assume that only one qubit eror at the same time; that is, weassume that the probability of measuring an error is proportional to some small parameter (and thus multiple simultaneous errorsare second order).It is evident that this error can be corrected by measuring the adjacent-qubits correlation operators S iz S i + z . Two neighboringqubits with the same sign tells us that an error has occurred, as well as revealing its location. Phase-flip errors , however, cannot be corrected, since a phase-flip on the the first four qubits coincides with the phase-flip onthe last four, in a nasty way: | ψ (cid:48) (cid:105) = ( − ε ) | ψ (cid:105) + ε ( a |−(cid:105)| + (cid:105) + b | + (cid:105)|−(cid:105) )+ ε ( a | + (cid:105)|−(cid:105) + b |−(cid:105)| + (cid:105) ) which is | ψ (cid:48) (cid:105) = ( − ε ) | ψ (cid:105) + ε (( a + b ) |−(cid:105)| + (cid:105) + ( b + a ) | + (cid:105)|−(cid:105) ) and since the separation between the populations, that is a and b, was destroyed, the error cannot be corrected. Decay errors occurring on the first bit with amplitude ε , will bring us to the following state: | ψ (cid:48) (cid:105) = (cid:32) ( − ε / √ ) a | (cid:105) + a | (cid:105)√ | + (cid:105) + ( − ε / √ ) b | (cid:105) − b | (cid:105)√ |−(cid:105) (cid:33) | n = (cid:105) + ε (cid:18) a √ | (cid:105)| + (cid:105) + b √ | (cid:105)|−(cid:105) (cid:19) | n = (cid:105) | ψ (cid:48) (cid:105) = ( − ε ) | ψ (cid:105)| n = (cid:105) + ε (cid:18) a | (cid:105)√ | + (cid:105) + b | (cid:105)√ |−(cid:105) (cid:19) | n = (cid:105) + ... + ε (cid:18) a | + (cid:105) | (cid:105)√ − b |−(cid:105) | (cid:105)√ (cid:19) | n = (cid:105) . This error can be resolved, again, by measuring the two spins correlation functions. If, for example, we measure the secondqubit to be the same as its neighbors, we necessarily arrived at the state | ψ (cid:48)(cid:48) (cid:105) = (cid:18) a | (cid:105)√ | + (cid:105) + b | (cid:105)√ |−(cid:105) (cid:19) | n = (cid:105) which indeed enables us to correct the error. The correction from here is straight forward. Now, let’s assume that we measuredno decay, that is, all the adjacent spin correlations turn out negative, in which case we reach the state: | ψ (cid:48)(cid:48) (cid:105) = | ψ (cid:105)| n = (cid:105) and thus the error has already been corrected. The conclusion is that since in this system it is possible to correct T errors, butnot phase-flip errors, then correcting T errors is not equivalent to correcting any error (that is, to correcting ”general errors”) . T errors are similar to general errors Despite the above demonstration, in many systems T errors appear to be almost as bad as general errors; in other words, theyare almost as hard to correct. Evidently, correction of T errors implies the correction of bit-flip errors. In addition, since theerror is a non Hermitian σ − error, it operates differently on the up and down states. Thus, in order to correct it, the code statesneed to have the same probability for being in the up state of each qubit. That is, the code states cannot differ in the probabilityof finding oneself in the up state of any of the qubits. Suppose for example the following state: | ψ (cid:105) = a | (cid:105) + b | (cid:105) Then a decay, occurring on the first bit for example, will take us to | ψ (cid:105) = (( − ε / √ ) a | (cid:105) + b | (cid:105) ) | n = (cid:105) + ε ( a | (cid:105) ) | n = (cid:105) . Evidently, this cannot be corrected. Also take a look at the code states | (cid:105) and | (cid:105) + | (cid:105)√ . This also is not correctable forthe same reason, since the decay rate of each state is different, and thus each correction induces a √ |±(cid:105) = | (cid:105) ± | (cid:105) ; then the states are | + (cid:105)| + (cid:105) and |−(cid:105)|−(cid:105) . Assume a decay, for brevity of the first qubit only, and we get from | ψ (cid:105) = a | + (cid:105)| + (cid:105) + b |−(cid:105)|−(cid:105) the state | ψ (cid:105) = ( a | (cid:105)√ | + (cid:105) − b | (cid:105)√ |−(cid:105) ) | n = (cid:105) + ( − ε / √ )( a | (cid:105)√ | + (cid:105) + b | (cid:105)√ |−(cid:105) ) | n = (cid:105) + ε ( a | (cid:105)√ | + (cid:105) + b | (cid:105)√ |−(cid:105) ) | n = (cid:105) By measuring the correlation operators it is easy to verify that a decay here is correctable. But look what happens when no decaywas measured: one can write | (cid:105) = | + (cid:105)−|−(cid:105)√ and thus we arrive at the state: | ψ (cid:105) = a (( − ε / ) | + (cid:105) + ε / |−(cid:105) ) | + (cid:105) − b ( − ε / ) |−(cid:105) + ε / |−(cid:105) ) |−(cid:105) This is a phase-flip and we know that phase-flips are not correctable in this system. It should be pointed out that this is only an”effective phase-flip” due to the structure of the system, since in our previous system no such phase-flip occurred. Also note thatin this system, apparently we need to be able to correct bit-flips, and subsequent bit and phase flips (on the same qubit) - whenwe measure decay. Furthermore, we have to be able to correct phase-flip when no decay was measured. Altogether this signifiesa ”general error”. Misleading us to believe that T errors are equivalent to general errors. Note that the ”effective phase-flip”appears only in second order in ε which is advantageous, since this is also the order of the non correctable ”two bit-flips” errors.4 In this section we present the error correction protocol applicable for both the models described in the article under ClassicalDrive Noise and Quantum Noise. In both models, we presented the mapping of the physical space into code states, in Eqs.( 1)and the error acts as a bit-flip that maps the states from the code space onto the error space. Denoting by | c , (cid:105) and | c , (cid:105) thecode states, the error operation will take these states to the states we denote by | c , (cid:105) and | c , (cid:105) , respectively.Preparing the system in an initial state | ψ ( ) (cid:105) = ( | c , (cid:105) + | c , (cid:105) ) / √ t , to be | ψ ( t ) (cid:105) = cos (cid:0) g + ν t (cid:1) √ ( | c , (cid:105) + | c , (cid:105) ) + i sin (cid:0) g + ν t (cid:1) √ ( | c , (cid:105) − | c , (cid:105) ) , (1)and g can be deduced by measuring the probability of the system will be still at the initial state, at time t . An error operationwill take the system into the error states defined above. We now measure the Hermitian operator Σ z = | c , (cid:105)(cid:104) c , | + | c , (cid:105)(cid:104) c , | −| c , (cid:105)(cid:104) c , | − | c , (cid:105)(cid:104) c , | , (2)which returns the eigenvalue (+ ) if the system had no error and ( − ))