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 M¨uller2, 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 f¨urExperimentalphysik, Universit¨atInnsbruck, Technikerstr. 25, A-6020 Innsbruck, Austria 5Departamento de F´ısica Te´orica, 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 f¨urQuantenoptik 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 Gr¨unberg Institute, Theoretical Nanoelectronics, Forschungszentrum J¨ulich,D-52425 J¨ulich,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 fj0i ; j1ig as j i = cos 2 j0i + e sin 2 j1i, 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 j 0i = h i been described in detail in reference [38]. The qubits are R t 0 0 exp −i 0 HG(t )dt j iL, leading to encoded in the 4S1=2(mj = −1=2) = j1i ground state and t the metastable excited state 3D5=2(mj = −1=2) = j0i θ Z θ j 0i = cos j0i + exp i φ + B(t0)dt0 sin j1i ; 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 = j i h j 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 j0i h0j + sin j1i h1j 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 j0i h1j + e j1i h0j): (8) in our experimental system is given by dephasing caused 2 by laser frequency noise and fluctuations in the bias mag- netic 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.