Relaxation Times Do Not Capture Logical Qubit Dynamics

Relaxation times do not capture logical qubit dynamics

Amit Kumar Pal,1, 2, 3 Philipp Schindler,4 Alexander Erhard,4 Angel´ Rivas,5, 6 Miguel-Angel Martin-Delgado,5, 6 Rainer Blatt,4, 7 Thomas Monz,4, 8 and Markus Müller2, 9, 10 1Department of Physics, Indian Institute of Technology Palakkad, Palakkad 678557, India 2Department of Physics, College of Science, Swansea University, Singleton Park, Swansea - SA2 8PP, United Kingdom 3Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL-02-093 Warszawa, Poland 4Institut fürExperimentalphysik, UniversitätInnsbruck, Technikerstr. 25, A-6020 Innsbruck, Austria 5Departamento de F´ısica Teórica, Facultad de Ciencias F´ısicas, Universidad Complutense, 28040 Madrid, Spain 6CCS -Center for Computational Simulation, Campus de Montegancedo UPM, 28660 Boadilla del Monte, Madrid, Spain. 7Institut fürQuantenoptik und Quanteninformation, Osterreichische¨ Akademie der Wissenschaften, Otto-Hittmair-Platz 1, A-6020 Innsbruck, Austria 8Alpine Quantum Technologies GmbH, 6020 Innsbruck, Austria 9Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany 10Peter Grünberg Institute, Theoretical Nanoelectronics, Forschungszentrum Jülich,D-52425 Jülich,Germany Quantum error correction procedures have the potential to enable faithful operation of large- scale quantum computers. They protect information from environmental decoherence by storing it in logical qubits, built from ensembles of entangled physical qubits according to suitably tailored quantum error correcting encodings. To date, no generally accepted framework to characterise the behaviour of logical qubits as quantum memories has been developed. In this work, we show that generalisations of well-established figures of merit of physical qubits, such as relaxation times, to logical qubits fail and do not capture dynamics of logical qubits. We experimentally illustrate that, in particular, spatial noise correlations can give rise to rich and counter-intuitive dynamical behavior of logical qubits. We show that a suitable set of observables, formed by code space population and logical operators within the code space, allows one to track and characterize the dynamical behaviour of logical qubits. Awareness of these effects and the efficient characterisation tools used in this work will help to guide and benchmark experimental implementations of logical qubits.

I. INTRODUCTION ensembles of physical qubits [2, 23, 24]. An important short-term goal is to reduce the effective error rates [25– 27], as a first step towards the long-term goal of protected High-quality physical qubits with long coherence times large-scale fault-tolerant quantum computation [28–30]. that allow one to reliably store fragile quantum states However, characterising the performance of logical form the backbone of currently developed quantum pro- qubits is naturally more involved, because fully charac- cessors [1,2]. Over the last decades, the development of terising the state of its constituents is not feasible for methods to characterise physical qubits and their coher- even intermediate-size quantum registers. It is tempt- ence properties has been subject of intense study. Here, ing to try to directly leverage the well-established figures widespread and popular figures of merit are the longi- of merit developed for physical qubits to logical qubits, tudinal and transverse relaxation time scales, known as guided by the intuition that the encoded information in T and T . They were originally introduced in the field 1 2 logical qubits should show qualitatively similar dynami- of nuclear magnetic resonance, describing a simple expo- cal behaviour as their physical constituents. In this work nential decay dynamics of spin states [2,3]. we illustrate that the analogy to a physical qubit does Such simple descriptions, however, become incomplete not hold generally, and that the characterisation of logi- in the presence of, e.g., temporal noise correlations giving cal qubits as quantum memories [23] comes with a num- rise to non-Markovian dynamics [4,5]. Similarly, spa- ber of unique challenges. In particular, spatial noise cor- tial noise correlations can play a role in larger quantum relations can strongly affect QEC performance [31–37] registers, where such correlations can be quantified and and influence dynamical behaviour of logical qubits in a measured [6,7] and sometimes also harnessed for noise counter-intuitive way. arXiv:2012.07911v1 [quant-ph] 14 Dec 2020 mitigation techniques, for instance by storing quantum For example, we show that generalisations of, e.g. T1 information in decoherence-free subspaces [8–13]. and T2 times to logical qubits fail, even for encodings Currently, we are witnessing enormous efforts to build consisting of no more than 3 or 4 physical qubits. We and reliably control increasingly larger quantum proces- theoretically discuss and experimentally observe rich de- sors - often termed noisy intermediate-scale quantum cay dynamics of small-scale logical qubits, due to leakage (NISQ) devices [14]. These devices are also used to imple- of quantum information from the code space, or temporal ment low-distance quantum error correcting codes [15– behavior governed by multiple time scales in contrast to 22], which allow one to encode and protect quantum in- simple exponential decay. We foresee that awareness of formation in so-called logical qubits formed of entangled these effects and the efficient characterisation tools used 2 in this work will guide the development and optimisation where we define γ = h[B(0)]2i. of logical qubits. We will, for completeness, now present a brief overview of the relevant results obtained when dephasing noise is applied to a single physical qubit. Writing a generic II. EXPERIMENTAL SYSTEM AND NOISE pure single-qubit state in terms of the computational θ iφ θ basis {|0i , |1i} as |ψi = cos 2 |0i + e sin 2 |1i, with θ The experimental setup consists of a trapped-ion quan- and φ being real parameters (0 ≤ θ ≤ π, 0 ≤ φ ≤ tum information processor with 40Ca+ ions, that has 2π), the dephasing noise acts on the state as |ψ0i = h i been described in detail in reference [38]. The qubits are R t 0 0 exp −i 0 HG(t )dt |ψiL, leading to encoded in the 4S1/2(mj = −1/2) = |1i ground state and t the metastable excited state 3D5/2(mj = −1/2) = |0i θ Z θ |ψ0i = cos |0i + exp i φ + B(t0)dt0 sin |1i , and transitions between these states are driven with 2 2 a narrow linewidth laser [38]. The system provides a 0 universal set of gate operations consisting of Mølmer- (7) Sørensen (MS) entangling gates and arbitrary local oper- h t i discarding a global phase exp − i R B(t0)dt0Z . ations [38, 39]. Any local operation can be implemented 2 0 by a combination of a resonant collective local operation Denoting the distribution of the random values of the P Ux(θ) = exp(−iθ/2Sx), with Sx = Xi being the sum magnetic field by P (B), the density matrix of the qubit i 0 R 0 0 over all single-qubit X Pauli operators[40], and single- is given by ρ = |ψ i hψ | P (B)dB. Assuming P (B) to qubit AC-Stark shifts, represented by rotations around be a Gaussian distribution, and using Eq. (6), the noisy (i) density matrix can be simplified as the z-axis of the Bloch sphere Uz (θ) = exp(−iθ/2Zi). The action of the entangling MS gate operation on the en- 0 2 θ 2 θ 2 ρ = cos |0i h0| + sin |1i h1| tire qubit register is described as MS(θ) = exp(−iθ/4Sx). 2 2 The dominating noise source for storing information 1 − 1 γt -iφ iφ + e 2 sin θ(e |0i h1| + e |1i h0|). (8) in our experimental system is given by dephasing caused 2 by laser frequency noise and fluctuations in the bias magnetic field [38]. In our system, the effect of fluctuations of For a physical qubit represented completely by its Bloch the laser frequency as well as the magnitude of the mag- vectors ~r = (rx, ry, rz), where rx ≡ X, ry ≡ Y , and netic field cannot be distinguished. We can thus describe rz ≡ Z, it is crucial to understand how the components the dephasing process using a single fluctuating variable of the Bloch vectors are modified under the application of the dephasing noise. The expectation values of the B(t), referred to in the following as effective magnetic 0 field: components of the Bloch vector in the state ρ evolve under dephasing as 1 HG(t) = B(t)Z. (1) 0 − 1 γt 2 hXi = Tr(Xρ ) = e 2 sin θ cos φ, (9) 0 − 1 γt In the following, we assume the random fluctuation in the hY i = Tr(Y ρ ) = e 2 sin θ sin φ, (10) values of the effective magnetic field to obey a Gaussian hZi = Tr(Zρ0) = cos θ. (11) distribution P (B), which implies that Z t This behavior is a special case of the most general qubit exp ±i B(t0)dt0 relaxation dynamics which is characterized by the longi- 0 tudinal and transverse the relaxation time-scales T1 and " * 2+# 1 Z t T2, as introduced in the early nuclear magnetic resonance = exp − B(t0)dt0 . (2) experiments. These relaxation times are defined as 2 0 t eq − T eq We also assume a stationary autocorrelation function of rz(t) = rz − e 1 [rz − rz(0)] , (12) t eq − T eq the noise source, implying r⊥(t) = r⊥ − e 2 [r⊥ − r⊥(0)] , (13)

hB(t + τ)B(t)i = hB(τ)B(0)i, (3) q 2 2 where r⊥ = rx + ry, and the superscript “eq” signifies and a further δ-correlation of the noise, such that the equilibrium time of the corresponding signal when the hB(τ)B(0)i = h[B(0)]2iδ(τ). (4) system has fully relaxed. Here, T1 represents the typical decay time of the eigenstates of the Z Pauli matrix, and Therefore, in the case of local dephasing, this implies T2 quantifies the lifetime of quantum coherence between 2 them. Comparing Eqs. (12)-(13) with Eqs. (9)-(10), one hBk(t + τ)Bl(t)i = h[Bk(0)] iδk,lδ(τ). (5) 1 obtains T1 = ∞, while T2 = 2γ for dephasing noise. Using these properties, one finds In a multi-qubit system, the spatial correlation of the noise needs to be accounted for. We concentrate on two *Z t 2+ B(t0)dt0 = h[B(0)]2it = γt, (6) extreme cases of spatial noise correlations: (i) local de- 0 phasing noise, where each qubit has its own, independent 3 noise source, and (ii) a global, i.e. collective dephasing values given by m ∈ {−N, −N + 2, ··· ,N − 2,N}. The where one noise source is affecting all qubits identically. logical basis state |1iL can then be written by grouping Local dephasing would be caused by local fluctuating the physical basis states by their magnetization: magnetic fields, where each of the physical qubits con- Nm stituting the logical qubit experiences a different random X X m m magnetic field, and the noise Hamiltonian is given by |0iL = bl |bil . (19) m l=1 1 X HL(t) = Bk(t)Zk, (14) 2 The state |1iL can also be written in a similar way. k Let us first consider the global dephasing noise represented by the noise Hamiltonian H (t). The effect where Bk(t) is the time-dependent strength of the mag- G of the global dephasing noise on a generic logical state netic field local to the physical qubit k, and Zk is the z- 0 h R t 0 0i component of the Pauli matrices corresponding to qubit |ψiL, given by |ψ iL = exp −i 0 HG(t )dt |ψiL, is de- k. On the other hand, the global dephasing noise is due termined by the eigenvalue equation to a randomly fluctuating effective magnetic field that acts on all of the physical qubits, such that the noise " # X m m Hamiltonian is given by Zk |bil = m|bil . (20) k 1 X H (t) = B(t) Z , (15) G 2 k Therefore, in the density matrix ρ = k R 0 0 (|ψ i hψ |)L P (B)dB of the logical qubit, the off- 0 where B(t) is the time-dependent strength of the global m m diagonal elements |bil hb|l0 have coefficients decaying fluctuating magnetic field. In typical ion-trap experi- with time as ments, global dephasing is dominating, as the typical " # length-scale of noise fields is much larger than the inter- 1 ∆m2 c = exp − γt , (21) ion distance [6, 16]. Global dephasing is also applicable to ∆m 2 2 any system that uses a common local oscillator as phase reference. where the difference in magnetization ∆m = m−m0 takes integer values. Note that the time-decays of these coefficients originate solely due to the difference ∆m = m−m0 III. A LOGICAL QUBIT UNDER DEPHASING in the magnetization values corresponding to different ba- m sis states |bil . For situations where ∆m = 0, no manifes- A logical qubit is constructed from N physical qubits, tation of the global noise in the form of the time-decay of and its generic pure logical state is denoted as |ψiL = the coefficients of the density matrix can be found. The θ iφ θ cos 2 |0iL + e sin 2 |1iL. The logical basis states {|0iL subspace of the Hilbert space of the N-qubit system host- and |1iL are, in general, N-qubit entangled states. A ing the basis states for which ∆m = 0, therefore, forms logical qubit is defined by the set of stabilizer generators a decoherence-free subspace (DFS) which is not affected {Si} and the set of logical operators {XL,YL,ZL} as by global dephasing noise. In contrast to Eq. (20), the effect of local dephasing X |0i = |1i ,X |1i = |0i , (16) L L L L L L noise governed by the Hamiltonian HL(t) on the logical ZL |0iL = |0iL ,ZL |1iL = − |1iL . (17) qubit state |ψiL is determined by the eigenvalue equation Each of these logical operators is acting on multiple " # " # X X physical qubits. Without any loss in generality, one Bk(t)Zk |bil = αkBk(t) |bil , (22) can express the logical state |0iL as a superposition k k of computational basis states of the physical qubits as P where the factors αk = ±1 are defined by Zk |ki = |0iL = l bl |bil. l The effect of dephasing noise on such a complex N- αk |bil, i.e., whether the kth qubit in |bil is in |0i or qubit state can be analyzed straightforwardly by group- |1i state. For uncorrelated dephasing of equal strength ing the physical basis states by their magnetization. The on the n qubits this leads to a decay of the off-diagonal 0 magnetization of a basis state is defined as the difference terms in the density matrix ρ as between the number of spins in the ground state |0i with ∆n eigenvalue +1, denoted as n, and the remaining number c∆n = exp − γt , (23) of spins in the excited state |1i with eigenvalue −1, N −n. 2 The magnetization is expressed as where ∆n is the number of positions in the basis states m = 2n − N. (18) |bil and |bil0 where the entries differ (Hamming distance). Note that in this case the dephasing dynamics is not gov- Each magnetization value has the multiplicity Nm = erned by the (differences in) magnetization m, and DFS N!/(m!(N − m)!). The magnetization has N + 1 possible does not exist in this case. 4

A. Assessing the quality of a logical qubit The logical basis states {|0iL , |1iL} are given by 1 We now discuss the relevant quantities to assess the |0iL = √ (|001i + |110i) , quality and to characterise decay dynamics of a logical 2 qubit. A natural choice of such quantities would be the 1 |1iL = √ (|000i − |111i) . (29) components of the logical Bloch vector R~ = (Rx,Ry,Rz), 2 where we identify Rx,y,z as m Evidently, the basis states |bil contributing in |ψiL have four specific values of m, given by m = −3(|111i), Rx = hXLi,Ry = hYLi,Rz = hZLi. (24) −1(|110i), 1(|001i), and 3(|000i). Therefore, following Here, hOi = Tr [Oρ0] is the expectation value of the op- the discussions in Sec.III, the dynamics of the coef- 0 erator O in the noisy state ρ0 of the logical qubit. ficients of the off-diagonal elements in ρ are governed A major issue for characterizing logical qubit dynamics by the exponential decay factors as given by Eq. (21), 1 is the fact that noise typically causes leakage from the namely exp − 2 γt (corresponding to the off-diagonal m m±2 code space. It is therefore useful to also quantify the terms of the form |bil hb|l0 ), exp [−2γt] (corresponding code-space population, p = hP i, where m m±4 code to the off-diagonal terms of the form |bil hb|l0 ), and exp − 9 γt (corresponding to the off-diagonal terms of N 2 1 Y m m±6 P = (I + S ) (25) the form |bil hb|l0 ). Explicit calculation of the expec- c 2N k k=1 tation values of the quantities discussed in Eqs. (24)-(26) in Sec.IIIA under global dephasing noise leads to: denotes the projector onto the code-space of an N-qubit −2γt stabilizer QEC code [2]. Here, {Sk} is the set of stabilizer Rx = e sin θ cos φ, (30) generators that define the code, and I is the identity − 1 γt Ry = e 2 sin θ sin φ, (31) operator in the Hilbert space of the N physical qubits. 1 − 9 γt 4γt 2 θ In order to incorporate the effect of leakage from the Rz = e 2 cos θ + 2e cos − 1 , (32) code space in the expectation values of the logical oper- 2 2 ators, we also consider the quantities {px, py, pz}, where 1h − 1 γt 2 θ − 9 γt 2 θ i p = e 2 cos + e 2 sin + 1 , (33) 2 2 2 px = hXLPci, py = hYLPci, pz = hZLPci. (26) − 5 γt 3γt px = e 4 sin θ cos φ cosh , (34) The relevant time-scales in the evolution of these quan- 4 − 5 γt 3γt tities under global dephasing noise are given by Eq. (21) py = e 4 sin θ sin φ cosh , (35) for magnetization differences ∆m. In Sec.IIIB, we de- 4 rive the theoretical results for the time evolution of the 1h − 9 γt 2 θ − 1 γt 2 θ i pz = cos θ − e 2 sin + e 2 cos . (36) expectation values of these quantities. We then also com- 2 2 2 pare this to our experimental results. The encoding of the logical qubit is a 3-qubit repeti- tion code and can be implemented by a single fully entangling MS gate with unitary MS(π/2), followed by a col- B. Dephasing noise on small QEC codes lective local operation collective operation Ux(π/2) [16]. The individual eigenstates of the logical Pauli opera- We now examine how the quantities discussed in tors can be prepared by applying single qubit operations Sec.IIIA evolve over time under global dephasing noise, UE(θ) = exp(−iθ/2Y1) on the first physical qubit be- for a single logical qubit in a variety of three- and four- fore applying the MS gate. The rotation angle of UE(θ) qubit QEC codes. is θ ∈ {0, π, π/2} to generate the {−1, +1, +1} logical eigenstates of the logical {ZL,ZL,XL} operators. The encoding circuit is shown in Fig.1. We thus prepare the 1. A three-qubit bit-flip code logical qubit in the +1 eigenstate of the logical X operator and the ±1 eigenstates of the logical Z operators. The first example we consider is that of a 3-qubit QEC In Fig.2 we present measured data of the dynamics af- code, whose stabilizer operators are given by ter preparing the logical state in the {+1, +1, −1} eigenstate of the logical {XL,ZL,ZL} operator. We estimate S1 = Y1X2Y3,S2 = X1Y2Y3, (27) the coherence time of the physical qubits by performing least-squares fits of the dynamics of the individual expec- and the logical operators are tation values according to Eqs. (30)-(36), where the experimental imperfections are modeled by multiplying the X = −Y Y Z , L 1 2 3 expectation value with a constant contrast factor. The ZL = X1X2X3, mean value of all individual fit results yields an exper- YL = iXLZL = Z1Z2Y3 (28) imental coherence time T2 = 78(32) ms and a contrast 5

a) b)

FIG. 1. Circuit to prepare the a) 3-qubit and b) 4-qubit encoded states |ψiL.

0.89(3), where the error describes the standard devia- a time dependence given by exp [−8γt]. These character- tion of the mean. All lines depicted in Fig.2 represent istic time-decays yield decays of the expectation values the theoretical models with the mean coherence time and (see Sec.IIIA), as given by the following equations: contrast estimated from experimental data. R = sin θ cos φ, (40) Notably, in Fig.2 a) the expectation value of the log- x R = e−2γt sin θ sin φ, (41) ical ZL operator initially vanishes but then grows with y −2γt increasing storage time, as predicted by Eq. (32). This Rz = e cos θ, (42) is counter-intuitive to the expectation from dephasing 1 p = 3 + e−8γt + e−8γt − 1 sin θ cos φ , (43) from physical qubits. An animation of the logical qubit 4 behavior on the Bloch sphere can be found in the online 1 −8γt −8γt supplementary material [41]. px = e − 1 + e + 3 sin θ cos φ , (44) 4 The expectation values of the +1 and -1 eigenstate of −2γt py = e sin θ sin φ, (45) the logical ZL operator depicted in figure2 b) and c) are −2γt expected to show drastically different dynamics accord- pz = e cos θ. (46) ing to Eq. (36), which is reflected in the experimental The procedure to generate the 4-qubit Grassl code con- data. Animations of the logical qubit behavior on the sists of two half-entangling MS gates MS(π/4) with an Bloch sphere can be found in the online supplementary interleaved spin echo pulse Uz(π) = exp(−iπ/2Sz), with material [41]. P Sz = i Zi. Preparing a particular logical eigenstate of the Pauli operators can again be achieved by a single local operation applied before the first entangling oper- 2. Four-qubit Grassl code ation. Again, the logical state is determined by a single local operation before the entangling operation. The pro- Next, we consider the four-qubit QEC code used for cedure is shown in Fig.1b). correcting erasure noise, as proposed by Grassl et al. [42], The experimental results for this four-qubit code for defined by the stabilizers the +1 eigenstates of the logical X operator are shown in figure3a). Here, it is notable, that the logical X expecta- S1 = X1X2X3X4, tion value does not decay, but the population in the code S = Z Z Z Z , space is decaying rapidly to the steady state value of 0.5. 2 1 2 3 4 Figure3b) shows the behavior for the +1 eigenstate of S3 = −X1X2Y1Y2. (37) the logical Z operator. Due to experimental drifts, the experimentally generated eigenstate has been rotated. The The computational basis corresponding to the logical theoretical description in Fig.3b) is based on a qubit qubit is given by {|0i , |1i }, with L L in the state |Ψi = cos(θ) |0i + sin(θ) |1i with θ = 0.16 |0i = |Φ+i |Φ+i , |1i = |Φ−i |Φ−i , (38) radian. It is notable that the expectation value of the L L X logical operator increases with the waiting time if the where |Φ±i = √1 (|00i ± |11i), and the logical operators code was initially close to the +1 eigenstate of the logi- 2 are cal Z operator. This behavior is predicted by Eq. (44). Animations of the logical Bloch vectors are shown in the online supplementary material [41]. XL = Z1Z3,ZL = X1X2,YL = −Y1X2Z3. (39) Note that one could work also with a variation of this The forms of {|0i , |1i } suggest that the different mag- code, by working with logical basis states given by L L m netization values corresponding to the basis states |bil + + |0iL = |Ψ i |Ψ i , contributing in |ψiL are m = 4, 0, −4. This implies that − − m m±4 |1iL = |Ψ i |Ψ i , (47) the coefficients of terms of the form |bil hb|l0 in the density matrix would decay as exp [−2γt], while the co- with |Ψ±i = √1 (|01i±|10i). Note that this code is up to 2 m m±8 efficients of the terms of the form |bil hb|l0 would have local single-qubit rotations equivalent to the investigated 6

Code space Logical Bloch projections vector a) 1.0

0.8

0.6

+X 0.4

0.2

Expectation value 0.0

b) 1.0

0.8 pz

py 0.6 px +Z 0.4 p

Rz 0.2 Ry Expectation value 0.0 Rx

c) 1.0

0.5

-Z 0.0

0.5 Expectation value

1.0 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.4 Time (T2) Time (T2)

FIG. 2. Expectation values of the logical Pauli operators and code space population for the 3 qubit code, initially in the a) +1 eigenstate of the logical X operator, b) +1 eigenstate of the logical Z operator, c) -1 eigenstate of the logical Z operator. The wait time for experimental data is given in units of T2 = 78(32)ms and the theoretical expectation values are multiplied by a constant value of 0.89(3). code as defined by the basis states given in Eq. (38), how- Awareness of these effects is particularly relevant for ever, it is expected to provide immunity against global quantum error correction protocols that protect quantum dephasing noise. memories, where a key goal is to extend the information storage time. Here, a careful choice of logical operators, and local-unitary equivalent stabilizer operators, actually IV. CONCLUSIONS matters, and should also be taken into account when an- alyzing the expected performance of longer algorithms on fault-tolerant hardware. In this work, we illustrated that simple physical noise models can lead to non-trivial dynamics of logical qubits, Extensions of the present work could include the analy- which are not captured by usual relaxation time scales. sis of spatial correlations which are not maximal through- As shown by the examples explored in this work, devia- out the entire register, the effect of temporal correlations from simple exponential decay dynamics of logical tions, and potential generalizations of spin-echo tech- qubits are possible even in Markovian systems. However, niques from physical qubits to logical qubits. In this re- the behavior of the encoded system can be described by gard, physically Markovian dynamics implies monotonic the logical Pauli expectation values in conjunction with decay of the physical Bloch volume element [5, 43]. This the code space population, given by the expectation value property can be translated to the logical level by con- of the code-defining stabilizers. sidering the logical Bloch volume element relative to the 7

Code space Logical Bloch projections vector a) 1.0

+X 0.5

Expectation value 0.0 py

px p b) 1.0 Rz

Rx 0.5 +Z

0.0 Expectation value

0.5 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 Time (T2) Time (T2)

FIG. 3. Expectation values of the logical Pauli operators and code space population for the 4 qubit code, initially in the a) +1 eigenstate of the logical X operator and b) the +1 eigenstate of the logical Z operator. The evolution time for experimental data is given in units of T2 = 25(2)ms and the theoretical expectation values are multiplied by a constant value of 0.93(1).

code population. Namely, the volume element induced der Grant Agreement no. 820495, and by the Office of c c by the mean values Rx = Tr[ρcXL], Ry = Tr[ρcYL] and the Director of National Intelligence (ODNI), Intelligence c Rz = Tr[ρcZL] for the conditional state ρc = PcρPc/p0, Advanced Research Projects Activity (IARPA), via the c which in our previous notation are nothing but Rx = U.S. ARO Grant No. W911NF-16-1-0070. ll statements c c px/p0, Ry = py/p0 and Rz = pz/p0. A nonmonotonic de- of fact, opinions or conclusions contained herein are those cay of this volume element certifies non-Markovian evo- of the authors and should not be construed as represent- lution at the logical level. ing the official views or policies of IARPA, the ODNI, Furthermore, one could aim at the development of or the U.S. Government. We acknowledge support from state preparation and measurement error insensitive ver- the Institut fürQuanteninformation GmbH (Innsbruck, sions of the characterization protocols used in this work. Austria). We acknowledge financial support from the Finally, an interesting and open challenge concerns the Spanish MINECO grants MINECO/FEDER Projects derivation of effective, efficiently simulatable noise mod- FIS 2017-91460-EXP, PGC2018-099169-B-I00 FIS-2018 els for logical qubits. This is not only relevant for the and from CAM/FEDER Project No. S2018/TCS- quantum memory scenario, but also for reliable numerical 4342 (QUITEMAD-CM). AKP acknowledges support predictions of the performance of logical gates or gadgets from the National Science Center (Poland) Grant No. like lattice surgery, state distillation and injection tech- 2016/22/E/ST2/00559. niques, which will be required for the operation of large, fault-tolerant quantum processors. Author contributions. AKP, PS, and MM wrote the manuscript and all authors provided revisions. AKP, Acknowledgements We gratefully acknowledge MM, PS, and TM developed the research based on funding by the U.S. Army Research Office (ARO) discussions with RB, AR, and MAMD. AKP and MM through grant no. W911NF-14-1-0103. We also ac- developed the theory. AE and PS performed the exper- knowledge funding by the Austrian Science Fund (FWF), iments and evaluated the data. AE, PS, RB, and TM through the SFB BeyondC (FWF Project No. F71), by contributed to the experimental setup. All authors con- the Austrian Research Promotion Agency (FFG) contributed to discussions of the results and the manuscript. tract 872766, by the EU H2020-FETFLAG-2018-03 un- 8

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