Pulse Techniques for Quantum Information Processing Gary Wolfowicz1 & John J.L

Pulse Techniques for Quantum Information Processing Gary Wolfowicz1 & John J.L. Morton2 1University of Chicago, Chicago, IL, USA 2University College London, London, UK

The inherent properties of quantum systems, such as superposition and entanglement, offer a new paradigm to the treatment and storage of information in physical systems, giving rise to potentially faster computation, more secure communication, and better sensors. Electron spins in a range of host environments are ideal systems for representing quantum information, and such systems can be both characterized and controlled through the arsenal of pulse electron paramagnetic resonance (EPR) techniques. In this article, we introduce basic concepts behind quantum information processing with spins, including how spin decoherence (i.e., the corruption of quantum information) has been studied and mitigated in various systems; how EPR methods can be used to manipulate quantum information in spins with high fidelity; how individual spins can be used as high-resolution sensors (e.g., of magnetic field); and how multiple spins can be coupled together to develop the building blocks of a larger scale quantumcomputer. Keywords: quantum bit (qubit), spin qubit, quantum information processing, quantum sensors, quantum memories, decoherence, gate fidelity, quantum state tomography

How to cite this article: eMagRes, 2016, Vol 5: 1515–1528. DOI 10.1002/9780470034590.emrstm1521

Since the early 1980s, it has been recognized1,2 that representing much richer state space to occupy than its classical counterpart. information within quantum systems offers the potential for We can write the state of the qubit in general as 3,4 profound enhancements in information processing. By 𝜃 𝜙 𝜃 |𝜓⟩ = cos |0⟩ + ei sin |1⟩ (1) initializing, controlling, and measuring arrays of coupled 2 2 quantum systems, it is possible, in principle, to build efficient which maps out the surface of the Bloch sphere (Figure 1a). tools for the simulation of quantum systems, ranging from A spin-1/2 in a magnetic field is a natural⟩ realization of purpose-built analog models through to ‘digital’ quantum | a qubit, with the eigenstates |m =−1 ≡ |↓⟩ ≡ |0⟩ and simulators, which follow algorithmic approaches to solve the ⟩ | S 2 | evolutionofaHamiltonian.So-calleduniversalquantumcom- |m =+1 ≡ |↑⟩ ≡ |1⟩ (representing, respectively, the spin | S 2 puters possess a sufficient set of logical operations, or gates, projection, the spin orientation, and the qubit state— (see to perform arbitrary quantum algorithms such as those for Spin Dynamics). Much early progress was made in exploring factoring large numbers and searching datasets, and represent quantum algorithms and control using liquid-state NMR, a generalization of the concept of a Turing machine. Motivated including the application of Shor’s factoring algorithm to the by these opportunities, the past 20 years has seen a great deal number 15 in Ref. 5. However, thermal initialization of NMR of experimental research into realizing quantum information systems becomes exponentially harder with increasing number processors, based on a wide range of physical systems, includ- of qubits, limiting the scalability of the approach. Focus on ing spins, atoms, ions, and superconducting circuits. More spin qubits has therefore shifted to a large extent to electron recently, applications of individual highly coherent quantum spins,thoughnuclearspinqubitsremainattractiveformemory systems as sensors have been identified, potentially yielding applications due to their long quantum state lifetimes of up to nearer-term technologies. In this article, we will review pulse 6h.6 Electron spins being explored for quantum information techniques in electron paramagnetic resonance (EPR) related applications include the nitrogen vacancy (NV) and other color to research in quantum information processing and sensing centers in diamond7,8 and SiC,9 donors in silicon,10 and quan- using electron spins. tum dots in silicon,11 GaAs,12 and other semiconductors.13 Molecular electron spins14–16 have also been explored as useful systems to test concepts in quantum information, owing Spin Qubits to their reproducible nature and the ability, in principle, to In analogy to digital binary logic, the fundamental unit of quan- engineer coupled systems. tum information is the quantum bit, or qubit, consisting of a two-level quantum system labeled with the basis states |0⟩ and Single-qubit Gates |1⟩. The ability of a quantum system to exist in a superposition In classical binary logic, the only nontrivial operation, or ‘gate’, of states, with some phase relationship, provides the qubit with a which can be applied to a single bit is the NOT gate (where

θx Measure Prepare −θz xxx x x y −x x | 〉 T π π Echo π π T π π Echo 0 2 2 2 2 z z z 1 1 θ |ψ〉 y y x x

y ϕ 0 0 x Echo intensity/a.u. Echo intensity/a.u. In-phase (y) In-phase (y) Quadrature (x) −1 −1 |1〉 0 500 1000 1500 0 100 200 300 400 500 (a) (b) T/ns (c) T/ns

Figure 1. Single-qubit representation and control. (a) The Bloch sphere representation of a qubit state |휓⟩,definedbyangles휃 and 휙.Spin-downand spin-up eigenstates can be mapped onto the qubit states |0⟩ and |1⟩.(b)Rotationsofanelectronspinqubitaboutthex-axis can be performed using a simple resonant microwave pulse, observable (using a subsequent 2-pulse echo sequence) as Rabi oscillations where the duration of the pulse is swept.(c) Rotations about the z-axis can be performed by decomposing into x-andy-rotations. Experimental data taken using P donors in Si, at 8 K

0 → 1and1→ 0). Thanks to the much richer state space of applying a selective 2π microwave pulse to an electron spin the qubit (Figure 1a), there is a much greater set of single-qubit transition, resulting in a nuclear spin phase gate that is orders gates, which (neglecting a global phase) can be represented as of magnitude faster than that which could be achieved using rotations of the Bloch sphere around some axis and by some conventional ENDOR/NMR pulse sequences.14 angle. In pulse EPR, and in the rotating frame, rotations about axes in the x–y-plane can be trivially performed through sim- Quantum State Tomography plemicrowavepulses,wherethepulsephasedeterminesthe rotation axis, while the pulse power and duration set the angle The complete state of a qubit, taking into account effects such as (Figure 1b). Rotations about other axes (and thus any unitary decoherence, must in general be described by the density matrix Spin Dynamics 𝛔̂ operation) can be performed by decomposition into a series (see ), ,whichcanbedecomposedasfollows: of x and y rotations (휃̂ ), or through more elaborate methods 𝟏 x,y 𝛔̂ = + r Ŝ + r Ŝ + r Ŝ (4) such as pulse shaping, as discussed in Shaped Pulses in EPR. 2 x x y y z z For example, a z-rotation (Figure 1c) can be implemented as 𝟏 ̂ where is the identity matrix, Sx,y,z are the spin operators, and 휃̂ 휃 r , , are the real coefficients that determine the state. In order to z =(π∕2)x( )y(π∕2)−x (2) x y z characterize the practical implementation of the gates described while the Hadamard gate, a common single-qubit gate used to above, it is necessary to measure the resulting state of the qubit, generate superposition states (and representing a rotation by determining its density matrix through a process known as state π aboutanaxisinthedirectionx + z), can be decomposed as tomography. Owing to the collapsing effect of measurement on follows: ( ) quantum systems, state tomography is achieved only by com- ( ) bining many repetitions of an experiment on a single qubit, or ̂ √1 11 π UHadamard = = i(π)x (3) using an ensemble of many such qubits. In conventional EPR, 2 1 −1 2 y values proportional to rx and ry can be determined upon forma- An additional approach to single-qubit operations employs tion of a spin echo (Figure 3). A corresponding value for rz can- the geometric phase acquired by spins driven through some not be directly measured simultaneously (at least not without closed trajectory. In particular, when a spin qubit is driven some additional measurement technique, such as a SQUID), through a cyclic evolution (i.e., the initial and final states but its value can be inferred if the echo is followed by an ideal coincide on the Bloch sphere), it acquires a geometric, or π∕2-pulse, effectively mapping the desired Ŝ component onto ̂ z ‘Berry’,phasethatisequaltohalfthesolidangleenclosedbyits the Sx observable. 17 ′ 훼 evolution on the Bloch sphere (Figure 2). Because this phase The measured values rx,y,z(= rx,y,z) must then be normal- is global (the same phase is acquired by both eigenstates), it ized to produce a physical density matrix–this can be done serves no purpose for an isolated spin-1/2 qubit. However, in various ways, but a common approach in ensemble EPR given the availability of additional levels (e.g., thanks to hyper- is to make the ‘pseudo-pure’ approximation,19 which works fine coupling, or for S > 1∕2), it can be exploited to create as follows: At typical temperatures and magnetic fields, elec- ≪ the so-called Aharonov–Anandan (AA) phase gate, which is tronspinsareinaweaklypolarizedstatesuchthatrx,y,z 1. an effective z-rotation of the qubit.18 AA phase gates have However, it is possible to neglect the large 𝟏 component of the been performed on coupled electron–nuclear spin systems by density matrix, as it is invariant under unitary transformations,

Prepare Measure S Measure S | 〉 x, y z |1〉 3 x y | 〉 , Φ 2 π Echo π Echo π π 2 2 Φ = 2π (a) ϕ = π MW +x +y +z Identity 1 1 |1〉 RF |0〉 0 0 Initial

Real Electron spin Nuclear spin −1 −1 Image (a) |2〉 (b)

Figure 3. Quantum state tomography using EPR pulses to obtain the | +1 1〉 density matrix of a qubit. (a) Two consecutive Hahn echo sequences provide separately the x, y,andz components of the qubit state, necessary to reconstruct the density matrix. (b) Extracted spin-1/2 density matrices (matrix elements plotted on a vertical scale), given different state prepara- 0 tion pulses

Decoherence | Nuclear spin polarisation (2π) (2π) (2π) 〉 −1 MW MW MW 0 One of the primary criteria in choosing a particular physical 0 100 200 300 400 representation of a qubit or quantum sensor is maximizing (b) Time/μs the coherence lifetime: decoherence, defined as undesired phase shifts in the qubit state, are errors in quantum com- Figure 2. Aharonov–Anandan geometric phase gates using coupled putation, while a short coherence time prevents long-phase electron and nuclear spins. (a) A microwave pulse may be applied reso- accumulation necessary for sensing applications. | ⟩ ↔ | ⟩ | ⟩ | ⟩ nant with the 1 2 transition to drive states 1 and 2 around a closed Although electron spin (2-pulse echo) coherence times (T ) trajectory on the corresponding Bloch sphere. The solid angle, 훷,enclosed 2 as long as several hundred microseconds have been observed by this trajectory defines the geometric phase 휙 = 훷∕2acquiredbyboth 20 thestates.Exampleclosedtrajectoriesareshownforthecaseswherethe in certain molecular systems, thelongestcoherencelifetimes transition is driven resonantly (white) or nonresonantly (yellow). (b) The have emerged from defect spins in solid-state materials, in par- effect of this phase shift on a qubit defined by states |0⟩ and |1⟩ can be ticularthosewithalownaturalabundanceofnuclearspinssuch observed by driving nuclear spin Rabi oscillations without (gray) and with as carbon- and silicon-based materials. The ability to grow pure (black) fast phase gates. After each 2π microwave pulse applied on the elec- crystals with low charge defect concentration and the possibil- π tron spin, the nuclear spin acquires a phase and is prevented from making ityofisotopicallyenrichingsamplestominimizethenuclear a complete rotation around the Bloch sphere. Experiment performed using spin concentration have led to measured electron spin coher- N@C60. (Reprinted by permission from Macmillan Publishers Ltd: Nature Physics, Ref. 14 © 2006.) ence times of up to 1.8 ms for NV centers in diamond at room temperature21 and up to 20 ms for P donors in silicon at 6 K.22 Notably, when measuring such long times in spin ensembles and treat the remaining part of the state as if it were pure. In by conventional EPR, instantaneous diffusion often becomes this way, we can assume a given state to be ‘pseudo-pure’ and the limiting decoherence mechanism23 (see Pulse EPR). This impose the normalization condition for a pure state, ||r|| = 1, can be confirmed by measuring the decoherence time as a in order to extract a value of the proportionality constant, 훼. function of the refocusing pulse angle in the 2-pulse echo This assumption is usually applied to the initial state in the sequence (Figure 4). Instantaneous diffusion23 can be avoided pulse sequence, such that subsequent states produced (which using alternative spin measurement techniques with greater in general will not be pure) are normalized using the same sensitivity such as optically detected magnetic resonance, value of 훼. wheremorediluteensembles,24 or indeed single spins,7,25 can As multiple qubits are coupled together, the state space expo- be measured. nentially expands. This gives an illustration of the increased The T2 values measured in bulk crystals can become orders power of quantum information processors, and also brings of magnitude shorter when the electron spins are brought near new challenges in performing state tomography due to the interfaces,astheyoftenmustforpracticalapplications.There number of operations required (see section titled ‘Multiple is good motivation therefore to develop methods for extend- Qubits’). Having discussed methods to extract the full state ing coherence times and protecting the spin qubit from noise ofaspinqubit,wenextturntowaysinwhichthisstatecan sources. In the following section, we describe two approaches: be corrupted–first through ‘natural’ processes in the system first, using a technique known as dynamical decoupling (DD), and its environment (decoherence) and second through the which is a natural extension of the 2-pulse (Hahn) echo to a nonideality of the applied operations (pulse errors). large number of refocusing pulses and second, by identifying

1 π τ/2 τ/2 Echo τ θ 2 θ = π π π π π π θ = π/2 (a) θ = π/4 θ = π/8 1 Echo amplitude/a.u. 1 pulse 0 00.05 0.1 5 pulses 40 pulses (a) τ/s

200 ) 150 T = Filter function/a.u. 1 lim 2 40 ms

− θ → 0 0 (s 100 2 0.1 1 10 T

1/ 50 (b) Frequency × 2 τ 0 0.0 0.2 0.4 0.6 0.8 1.0 104 2 θ 300 K (b) sin ( /2) 240 K 190 K 160 K Figure 4. Instantaneous diffusion.(a)Measuringechodecaycurvesfor 103 120 K different refocusing pulse angles (휃),asshowninthepulsesequence(inset), 77 K can be used extract the instantaneous diffusion contribution to T2.(Note: Curves are normalized to unit intensity (for 휏 ≈ 0) and the actual echo 102 intensity falls as 휃 decreases). (b) Extrapolating the measured decay con- /ms 2

휃 → T stants (T2)forthecaseof 0providesanestimateforT2 in the absence of instantaneous diffusion (i.e. in the dilute spin limit).22 Experiment per- 1 formed at 5 K using Bi donors in silicon, at a concentration of 4 × 1015 cm−3 10 spin transitions that are inherently insensitive to external per- 100 turbations such as magnetic field noise. These methods can be used to achieve gains in coherence time between one and two 100 101 102 103 104 105 orders of magnitude. (c) Number of pulses n

Figure 5. Dynamical Decoupling (DD) Dynamical decoupling (DD).(a)AtypicalDDsequencecon- sists of repeated π refocusing pulses whose inverse period 1∕휏 must be The single refocusing π-pulseintheHahnechoisthemostbasic smaller than the frequency of the noise (black curve). (b) DD acts as a filter exampleofDD,abletoprotectthespinqubitfromdephasing function, where larger pulse numbers lead to narrower bandpass filters. (c) caused by variations in the spin transition frequency that are Electron spin coherence time T2 as a function of pulse number in an NV center (Reprinted by permission from Macmillan Publishers Ltd: Nature static (at least, on the timescale of the experiment). More gen- Communications, Ref. 29 © 2013.) showing an increase of up to three orders eral DD sequences can be used to protect the spin from more of magnitude using 10 000 DD pulses elaborate noise sources and are adaptable to various noise spec- 26 tra and spin interactions. isappliedtoaninputstateorientedalongx,thepulseerrorsare Decoherence from dynamic noise can be suppressed by additive leading to rapid corruption of the state. On the other repeated application of refocusing π-pulses, with an inverse hand, if the same sequence is applied to a ±y input state, it is 휏 period 1∕ chosen to be less than the high-frequency cut- robust to pulse errors as they cancel out to first order–indeed, 26 off of the noise spectrum. Such a sequence corresponds 31 32 this is why Meiboom and Gill adapted( ) the( CP sequence) to n to a bandpass filtering effect centered around harmonics of π 휏 휏 1 produce what we now term CPMG: − −π − − . f = , and with a bandwidth narrowing with large pulse 2 2 y 2 2휏 x number,27,28 as shown in Figure 5. The higher harmonics of DD sequences of most interest in quantum information pro- f have a lesser contribution to the decoherence as the noise cessing possess robustness to pulse errors for all input states, 33 frequency becomes too fast compared to the spin evolution. and examples include the XYXY sequence and higher orders : DD sequences have been applied with millions of pulses in ≡ XY − 4 πyπxπyπx (5) NMR30 and up to 10 000 pulses in EPR29 (Figure 5), and in such ≡ cases, careful consideration must be given to the effect of pulse XY − 8 πyπxπyπxπxπyπxπy (6) errors. In order for the DD pulses not to significantly intro- where the delay times 휏 between the pulses are omitted for duce errors themselves, either pulses with very high fidelity are clarity. required or the DD sequence should be robust( to pulse error.) 휏 휏 n Theabovesequencesaresuitablefordecouplinga‘cen- For example, when the basic DD sequence − −π − − 2 x 2 tral’ spin from some noise source, such as a bath of different

spins, but they are not able to refocus interactions between 10 identical spins, which also evolve under the DD pulses. Sequencestodecouplesuchinteractionsinvolveπ∕2rota- tions, and examples to decouple dipolar interactions include 5 X-band the Waugh–Huber–Haeberlen sequence (WAHUHA) and its Transition ESR-type CTs higher order extension MREV.34 frequency/GHz DD sequences have applications beyond simply preserving 0 100 200 300 400 500 600 spin-qubit states–their filtering properties can also be used in (a) Magnetic field/mT spectroscopic applications as a ‘lock-in amplifier’ to pick out 35 10 dω certain interactions (see section titled ‘Sensing’), while they 7 GHz/dB (d 9.7 GHz 3.6 × 1014 cm−3 ω can also be used to indirectly drive rotations of coupled spins. 0 /d ̂ B 0 For example, given an electron spin (with operator S)anda ) 2 1 − nuclear spin (with operator Î) coupled by an anisotropic inter- 2.0 × 1015 cm 3 action, the repeated π-pulses on the electron spin under a DD 4.4 × 1015 cm−3 ̂ ̂ ̂ ̂ sequence can, as a result of the SzIx or SzIy terms in the spin 0.1 1 /s

Hamiltonian, be seen by the nuclear spin as an oscillating field 2 T = T 2 2.7 s along the x-ory-axis. If the decoupling period matches the Larmor precession of the nuclear spin, there is a resonant condi- 0.01 0.5 tion, and the nuclear spin rotates while the electron spin coherence is simultaneously preserved.36 Echo signal/a.u. 0 062 4 Time, τ/s Parameter-insensitive Transitions 10−4 10−3 10−2 10−1 1 Decoherence can be directly related to the loss of phase caused (b) |dω/dB |, in units of γ byuncontrolledfluctuationsinthequbittransitionfrequency, 0 e altering the Larmor precession rate. Such fluctuations further Figure 6. EPR clock transitions. (a) The allowed spin transition frequen- originate from variations in free (i.e., externally controllable) cies for Bi donor spins in silicon (S = 1∕2, I = 9∕2, A ≈ 1.5GHz)showfour parameters present in the spin Hamiltonian, such as the exter- ‘clock transitions’ (CTs) at microwave frequencies (open circles) at which nal magnetic or electric field. An important characteristic the first-order dependence on magnetic field goes to zero. (b) Electron is the frequency sensitivity of the relevant spin transition to spin coherence time, T2, shown as a function of the first-order sensitiv- 휔 these parameters. A simple and obvious example here is the ity of the transition frequency to magnetic field (d ∕dB0) in Bi donors in 휔∕ ≈ 훾 difference in coherence times typically seen for electron and silicon, for 3 different Bi concentrations. Measurements for d dB0 e nuclear spins: the greater decoherence rate of the electron spin were obtained from the 10 X-band (9.7 GHz) transitions (100–600 mT), while the remaining points were taken in the vicinity of the clock transition is attributed to its much higher gyromagnetic ratio, which 휔 . highlighted in panel (a) (B0 ≈ 80 mT, 0∕2π=7 03 GHz). Lines are fit to (for a system with only the Zeeman interaction) is equal to a model with a quadratic dependence in d휔∕dB due to instantaneous dif- 휔 0 d ∕dB0, the first-order dependence of the transition frequency fusion and a linear dependence due to spectral diffusion. As d휔∕dB → 0, 휔 0 with respect to the external magnetic field B0.Insystems the coherence time becomes limited by direct flip-flop processes that are where various interactions compete, this dependence can be not suppressed at the clock transition and are spin concentration depen- less than the free-spin gyromagnetic ratio, and even reach zero, dent. The inset shows a Hahn echo decay measured at 4.3 K using [Bi] = . × 14 −3 to first order, for certain parameter values (e.g., orientation and 3 6 10 cm magnetic field strength). In the case of magnetic field insen- sitivity, such transitions are called ‘zero first-order Zeeman 휔 shift (ZEFOZ)’ or ‘clock transitions (CTs)’ and can be used to with decreasing d ∕dB0, leading to coherence times in Bi suppress decoherence (Figure 6). donors as long as 2.7 s at 4.3 K, measured using a 2-pulse 38 CTs at, or close to, EPR frequencies have been studied in Hahn echo (Figure 6b, ). In addition to being a strategy to two separate electron–nuclear spin systems, first theoretically mitigate decoherence, the ability to tune the sensitivity of a in rare-earth-doped crystals37 and later demonstrated for Bi spin transition to various parameters can also be used as a way donors in silicon.38,39 InthecaseofBidonors,thehyperfine to distinguish various decoherence mechanisms. interaction between the electron spin (S = 1∕2) and nuclear spin (I = 9∕2) is close to 1.5 GHz, resulting in the presence of four CTs at frequencies between 5.2 and 7.3 GHz (Figure 6a). Measuring Gate Fidelities For rare-earth ions (Er3+), the weaker hyperfine interaction (|A| < 700 MHz, anisotropic) provides CTs below 1 GHz. In As described above, the decoherence times in many candidate both these systems, the limiting decoherence mechanism is electron spin-qubit systems have reached such a level that natu- spectral diffusion (see Relaxation mechanisms; Pulse EPR) ral decoherence can cease to be a major error source in a quan- from either nuclear or electron spins in the environment, tum algorithm. In such cases, imperfections of the qubit control arising from dipolar or hyperfine interactions, which can be place a limit on the achievable error rate–for example, the decay treated as magnetic field fluctuations. At the clock transition, in the Rabi oscillations of Figure 1(b) and (c) is due to the accu- decoherence due to spectral diffusion is suppressed linearly mulation of pulse error, and not intrinsic decoherence.

𝛔̂ PracticalgateerrorsinEPRaretypicallysystematicinnature, given input state described by the density matrix, in,is arising, for example, from nonidealities in the microwave com- transformedintoanoutputstate: ponents (switches, IQ mixers, etc.) used to form the pulses, ∑3 or the finite bandwidth of the cavity. In the usual case in EPR 𝛔̂ 𝛘 ̂ 𝛔̂ ̂ † out = mnAm inAn (8) where spin ensembles are addressed, the dominant source of m,n=0 error often arises from the inhomogeneity of the microwave ̂ 𝟏̂,̂휎 ,̂휎 ,̂휎 field intensity across the sample, leading to a variation in the where Ai is the Pauli basis { x y z}.Quantumprocess 𝛘 achievedrotationangleacrosstheensemble.Inprinciple, tomography (QPT) aims to obtain this matrix, , by per- errors arising from field inhomogeneity are not relevant at the forming state tomography of the output state, given a suitably 3 level of single-spin control.25 However, owing to the challenges representative set of input states. Owing to state preparation of confining microwave magnetic fields to small regions of and measurement (SPAM) errors, or variations in the applied space, many quantum computing schemes based on spins operation, the process matrix directly constructed by linear employ microwaves applied to large arrays of spins, each of inversion may not be physical (Hermitian positive). In such which can be individually tuned into resonance with the field.40 cases, the closest physical operation can be found by least 44,45 Assuming the experimentally applied operator B̂ is uni- squared fitting. Some example process matrixes are shown tary, its fidelity  with respect to an ideal operator Â can be in Figure 7. expressed as (for a spin-1/2)3 Randomized Benchmarking 1  = Tr(Â B̂ −1) (7) 2 The above methods to characterize gates have a number of lim- In the following sections, first, we describe the methods that itations: QPT is sensitive to SPAM errors (although in principle have been used to characterize gate fidelities in EPR and then they can be separated from the applied process with appropri- we discuss the strategies for maximizing the fidelity. ate calibration steps) and requires an exponentially increasing number of operations as the number of qubits grows. On the CP vs CPMG other hand, the method of comparing CP and CPMG assesses only one kind of error and gives an unrealistic picture of how Minimum gate fidelities for fault-tolerant quantum compu- errors may combine in a practical computation. tation are around 99% (based on the latest, most forgiving Randomized benchmarking is a technique that solves these 41 error-correction schemes identified so far )butareideallyat issues by applying a sequence with increasing number of ran- the level of 99.99% and above. When the error per gate is so domlyselectedpulsesthateventuallydepolarizethequbit.The low, accurately measuring the gate fidelity requires repeated pulsesarechosenfromtheCliffordgroup,asetofrotationsthat application of the gate so the errors can build up and be more span uniformly the Hilbert space, thus providing an average readily measured. For example, we have already seen how pulse error.47 For a single qubit, the Clifford gates are the different DD sequences, such as CP and CPMG, have different identity, π, π∕2, and 2π∕3rotations.48 After each sequence, the tolerances to rotation angle errors and this difference can be qubit state is measured along some axis (e.g., x or y). The decay exploited to measure pulse errors associated with rotation in fidelity as a function of pulse number gives the average pulse 42,43 angle. Measuring the decay of echo intensity in a train of error, with randomization ensuring the fidelity is not corre- spin echoes can provide a basic estimate for rotation angle latedtoanyparticularpulse.Inordertotestaspecificgate,the errors (from which gate fidelity can be derived). However, for fidelity decay due to an increasing number of Clifford gates is a more complete characterization of the possible gate errors, compared to the decay with this gate interleaved between the more elaborate techniques are required. Clifford gates. Randomized benchmarking was first performed using Quantum Process Tomography liquid-state NMR,49 followed by EPR experiments on single- An experimentally applied operation can in general be electron spins of NV centers in diamond50 and donors in described by a process matrix, 𝛘,whichdescribeshowa silicon,11 and in ensemble EPR using a loop-gap resonator.51

1.0

1.0 1.0 1.0 1.0

0.5 0.5 0.5 0.5 0.5

0.0 0.0 0.0 0.0 i i i i x x x x z z z z y y y y y y y y z x z x z x z x 0.0 (a) i (b) i (c) i (d) i

휒 Figure 7. Quantum process tomography. Experimentally obtained processes matrices, , for (a) the identity operator, (b) πy rotation, (c) a Hadamard gate, and (d) a combination of processes, such as T1, T2 and pulse errors, leading to a partially mixed state. Red outlines correspond to the ideal processes. Experiments performed using (a, b) Sb : Si,44 (c) P : Si; and (d) Nd3+ :YSO46

High-fidelity Operations concatenation levels can theoretically achieve arbitrary robustness. In practice, however, higher levels are rarely applied as As mentioned above, and discussed in more detail in section they require exponentially more pulses. titled ‘Spin-qubit Networks’, required gate fidelities for quan- BB1 (short for ‘broadband 1’) is a common composite pulse tum computation are typically 99.9–99.99%, much higher than to tackle pulse amplitude error,54 where the sensitivity of fidelity is typically required for spectroscopic applications. In conven- to rotation angle errors in each pulse is reduced to sixth order. tional dielectric and loop-gap cavities in EPR, inhomogeneities It is defined with four pulses: of 10–20% in the microwave field strength are common, leading to fidelities of less than 95–99% for a simple π-pulse. In stripline (휃)ideal =(π) (2π) (π) (휃) (9) resonators and microresonators, the fidelities can be substan- 0 휙 3휙 휙 0 tially less.52 with 휙 = cos−1(−휃∕4π) (10) Thefidelityofagateachievedbyasinglerectangularpulse can be increased using composite pulses (comprising several where 휃 is the desired rotation angle, while the phases 휙 and pulses) or by various pulse shaping methods (see Shaped 3휙 aredefinedwithrespecttothephaseofthedesiredideal Pulses in EPR) including adiabatic pulses and the use of geo- rotation. Using BB1, fidelities above 99.9% have been measured metric phases. In each case, common considerations when in conventional dielectric EPR resonators, despite amplitude designing a pulse shape or sequence to yield a high-fidelity errors up to 10%.43 Other variations exist such as NB1 or PB1 operation include (i) minimizing the pulse duration, given (short for ‘narrowband 1’ and ‘passband 1’) that provide high the available power and bandwidth of cavity and microwave fidelitybutoverarestricteddeviationoftheamplitude,54 which components; (ii) achieving the desired frequency selectivity can be useful to spatially select spin ensembles in a sample (e.g., to perform a conditional operation); (iii) maximizing the using microwave field gradients. robustness to the dominant imperfections in the system (such For robustness against frequency detuning (e.g., given inho- as microwave field amplitude, or resonant frequency in the mogeneous broadening), the CORPSE (compensation for off case of inhomogeneous broadening in the spin system); and resonance with a pulse sequence) pulse can be used,55 defined, 휃 (iv) independence on the initial spin state–in other words, the for an ideal rotation ( )0,as ( ) ( ) sequence should be correcting the operator, not the final state. 휃 휃 휓 휓 휓 Many studies, often originating in NMR, have attempted − (2π−2 )π 2π+ − (11) to fulfill these criteria, offering robustness to errors in fre- 2 0 2 0 −1 quency/detuning and pulse amplitude, at the cost of longer with 휓 = sin (sin(휃∕2)∕2) (12) pulses.Inthefollowingsection,wediscussvariousapproaches to performing high-fidelity gates in EPR. If the desired pulse is a π-pulse (as is commonly used in many DD sequences), the so-called Knill pulse56,57 can be very effective, with robustness to both detuning and pulse amplitude Composite Pulses errors. The Knill pulse, which results in a π∕3 phase shift in Composite pulses attempt to minimize errors by concatenating additional to the desired rotation, consists of a symmetric multiple (rectangular) pulses with various phases and rotation sequence of five π-pulses: angles,53 based on the assumption that the error is systematic, at ( ) ideal π least over the duration of the composite pulse. The overall evo- π0 =(π)π (π)0(π) π (π)0(π) π (13) 3 z 6 2 6 lution is equivalent to a single rotation whose dependence on variations in parameters such as frequency detuning or pulse The performance of BB1, CORPSE, and Knill sequences is com- amplitude cancels up to a desired order (in average Hamilto- paredinFigure8.Compositepulsescanthemselvesbeusedas nian theory54).Compositepulsesareanappealingsolutionas the base pulses of other composite pulses, in order to combine they are relatively simple to apply, often universal, and for high different type of robustness.58

BB1 CORPSE Knill 0.6 0.6 0.6 Fidelity 0.4 0.4 0.4 99.999% 0.2 0.2 0.2 1 1 1 99.97% ω ω ω / / / 1 0.0 1 0.0 1 0.0 99.9%

Δω −0.2 Δω −0.2 Δω −0.2 99.7% 99% −0.4 −0.4 −0.4 97% −0.6 −0.6 −0.6 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 Δω ω Δω ω Δω ω s/ 1 s/ 1 s/ 1

Figure 8. Robustness of composite pulses for frequency detuning and pulse amplitude errors. The fidelity of BB1, CORPSE, and Knill sequences are 훥휔 훥휔 휔 훾 compared as a function of frequency detuning ( S)andpulseamplitudedetuning( 1)expressedasafractionof 1(= eB1),themicrowavefieldstrength (in angular frequency units). Both the BB1 and Knill sequences are five times the duration of a simple π-pulse (5t ), while the CORPSE sequence is 13 t 180 3 180 in duration

Adiabatic Pulses applicability of such methods at the single-spin level holds As discussed in Spin Dynamics and Pulse EPR,anapplied the potential for high-spatial resolution studies, for example, microwave field resonant with a spin appears, in the rotating using scanning probe techniques to create spatial magnetic frame, as a static magnetic field in the x–y-plane about which field images. thespinthenprecesses.Ifthemicrowavefieldisoff-resonant, The development of electron spin-based ‘sensors’ has pri- the static field in the rotating frame acquires a residual marily arisen out of research on NV centers in diamond, z-component. As an alternative to abruptly turning on and where the electron spin of a single center can be optically measured (see ODMR) at room temperature with coherence off such fields to drive spin rotations, the microwave field 21 amplitude and frequency can be gradually adjusted to change times around a millisecond (Figure 9, ). Figure 9 (a, inset) the direction of the effective magnetic field in the rotating showstheenergyleveldiagramofthe(S = 1) NV center, in frame. In this way, thanks to the adiabatic theorem in quantum the low-field limit–the difference of the fluorescence inten- mechanics,59 spin eigenstates can be moved along determined sities of the mS =±1andmS = 0 states form the basis for trajectories around the Bloch sphere, defining some unitary the optical measurement. Such systems couple to magnetic operation. (The adiabatic theorem states that a physical system fieldsviatheZeemaninteractionandtootherspinsviadipolar remains in its instantaneous eigenstate if a given perturbation and hyperfine interactions, while, thanks to the crystal field is acting on it slowly enough and if there is a gap between the splittingterm,theNVcenterspinHamiltonianalsodepends eigenvalueandtherestofthespectrumoftheHamiltonian.) on electric field, temperature, and strain. Energy shifts arising from variations in these parameters impart phase shifts to A prototypical adiabatic pulse is the adiabatic fast pas- ∗ sage (AFP), introduced by Bloch60:achirppulsewherethe the electron spin, and thus the longer the T2 and T2 times of microwave frequency is swept across the spin resonance the system is, the weaker will be the perturbation that can be frequency. In the rotating frame, all spins whose resonant measured.TheNVcentersarebroughtincloseproximityto frequency lies within the range swept by the chirp pulse expe- thesamplethatistobeimaged,eitherbyimplantingthemnear 67 rience an effective magneticld fie that smoothly rotates from thesurfaceofadiamondsubstrate(Figure9b, )orwithin 61 up to down, creating an effective π-pulse (see Shaped Pulses in diamond nanocrystals on the tip of an AFM (Figure 9a, ). EPR). Recent achievements and proposals include nanoscale imaging 66 AFPs are robust to variations in the microwave irradia- of living cells (Figure 9c, ), measurement of action potentials 68 tionamplitudeaswellasdetuning;however,caremustbe in neurons, and NMR-like spectroscopy distinguishing mul- 69 takenwhenapplyingthemtocoherentstates–forexample, tiple nuclear species. Sensing techniques can be categorized the AFP includes a inhomogeneous phase shift due to fre- intotwoapproaches:DCandACsensings. quency broadening. This phase shift is canceled when pairs ofAFPsareapplied,makingthemsuitableformultipulseDD DC Sensing 38,61 sequences. DC sensing aims at measuring parameters that are static, at least WhileanAFPcanbeusedtoachieveaπ-pulse, other rota- on the timescale of the measurement. Measurements are real- tions can be performed with more complex adiabatic pulses ized either by detecting changes in the electron spin resonance such as the BIR-1 or BIR-4 pulse, which are composite AFP frequencyintheCWEPRspectrumorinthetimedomainvia 62 pulses. These have the disadvantage (in samples with short the free induction decay (π∕2 − 휏). In both cases, the sensi- T2) of being even longer than AFPs, which are already an tivity is determined by the inhomogeneous spin linewidth or order of magnitude or more longer than a standard rectangular ∗ T2 –thisincludestheeffectivebroadeningcausedbyrandomly pulse, and are only robust to pulse amplitude errors and not fluctuatingfields(asidefromthattobedetected).Depending frequency. BIR pulses have been used, for example, for EPR on the source of inhomogeneous broadening and the parame- using coplanar waveguides in Ref. 52 to perform 2-pulse echo ter to be measured, this limit can be extended to T2 when the experiments in the presence of over one order of magnitude refocusing pulse in a Hahn echo sequence does not reverse the variationinpulseamplitude. interactionbetweenthespinandthemeasuredquantity.For example, using NV centers in diamond, DC sensing of the tem- Sensing perature is possible with measurement times limited only by T2 (Figure 9c and Ref. 66), because while phase shifts caused by Research toward developing spin qubits has helped motivate magnetic field inhomogeneities are reversed following the refo- the identification and control of spins with long coherence cusing π-pulse, those due to temperature-dependent shift in the lifetimes, as well as their measurement at the single-spin zero field splitting are not. level (e.g., by optical63 or electrical25 means). These achievements have opened up new possibilities for single spins to act as sensitive local probes for parameters such as mag- AC Sensing netic field,64 electric field,65 or even temperature.66 The use In AC sensing, the sensor spin is tuned to sense the local of an electron spin as a probe of local environment is well environmental noise at some defined frequency. This can be known in EPR spectroscopy, for example, through techniques achieved, for example, by applying DD to the sensor spin and such as ENDOR (see Hyperfine Spectroscopy—ENDOR) exploitingthebandpassfilteringeffectofrepeatedπ-pulses and DEER (double electron-electron resonance, see Dipolar in DD, as was described in the section titled ‘Dynamical Spectroscopy—Double-Resonance Methods). However, the Decoupling (DD)’ and Figure 5(a, b). As the pulse periodicity

(×104 c.p.s) 25 Excitation laser | 〉 −10 1 Hydrocarbon 2γB 20 − Scanning diamond |−1〉 5 NV2 platform ν Heat 15 L Detection volume 1 m ω Sample H 3 0 MW ∼ (5 nm) μ / Sensor y 10 ∼5 nm 5 |0〉 MW coil Sensor NV 13C Microwave 10 NV1 50 z NV center 10.0 15 x y −5 0 5 10 15 x/μm Target spins 8.0 Noise ) ττ Hz √ 6.0 π/2 2π π/2

) (nT/ Oil 1 ω

( 2τ = 250 μs π π B 4.0 Sensor: x ππππy S 0.5 2 2 1H PMMA 2.0 Population 0 Target: −60.0 −30.0 0.0 30.0 60.0 24.2 24.22 24.24 24.26 24.28 ν − ν T ° (a) (b)( L)/kHz (c) / C

Figure 9. Quantum sensing using single-electron spins of nitrogen-vacancy (NV) centers in diamond. (a) Magnetic sensing with an NV center embed- ded in a scanning diamond nanopillar, allowing for 2D imaging with nanometer resolution (Reprinted by permission from Macmillan Publishers Ltd: Nature Physics, Ref. 61 © 2013). Here, a single-electron spin within the sample, also diamond, is targeted using DD (bottom) on both sensor and target spins (lock-in method). (b) (Top) Schematic of a shallow NV center asamagneticsensorformoleculesatthesurfaceofthediamond67 (From T Staudacher, F Shi, S Pezzagna, J Meijer, J Du, C A Meriles, F Reinhard, and J Wrachtrup. Nuclear Magnetic Resonance Spectroscopy on a (5-Nanometer)3 Sample Volume. Science, 339(6119):561–563, feb 2013. Reprinted with permission from AAAS. ). The detection sensitivity can reach volumes down to3 5nm , corresponding to about 104 protons. (Bottom) Frequency spectra obtained using DD on the NV electron spin and shown as sensitivity against detuning 휈 1 fromtheexpectedLarmorfrequency( L) of spins in the test molecules: protons in oil (yellow) and Hpolymethylmethacrylate(PMMA)(green).The spectrum from an external magnetic field noise source at a fixed frequency,mulating si a precessing spin, is also shown (red) for reference. (c) (Top) Confocal microscopy measurement of a single human embryonic fibroblast cell, outlined by the dotted line, with injected nanodiamonds (NV) as local thermometers (Reprinted by permission from Macmillan Publishers Ltd: Nature, Ref. 66 © 2013). (Bottom) Spin-state projection as a function of temperature, using the given modified Ramsey sequence for the spin-1 NV system, whereπ the2 -pulse here refocuses phase shifts from static magnetic fields but not from temperatures. The projection amplitude oscillates with temperature, hence this technique is used only for relative temperature sensing. In addition, because the delay 휏 determines the oscillation period, it can be varied to adjust the balance between greatest dynamic range (small 휏)andmaximumsensitivity(large휏) is swept, so is the filter frequency (centered around 1∕2휏 in where C is the measurement efficiency, which is the fraction Figure 5b); and when the filter coincides with a dominant com- of spins that can be measured in a single experiment with a ponent in the noise spectrum, a reduction is observed in the signal-to-noise ratio of 1. In EPR, C encompasses factors such echo intensity. Mapping out the noise spectrum experienced as the efficiency of the microwave detection, the resonator by a spin can be useful to investigate sources of decoherence. quality and filling factors, and inhomogeneous broadening. For sensing, the technique has been described as the quantum The relation above assumes the sensed field is fully synchro- analog of a lock-in amplifier.35 The detection frequency of nizedwiththeappliedDDsequence(i.e.,samefrequencyand the sensor spin could be set, for example, to match the nat- phase, see bottom of Figure 9b). ural Larmor precession of a nuclear spin.36 In Figure 9(b), 1H spins in hydrocarbon molecules were detected using this method with NV electron spins. Alternatively, DD can be Multiple Qubits applied simultaneously to some spin that is to be probed, as Up to this point, we have been concerned primarily with single well as to the sensor spin, in order to single out that particular qubits, the measurement and control of their states, and their interaction61 (pulse sequence shown in Figure 9a, bottom), as application as quantum sensors. Applications in quantum infor- an extension of multipulse DEER methods. mation processing rely on the ability to engineer interactions Whilemeasurementsinthefrequencydomainarethemost betweenmultiplespinqubitsinordertoperformusefulcom- common, time-resolved sensing is also possible. In this case, putations. DD sequences are used to obtain the different components in a time series expansion (e.g. the polynomial or Walsh series) Multiple Qubit Gates from which the signal can be reconstructed.68 This has been proposed to monitor the magnetic field produced by action In classical logic, the so-called universal gates such as NAND potentials in neurons. and NOR operations can be used to construct any combi- natorial logic function, and thus to perform any classical For√ magnetic field sensing (or magnetometry), the sensitivity algorithm. Analogously, although there exist a vast set of pos- (in T∕ Hz)ofanensembleofN independent electron spin-1/2 sible two-qubit operations that can be performed, we can focus is approximately given by the relation64: onasubsetthatpossessesthis‘universal’propertyandthus π provides the necessary toolbox to perform universal quantum 휂 ≈ √ (14) |훾 | 2 e C NT2 computation. One such universal set of gates is combination of

the single-qubit Hadamard gate (see section titled ‘Single-qubit State Tomography of Multiple Qubits Gates’) and the two-qubit controlled-NOT, or C-NOT gate, Extracting the density matrix for a set of coupled qubits expressed as requires going beyond the basic approach outlined in the ⎛1000⎞ sectiontitled‘QuantumStateTomography’andcarriessome ⎜0100⎟ U = ⎜ ⎟ (15) fundamental challenges. First, the number of independent C-NOT 0001 ⎜ ⎟ elements increases exponentially with the number of qubits, ⎝0010⎠ making full density matrix tomography practical for only small in the basis {|00⟩, |01⟩, |10⟩, |11⟩}. In other words, the C-NOT numbers of qubits. Second, these elements must be mapped to corresponds to an inversion (or π rotation) of the second observable quantities in the system (in the same way that the x,y ̂ ̂ (‘target’) qubit if the first (‘control’) qubit is in state |1⟩, and no Sz component was mapped onto Sx in the single-qubit case). ̂ operation (the 𝟏 process)ifthecontrolqubitisinstate|0⟩.A Given imperfect pulses, this mapping process can become C-NOTgatecanbeperformedinEPR(modulosomephase increasingly sensitive to pulse errors as the number of qubits increases. factor) as a selective πx-pulse, which can be readily implemented between pairs of spins if their coupling is large enough Whilethesechallengescannotbereadilyovercome,itis to be resolved. Another universal gate (when combined with at least possible to reduce the sensitivity to pulse error, to the H-gate) is the controlled-phase (C-PHASE) gate: someextent,by‘labeling’elementsofthedensitymatrixusing phase shifts, before mapping them onto observables.70 In this ⎛100 0⎞ 휙 method, illustrated in Figure 10, a phase shift i is applied to ⎜010 0⎟ | ⟩ | ⟩ | ⟩ U = ⎜ ⎟ (16) state i , such that coherences between such states i and j CPHASE 001 0 휙 휙 휙 ⎜ ⎟ acquire phases i ± j.Thephases i are each incremented by ⎝000−1⎠ 훿휙 some fixed amount i from measurement to measurement, leading to phase oscillations, and the phase increments 훿휙 which has the effect of applying a π phase shift to |11⟩ with i are selected so that each type of coherence acquires phase at a respect to the other three basis states. A C-PHASE can be con- distinct effective frequency. Various approaches can be used to verted to a C-NOT gate by applying a Hadamard operation to apply the phase shift, for example, using geometric phases and the ‘target’ qubit before and after the C-PHASE, and both the a (π) (−π) sequence.71 C-NOT and C-PHASE gates can be used to generate entan- 0 휙 gled states.70,71 AfastC-PHASEgatebetweentwonuclearspin qubits coupled to an electron spin can be performed using the Quantum Memory principle of the AA phase gate described in the section titled The ability to transfer a qubit state between different degrees of ‘Single-qubit Gates’ and has been demonstrated using optically freedom (such as between electron spin and nuclear spin transi- excited triplet states72 and NV centers.73 tions) has numerous applications, from enhanced fidelity spin

π −π |3〉 |3〉 |3〉 0 ϕ

| 〉 1 |2〉

|4〉 |1〉 |1〉 |1〉 |3〉

(a) ϕ = 0 ϕ = π/4 ϕ = π/2

π −π |1〉〈3| π −π 0 i⋅δϕ 0 i⋅δσ In-phase | | Quadrature 4〉〈3 | 〉〈 | 4 3 |2〉〈3| | 〉〈 | In-phase 1 3 Quadrature

|2〉〈3| FT/a.u. Amplitude/a.u. In-phase Quadrature 20 40 60 80 100 120 −0.25 −0.15 −0.05 0.05 0.15 0.25 (b) Increment number (i) (c) Frequency (phase acquired / i)

Figure 10. State tomography of a coupled electron and nuclear spin.(a)Ageometricphase휙 can be imparted onto two states driven by a pair of pulses: | ⟩ ↔ | ⟩ | ⟩ ↔ | ⟩ (π)0(−π)휙. (b) Such phase gates are applied to both the 1 3 and 3 4 transitions, where the phase applied is incremented from shot to shot by 훿휙 and 훿휎, respectively (thus, the phases applied on the ith shot are i훿휙 and i훿휎). The effect of these phase gates is shown for three different initial states: |1⟩⟨3| (an electron spin coherence); |3⟩⟨4| (a nuclear spin coherence); and |2⟩⟨3| (a zero quantum coherence)–in each case, the state is first prepared, then the phase gate is applied, and then the state is transformed to be observable in the electron spin echo. (c) The Fourier transform of the resulting phase oscillations yields distinct frequencies for the different quantum coherences, equal to 훿휙 for the |1⟩⟨3| coherence, 훿휎 for |3⟩⟨4|,and−훿휙 − 훿휎 for |2⟩⟨3|. (Reprinted by permission from Macmillan Publishers Ltd: Nature, Ref. 71 © 2011.)

(a) ‘Store’ ‘Retrieve’ |ψe〉 ≡ C-NOT

|ψn〉 = |0〉 (Electron echo) (Nuclear echo) Electron echo τ τ τ τ τ τ e1 e1 n n e2 e2 π/2 π π π π π π π Sx Sy

Generate Measure E-coherence Transfer to N Refocus Transfer to E of phase φ N-coherence (b) +X, φ = 0°

3 +Y, φ = 90°

2 −X, φ = 180° T n = 1.75 s 1 2 Electron spin echo Electron

Echo intensity/a.u. −Y, φ = 270° 0 0 0.5 1 τ Time/μs 020 Storage time, 2 n/s

Figure 11. SWAP operations between electron and nuclear spin degrees of freedom. (a) (top) Two C-NOTs approximate a SWAP and can be used as the |휓 ⟩ basis for a quantum memory where a coherent state of the electron spin e is transferred to a nuclear spin degree of freedom and subsequently retrieved. In the implementation of this scheme (bottom), C-NOT gates are achieved using selective π rotations, while additional refocusing pulses are needed to account for inhomogeneous broadening. The final electron spin echo is shown to vary its phase in accordance with the original π∕2-pulse applied at the beginning of the sequence (50 ms earlier) (Reprinted by permission from Macmillan Publishers Ltd: Nature, Ref. 76 © 2008). (b) Measuring the recovered 휏 echo intensity as a function of the total storage time 2 n yields a nuclear spin T2 of almost 2 s, compared to the electron spin T2 of5msunderthesame conditions. In later experiments, nuclear spin coherence times as long as 3 min were measured for neutral P donors in silicon at 1.7 K.24 measurement,74 initialization and sensing, quantum repeaters mid-1990s to identify and correct errors in qubits without , for quantum communication,75 and to preserve qubit states for learning about their state.78 79 These strategies have the com- longer periods of time.76 Thisstatetransfercanbeachievedby mon feature of redundancy, representing one logical qubit the SWAP logical operation, which exchanges the states of two using several physical qubits, and have different fault-tolerant qubits: thresholds, corresponding to the maximum intrinsic error ⎛1000⎞ rates per gate that can be effectively corrected (see section ⎜0010⎟ titled ‘High-fidelity Operations’). USWAP = ⎜ ⎟ (17) ⎜0100⎟ One approach to quantum error correction is based on ⎝0001⎠ ‘majority voting’ and requires at least three physical qubits per logical qubit. This scheme has been implemented using NV and can be performed in general via three C-NOT operations centers in diamond, employing the NV center electron spin (although typically in practice only two are used), as illustrated and two nearby nuclear spins to represent the logical qubit.73,80 in Figure 11. Aqubitstateisfirstmappedfromtheelectronspintoallthree In NV centers in diamond, nuclear spin quantum memory spins via two C-NOT gates, at which point it becomes robust to operations have also been achieved using fast passages across spin-flip errors (Figure 12). After some period, further C-NOT Landau-Zener transitions,77 or exploiting anisotropic hyperfine gates are applied to distill any single spin-flip error onto the interactions in the spin system.8 nuclear spins, leaving the electron spin qubit in its original state. The nuclear spins must then be refreshed into an initial state before being reused. Quantum Error Correction Notably, among the assumptions in such error-correction Identifying and correcting errors is essential for any practi- schemes is that of uncorrelated errors, and in the case of several cal realization of an information processor–this is especially nuclear spins with significant coupling to an electron spin, this important in quantum information processing where the assumption may not be justified. Nevertheless, such demon- expected error rates due to decoherence and gate fidelity are strations highlight the potential benefit of small ‘registers’ of expected to be high, and also particularly demanding, as we nuclear spins coupled to an electron spin. are prohibited from directly measuring the state of any logical More recent approaches to quantum error correction based qubit during the calculation (to avoid collapse of the qubit on topological schemes such as the surface code41 have high state). Fortunately, strategies have been put forward since the threshold error rates, although at the expense of even larger

‘Encode’ ‘Decode’ as 1.3 km.75 Suchschemesrequirespinssystemswithstrong coupling to light, for example, quantum dots or certain defect |0〉 centers in solids. Another possibility, albeit over shorter length scales, is to use single microwave photons strongly coupled to spins in microwave cavities. This represents a new regime for |ψ〉 E EPR, but one in which there is increasingly progress and many opportunities.84–86

|0〉 (a) Conclusions

0.50 PulsetechniquesfromNMRandEPRhaveplayedamajorrole in the development of spin qubits, from the characterization of candidate systems and measurement of coherence lifetimes, 0.45 to the high-fidelity control of spins to achieve logical gates. We are now at the stage that innovations arising from research 0.40 into quantum information processing may feed back into EPR p

F methodology, whether in the form of instrumentation such 86 0.35 as quantum-limited microwave amplifiers, magnetic field sensing techniques to complement structural characterization by DEER, and the use of DD methods as a spectroscopic 0.30 technique, and, ultimately, it may provide advanced simulators

Process fidelity to model the dynamics of coupled spin systems at scales not Error on: 0.25 possible today. Electron Nucleus 1 0.20 Nucleus 2 Acknowledgments Electron (+Nucleus 1) 0.15 We thank Richard Brown and Stephanie Simmons for the data 0.00.5 1.0 1.5 shown in Figure 1(c), as well as Sofia Qvarfort, Padraic Calpin, (b) Error angle θ/π DavidWise,andPhilippRossforthedatashowninFigure7.

Figure 12. Quantum error correction using an electron spin coupled to two nuclear spins.80 (a) Theoretical error-correction protocol for bit-flip Related Articles error (box labeled ‘E’). The initial ‘encoding’ step consists of entangling Spin Dynamics; Relaxation mechanisms; Pulse EPR thequbit(instate|휓⟩)withtwonuclearspinancillae,whilethedecod- ing stage leaves the qubit in its original state, |휓⟩, regardless of whether a single bit-flip error occurred. (b) Process fidelity of the protocol consider- ing a single or two flip errors among the three spins. The gray line gives the Biographical Sketches average fidelity without correction Gary Wolfowicz. b 1987; B.E. 2007, M.E. 2011, Applied Physics and Electrical Engineering, Ecole Normale Superieure resource overheads (e.g., in terms of the number of physical de Cachan, France. D.Phil (PhD) 2015, Materials, University qubits used to represented one logical qubit). of Oxford, UK. Postdoctoral scholar, 2015–present, University of Chicago, USA. Approximately 10 publications. Current interests: control of electron and nuclear spin qubits. Spin-qubit Networks John J.L. Morton. b 1980; B.E. M.E. 2002, Electrical Engi- Controlling scalable arrays of interacting spin qubits remains neering, University of Cambridge, UK. D.Phil (PhD) 2005, one of the greatest challenges in realizing spin-based quantum Materials, University of Oxford, UK. Junior Research Fellow information processors. Approaches such as those based on 2005–2009 and Science Research Fellow 2010–2012, St. John’s electrically tuneable exchange couplings in nanoelectronic College, University of Oxford, UK. Royal Society University devices12,81 and dipolar couplings between spins82,83 are being Research Fellow 2008–2016. Reader at UCL, UK 2012–2014 explored; however, the short-range nature of the spin–spin and Professor of Nanoelectronics & Nanophotonics at UCL interactions, combined with the need to address individual 2014–present. Approximately 90 publications. Current inter- spins for measurement and control, provide considerable tech- ests:electronandnuclearspinqubitsinsemiconductorsand nical challenges. For these reasons, quantum ‘interconnects’ molecular materials. to couple distant spins are an attractive ingredient to create scalable networks of spin qubits. References Candidates for such interconnects include optical photons, whichcanbeentangledwiththestateofanelectronspin 1. R. P. Feynman, Int. J. Theor. Phys., 1982, 21, 467. andtheninterferedtoentanglespinsseparatedbyasmuch 2. D. Deutsch, Proc. R. Soc.A: Math. Phys. Eng. Sci., 1985, 400, 97.

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