Universal and Operational Benchmarking of Quantum Memories ✉ ✉ ✉ Xiao Yuan 1,2,3 , Yunchao Liu 4,5 , Qi Zhao 5, Bartosz Regula 6, Jayne Thompson 7 and Mile Gu6,7,8

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ARTICLE OPEN Universal and operational benchmarking of quantum memories ✉ ✉ ✉ Xiao Yuan 1,2,3 , Yunchao Liu 4,5 , Qi Zhao 5, Bartosz Regula 6, Jayne Thompson 7 and Mile Gu6,7,8

Quantum memory—the capacity to faithfully preserve quantum coherence and correlations—is essential for quantum-enhanced technology. There is thus a pressing need for operationally meaningful means to benchmark candidate memories across diverse physical platforms. Here we introduce a universal benchmark distinguished by its relevance across multiple key operational settings, exactly quantifying (1) the memory’s robustness to noise, (2) the number of noiseless qubits needed for its synthesis, (3) its potential to speed up statistical sampling tasks, and (4) performance advantage in non-local games beyond classical limits. The measure is analytically computable for low-dimensional systems and can be efficiently bounded in the experiment without tomography. We thus illustrate quantum memory as a meaningful resource, with our benchmark reflecting both its cost of creation and what it can accomplish. We demonstrate the benchmark on the five-qubit IBM Q hardware, and apply it to witness the efficacy of error-suppression techniques and quantify non-Markovian noise. We thus present an experimentally accessible, practically meaningful, and universally relevant quantifier of a memory’s capability to preserve quantum advantage. npj Quantum Information (2021) 7:108 ; https://doi.org/10.1038/s41534-021-00444-9 1234567890():,;

INTRODUCTION memory platforms and error sources, as well as providing a Memories are essential for information processing, from commu- witness for non-Markovianity. We experimentally test our bench- nication to sensing and computation. In the context of quantum mark on the five-qubit IBM Q hardware for different types of error, technologies, such memories must also faithfully preserve the demonstrating its versatility. In addition, the generality of our uniquely quantum properties that enable quantum advantages, methods within the broad physical framework of quantum 12–14 including quantum correlations and coherent superpositions1. resource theories ensures that many of our operational interpretations of the RQM can also extend to the study of more This has motivated extensive work in experimental realisations 15–20 across numerous physical platforms2,3, and presents a pressing general quantum processes , including general resource need to find operationally meaningful means to compare theories of quantum channels, gate-based quantum circuits, and quantum memories across diverse physical and functional dynamics of many-body physics. Our work thus presents an settings. In contrast, present approaches towards detecting and operationally meaningful, accessible and practical performance- benchmarking the quantum properties of memories are often ad based measure for benchmarking quantum processors that is ’ hoc, involving experimentally taxing process tomography, or only immediately relevant in today s laboratories. furnishing binary measures of performance based on tests of entanglement and coherence preservation4–11. Our work addresses these issues by envisioning memory as a RESULTS physical resource. We provide a means to quantify this resource by Framework of quantum memories asking: how much noise can a quantum memory sustain before it Any quantum memory can be viewed as a channel in time— is unable to preserve uniquely quantum aspects of information? mapping an input state we wish to encode into a state we will Defining this as the robustness of a quantum memory (RQM), we eventually retrieve in the future. An ideal memory preserves all demonstrate that the quantifier has diverse operational relevance information, such that proper post-processing operations on the in benchmarking the quantum advantages enabled by a memory output state can always undo the effects of the channel. Such —from speed-up in statistical sampling to non-local quantum channels preserve all state overlaps, in the sense that any pair of games (see Fig. 1). We prove that RQM behaves like a physical distinguishable input states remain distinguishable at output. In resource measure, representing the number of copies of a pure contrast, this is not possible with classical memories that store idealised qubit memory that are required to synthesise the target only classical data. To distinguish orthogonal states in some basis memory. We show the measure to be exactly computable for jik , we are forced to measure in this basis and record only the many relevant cases, and introduce efficient general bounds classical measurement outcome k. Such a measure-and-prepare through experimental and numerical methods. The quantifier is, in process will never distinguishji 0 þ ji1 fromji 0 À ji1 . In fact, this particular, experimentally accessible without full tomography, procedure exactly encompasses the class of all entanglement- enabling immediate applications in benchmarking different breaking (EB) channels21,22: if we store one part of an entangled

1Center on Frontiers of Computing Studies, Department of Computer Science, Peking University, Beijing 100871, China. 2Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA. 3Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK. 4Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720, USA. 5Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China. 6Nanyang Quantum Hub, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371 Singapore, Singapore. 7Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore, Singapore. 8Complexity Institute, Nanyang Technological University, 637335 Singapore, ✉ Singapore. email: [email protected]; [email protected]; [email protected]

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Fig. 1 A schematic of the robustness of quantum memory (RQM) and its operational meanings. This work focuses on the resource theory of quantum memories. We deﬁne a entanglement-breaking memories as free resources and propose the RQM as a resource measure of b a quantum channel. We consider three operational interpretations of the measure in c one-shot memory synthesis with resource non- 1234567890():,; generating (RNG) transformations; d classical simulation of the measurement statistics of quantum memory; and e a family of two-player non- local quantum games generalising state discrimination.

bipartite state within classical memory, the output is always (entanglement cost)23,24. For quantum memories, we consider an separable. As such, classical memories are mathematically ideal qubit memory I 2 as the identity channel that perfectly synonymous with EB channels. preserves any qubit state. The task of single-shot memory To systematically characterise how well a general memory synthesis is then to convert n copies of ideal qubit memories n preserves quantum information, we consider how robust it is I 2 to the target memory N via a free transformation. We show against noise. We define the robustness of quantum memories that the robustness measure lower bounds the number n of the (RQM) as the minimal amount of a classical memory that needs to requisite ideal memories, i.e., n dlog2ðdRðN Þe þ 1Þe. There- be mixed with the target memory N such that the resultant fore, a larger robustness indicates that the memory requires more probabilistic mixture is also classical: ideal resources to synthesise. Furthermore, we show that there NþsM always exists an optimal RNG transformation that saturates this RðN Þ ¼ min s 0j 2 EB ; (1) lower bound, and thus the robustness tightly captures the optimal M2EB 1 þ s resource cost for this task. We summarise our first result as follows. where the minimisation is over the set of all EB channels EB. We Theorem 1 The minimal number of ideal qubit memories fi explicitly prove that the robustness measure is a bona de required to synthesise a memory N is n ¼dlog2ðdRðN Þe þ 1Þe. resource measure of quantum memories, satisfying all necessary In “Methods” section, we further consider imperfect memory operational properties. Crucially, we show that the robustness synthesis by allowing an error ε and show that the optimal satisfies monotonicity—a memory’s RQM can never increase resource cost is characterised by a smoothed robustness measure under any resource non-generating (RNG) transformation, that is, with smoothing parameter ε. Theorem 1 thus corresponds to the any physical transformation of quantum channels that maps EB special case of ε = 0. Complementing our result on memory channels only to EB channels. We thus refer to such transforma- synthesis, recent works25,26 studied the task of single-shot tions as free within the resource theory of quantum memories. memory distillation with another resource measure based on Commonly encountered free transformations include pre- or post- the hypothesis testing entropy. processing with an arbitrary channel or, more generally, the class 10 In the second task, we consider the classical simulation of of so-called classically correlated transformations . quantum memories. The motivation here is analogous to computational speed-up—the observational statistics of any Operational interpretations quantum algorithm can be simulated on a classical computer, We illustrate the operational relevance of RQM in three distinct albeit at an exponential overhead. Similarly, one strategy for settings. The first is memory synthesis. From the perspective of simulating quantum memories is to perform full tomography of physical resources, one important task is to synthesise a target the input state and store the resulting classical density matrix. resource by expending a number of ideal resources, which can be Then, at the output of the memory, the input state ρ is thought of as the “currency” in this process. Intuitively, a more reconstructed and any observational statistics on ρ can be directly resourceful object would be harder to synthesise and hence obtained. This method clearly requires an exponential amount of require more ideal resources, allowing us to understand the input samples for an n-qubit memory—and thus results in an required number of ideal memories as the resource cost of a given exponential overhead in resources and speed. memory. In entanglement theory, an analogous concept involves Formally, the functional behaviour of any memory is fully determining the minimum number of Bell pairs that are required described by how its observational statistics vary as a function of to engineer a particular entangled state using free operations input, i.e., the set of expectation values TrðONðρÞÞ, for each

npj Quantum Information (2021) 108 Published in partnership with The University of New South Wales X. Yuan et al. 3 possible observable O and input state ρ. In order to estimate Tr½ONðρÞ to an additive error ε with failure probability δ, we need 2 −1 T0 ∝ 1/ε log(δ ) samples when having access to N due to 27 Hoeffding’s inequality . Alternatively,P we can linearly expand the ; R target memory as N¼ iciMi ci 2 and obtain the target statistics byP using Mi and measuring Tr½OMiðρÞ as ρ ρ Tr½ONð Þ ¼ iciTr½OMið Þ. When only havingP access to free fi resources in a speci c decomposition N¼ iciMi, we need T / 2=ε2 δÀ1 kck1 logð Þ samples. The simulation overhead is thus given 2 = by C ¼kck1 / T T0. We then prove that the optimal overhead that minimizes over all possible expansions is given exactly by the RQM of the quantum channel. Theorem 2 The minimal overhead—in terms of extra runs or input samples needed—to simulate the observation statistics of a 2 quantum memory N is given by Cmin ¼ð1 þ 2RðN ÞÞ . For EB channels M, the robustness RðMÞ vanishes and hence CminðMÞ ¼ 1, aligning with the intuition that classical memories require no extra simulation cost. For n ideal qubit memories, n n RðI 2 Þ¼2 À 1 and hence the classical simulation overhead scales exponentially with n. In the third setting, we consider the capability of quantum memories to provide advantages in a class of two-player non-local quantum games. Related games of this type have previously been employed in understanding features of Bell nonlocality28 and 10 detecting quantum memories . Consider then a set of states {σi}, from which one party (Alice) selects one state uniformly at random and encodes it in a memory N . Her counterpart Bob is given this memory and tasked with guessing which of the states {σi} was encoded by performing a measurement {Oj}. The probability that Bob guesses σj when the input state is σi is given by Tr½N ðσiÞOj. Thus, by associating with each such guess a coefficient αij 2 R,we can define the payoff of the game—this can be used to give different weights to corresponding states, or to penalise certain guesses. The performance of the two players in the game defined by G¼ffαijg; fσig; fOjgg is then evaluated using the average payoff function, X ; α σ : PðN GÞ ¼ ijTr½N ð iÞOj (2) i;j Such games can be considered as a generalisation of the task of quantum state discrimination, as can be seen by taking αij = δijpi for some probability distribution pi. We see that the players’ maximum achievable performance is limited by Bob’s capacity to Fig. 2 Numerical evaluation of the exact value of robustness of ’ quantum memories. a Robustness of memories with qubit inputs discern Alice s inputs, and thus each such game serves as a gauge ; for the memory quality of N . In order to establish a quantitative and computational basis fji0 jig1 for dephasing channels Δ (ρ) = pρ + (1 − p)ZρZ, stochastic damping channels benchmark for the resourcefulness of a given memory, we can p DpðρÞ¼pρ þð1 À pÞji0 hj0 , and erasure channels EpðρÞ¼pρ þ then compute the best advantage it can provide in the same ð1 À pÞji2 hj2 with ji2 orthogonal to fji0 ; jig1 . b Memory robustness game G over all classical memories. To make such a problem well- under dynamical decoupling (DD) and its quantification of non- defined, we will constrain ourselves to games for which the payoff Markovianity. We consider a qubit memory (M) coupled to a qubit ; “ ” ρ : : PðM GÞ is non-negative. In the Methods section, we then show bath (B) with an initial state Bð0Þ¼0 40jihj0 þ 0 61jihj1 and an that the maximal capabilities of a quantum memory in this setting interaction Hamiltonian H = 0.2(XM ⊗ XB + YM ⊗ YB) + ZM ⊗ ZB. Here are exactly measured by the robustness. X, Y, Z are the Pauli matrices. We consider the evolution with time t π Theorem 3 The advantage that a quantum memory N can from 0 to . To decouple the interaction, we apply X operations on provide over classical memories in all non-local quantum games is the memory at a constant rate. We show that the memory given by robustness can be enhanced via dynamical decoupling (DD). Furthermore, as the memory robustness can increase with time, PðN; GÞ we calculate the non-Markovianity using the robustness derived max ¼ RðN Þ þ 1: (3) measure as defined in Eq. (5). G max PðM; GÞ M2EB We will shortly see that such games, in addition to showcasing such game G provides a lower bound PðN; GÞ À 1onRðN Þ,akin another operational aspect of the robustness, allow us to to an entanglement witness quantitatively bounding measures of fi ef ciently bound RðN Þ in many relevant cases. entanglement29. This provides a physically accessible way of bounding the robustness measure by performing measurements Computability and measurability on a chosen ensemble of states, and in particular there always exists We can efficiently detect and bound the robustness of a memory a choice of a quantum game G such that PðN; GÞ À 1isexactly through the performance of the memory in game scenarios. equal to RðN Þ. This approach makes the measure accessible also in Specifically, consider games G such that all classical memories experimental settings, avoiding costly full process tomography. We achieve a payoff in the range [0, 1]. By Theorem 3, we know that any use this method to explicitly compute the robustness of some typical

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quantum memories in Fig. 2a, with detailed construction of the For any Markovian process N t, the robustness measure RðN tÞ is a quantum games deferred to the Supplementary Notes. decreasing function of time owing to monotonicity of R (see In addition to the above linear witness method, we also give “Methods” section). Thus IðTÞ¼0 for any Markovian process N t, non-linear witnesses of a memory N based on the moments of its and nonzero values of IðTÞ directly quantify the memory’s non- Choi state. Consider channels N with input dimension d and k = Markovianity in a similar way to ref. 31. Meanwhile, the goal of any 0, 1, …, ∞, in Supplementary Notes, we prove error-mitigation procedure is to preserve encoded qubits. Thus, hi1 k À 1 k k the characterisation of an increase in the RQM of relevant k Φ Φ RðN Þ d Tr ðÞN À 1, where N is the Choi state of encoded sub-spaces provides a universal measure of the efficacy N . Higher values of k provide tighter lower bounds, which can be for any such behaviour. measured in experiment by implementing a generalised swap test In Fig. 2b, we illustrate these ideas using a single-qubit memory on k copies of the channel. In the limit k → ∞, we obtain the subject to unwanted coupling from a qubit bath. The RQM strongest bound, which depends only on the maximal eigenvalue degrades over time (yellow-starred line)—but has a revival around of the Choi state. Remarkably, the bound is actually tight for all t = 1, indicating non-Markovianity. Indeed, plotting IðtÞ, we see qubit-to-qubit and qutrit-to-qubit channels. clear signatures of non-Markovian effects arise at this moment Theorem 4 The RQM of any quantum channel N with input (cyan-crossed line). Meanwhile, the green-dotted line quantifies dimension dA and output dimension dB can be lower bounded by how dynamical decoupling improves this memory through increased RQM. This improvement has a direct operational RðN Þ maxf0; d max eig ðΦ ÞÀ1g; (4) A N interpretation. For example, the approximately fourfold increase = and equality holds when dA ≤ 3 and dB = 2. in robustness around t 0.8 indicates that a quantum protocol We stress that this provides an exact and easily computable that runs on a dynamically decoupled quantum memory could be expression for the robustness for low-dimensional channels. This much harder to simulate than its counterpart. contrasts with related measures of entanglement of quantum 30 states, such as the robustness of entanglement , for which no Experiment general expression exists even in 2 × 2 dimensional systems. We experimentally verify our benchmarking method on the ‘ibmq- Given a full description of the memory, we can also provide ourense’ processor on the IBM Q cloud. We first consider a proof-of- efficiently computable numerical bounds on the robustness via a fi fi principle veri cationoftheschemebyestimatingtheRQMofthree semi-de nite programme, which we show to be tight in many types of single-qubit noise channels—the dephasing channel, relevant cases. We leave the detailed discussion to stochastic damping channel, and erasure channels. We synthesise Supplementary Notes. the noise channels by entangling the target state with ancillary qubits. For example, the dephasing channel Δp(ρA) = pρA + (1 − p) Applications ZρAZ can be realised by the circuit in the dashed box of Fig. 3a, The RQM, being information theoretical in nature, applies across where we input an ancillary stateji 0 E,rotateitwith all physical and operational settings. This enables its immediate RY exp iθY=2 , and apply a controlled-Z. Here θ θ ¼ pðÀffiffiffi Þ ¼ applicability to many present studies of quantum memory. For 2arccosð pÞ and Y is Pauli-Y matrix. We exploit the quantum game example, non-Markovianity and mitigation of errors resulting from approach to estimate the RQM of the three types of noise channels. non-Markovianity are widely studied problems in the context of We choose a normalised quantum game G with the maximal payoff quantum memories. RQM can be used both to identify the former, for EB channels of maxM2EB PðM; GÞ ¼ 1, so that the robustness of and measure the efficacy of the latter. memory N can be lower bounded by RðN Þ PðN ; GÞ À 1. For In particular, considering a memory Nt that stores states from each input-output setting (σi, Oj), we measure the probability pðjjiÞ¼ ≥ time 0 to t 0, we can quantify its non-Markovianity as Tr½OjNðσiÞ with 8192 experimental runs. The payoffP is obtained as a Z ; α T linear combination of the probabilities PðN GÞ ¼ i;j i;jpðjjiÞ with dRðN tÞ fi α IðTÞ¼ dt max 0; : (5) real coef cients i,j. As shown in Fig. 3b, the experimental data 0 dt (circles, upper and lower triangles) align well with the theoretical

Fig. 3 Experimental veriﬁcation of the benchmark with IBM Q hardware. a Circuit diagram for realising the dephasing channel. b The RQM of dephasing channels Δp(ρ) = pρ + (1 − p)ZρZ, stochastic damping channels DpðρÞ¼pρ þð1 À pÞji0 hj0 , and erasure channels EpðρÞ¼ pρ þð1 À pÞji2 hj2 with ji2 orthogonal to the basis fji0 ; jig1 . We synthesise the noise channels by interacting the target system with up to two ancillary qubits. We measure the payoff of quantum games PðN; GÞ which lower bounds the RQM as RðN Þ PðN ; GÞ À 1. c Benchmarking IBM Q hardware via the RQM of sequential controlled-X (CX) gates. We interchange the control and target qubit so that two 0 sequential CX gates will not cancel out. For example, denote CX1 to be the CX gate with control qubit 0 and target qubit 1; the three CX gates 0 1 0 1 0 1 0 1 0 are the swap gate CX1CX0CX1 SWAP and the six controlled-X gates become the identity gate CX0CX1CX0CX1CX0CX1 I4. The error bar is three times the standard deviation for both plots.

npj Quantum Information (2021) 108 Published in partnership with The University of New South Wales X. Yuan et al. 5 result (solid lines), with a deviation of less than 0.13. The deviation experimentally inaccessible, and obscuring a direct quantitative mostly results from the inherent noise in the hardware, especially the connection to tasks of practical relevance—the robustness notable two-qubit gate error and the read-out error. explicitly addresses all of these issues. Furthermore, the generality Next, we show that the RQM can be applied to benchmark of the resource-theoretic framework ensures that the tools quantum gates and quantum circuits. Conventional quantum developed here for quantum memories can be naturally extended process benchmarking approaches, such as randomised bench- to other settings, including purity, coherence, entanglement of marking32,33, generally focus on characterising the similarity channels18,36–46, and the magic of operations47,48. between the noisy circuit and the target circuit. In contrast, our There are a number of interesting future considerations. One is method is concerned with the capability of the noisy quantum to consider how memory robustness relates to another opera- processor in preserving quantum information, which can be thus tional task: the storage and retrieval of encoded quantum states, regarded as an alternative operational approach for benchmark- whose performance can thought to represent some sort of ing processes. In the experiment, we focus on the two-qubit memory capacity. This latter quantifier can be thought in the controlled-X (CX) gate, a standard gate used for entangling qubits. context of resource distillation. It asks how much many imperfect We sequentially apply n (up to six) CX gates with interchanged memories in question can be used to synthesize an ideal qubit control and target qubits for two adjacent gates. For example, memory. Thus, memory capacity is essentially the dual to memory one, three, and six CX gates correspond to the CX gate, the swap robustness, which was shown above to quantify the reverse gate, and the identity gate, respectively. process of resource dilution (quantifying the amount of ideal qubit Assuming that the dominant error is due to depolarising or memory needed to synthesis a target memory). Indeed, recent dephasing effects, we estimate the RQM of each circuit via the follow-up works suggest that memory distillation could also be correspondingly designed quantum game. As shown in Fig. 3c, we described within the same resource-theoretic framework, albeit can see that although the robustness with one CX is 2.667 ± 0.106, it with a resource monotone alternative to robustness25,26,49. Other only slowly decreases to 2.497 ± 0.115 for six CX gates. Our results interesting potentials include consideration of infinite-dimensional thus indicate that while the CX gate is imperfect (with an average quantum systems, where one may be interested in quantum 0.0340 decrease of robustness for each CX gate), the dominant noise memories that preserve non-classicality or non-Gaussianity— of the two-qubit circuit may instead stem from imperfect state potential sources of quantum advantage in continuous-variable preparation and measurement (roughly leading to a 0.3 decrease in quantum computation50–54. Finally, memories are essentially a robustness). We also note that the large robustness loss of a single question of reversibility, and thus naturally relate to heat CX gate might also be due to the existence of other errors, which dissipation in thermodynamics55,56. Indeed, recent results show would imply that the choice of the quantum game could be further connections between free energy and information encoding57, optimised. However, whenever the quantum game gives a large opening interesting possibilities toward understanding the lower bound for the robustness, this is sufficient to ensure that the thermodynamic consequences of such memory quantifiers. quantum process performs well in preserving quantum information. 0 0 To demonstrate this, we consider the circuit CX2 Á CX 1 for preparing the three-qubit GHZ state. We lower bound the robustness as METHODS 5.837 ± 0.548, verifying that the three-qubit noisy circuit can preserve Here we present properties of the robustness measure, formal statements more quantum information than all two-qubit circuits, whose of Theorems 1–4 and sketch their proofs. Full version of the proofs and robustness is upper bounded by 3. We leave the detailed details on the numerical simulations can be found in Supplementary Notes. experimental results and analysis to the Supplementary Notes. Properties of RQM Recall the definition of RQM DISCUSSION NþsM In this work, we introduced an operationally meaningful, practically RðN Þ ¼ min s 0 : 9M 2 EB; s:t: 2 EB ; (6) measurable and platform-independent benchmarking method for 1 þ s quantum memories. We defined the RQM and showed it to be an where our chosen set of free channels are the EB channels. Define free operational measure of the quality of a memory in three different transformation O as the set of physical transformations on quantum practical settings. The greater the robustness of a memory, the more channels (super-channels) that map EB channels to EB channels, i.e. ideal qubit memories are needed to synthesise the memory; the O¼fΛ : ΛðMÞ 2 EB; 8M 2 EBg. This class includes, for instance, the more classical resources are required to simulate its observational family of classically correlated transformations, which were considered in10 statistics; and the better the memory is at two-player non-local as a physically motivated class of free transformations under which quantum memories can be manipulated. In particular, transformations quantum games based on state discrimination. The measure can be Λ fi ðN Þ ¼ M1 N M2 with arbitrary pre- and post-processing channels evaluated exactly in low-dimensional systems, and ef ciently ; fi fi M1 M2 are free. We show that RQM satis es the following properties. approximated both numerically by semi-de nite programming and Non-negativity. RðN Þ 0 with equality if and only if N2EB. experimentally through measuring suitable observables. This thus Monotonicity. R does not increase under any free transformation, constitutes a promising means to quantify the quantum mechanical RðΛðN ÞÞ RðN Þ for arbitrary N and Λ 2O. aspects of information storage, and provides practical tools for ÀÁConvexity.P PR does not increase by mixing channels, benchmarking quantum memories across different experimental R ipiN i ipiRðN iÞ. platforms and operational settings. The theory is applicable across Additional properties, such as bounds under the tensor product of different physical platforms exhibiting any known type of error channels are presented in Supplementary Notes. Proof fi source, as we experimentally confirm on the five-qubit IBM Q Non-negativity follows directly from the de nition. For mono- hardware. With the development of near-term noisy intermediate- tonicity, suppose s ¼ RðN Þ with the minimisation achieved by M such that scale quantum technologies34,35, we anticipate that our quantifier 1 s canbecomeanindustrystandardforbenchmarkingquantum Nþ M¼M0 2 EB: (7) devices. s þ 1 s þ 1 From a theoretical perspective, our work also constitutes a Apply an arbitrary free transformation Λ on both sides and using linearity, significant development in the resource theory of quantum 1 Λ s Λ Λ 0 fi we obtain s þ 1 ðN Þ þ s þ 1 ðMÞ ¼ ðÞ2M EB. Therefore by de nition memories. The only previously known general measure of this RðΛðN ÞÞ s ¼ RðN Þ. For convexity, suppose si ¼ RðN iÞ with the resource involved a performance optimisation over a large class of minimisation achieved by Mi Pfor each iP and let 10 fi 0 = = possible quantum games , thus making it dif cult to evaluate, Mi ¼ðNi þ siMiÞ ð1 þ siÞ. Let s ipisi, N¼ ipiN i and

Published in partnership with The University of New South Wales npj Quantum Information (2021) 108 X. Yuan et al. 6 P 1 ϕþΦ Φ M¼s ipisiMi, then by convexity of the set of EB channels with q ¼ dcTr½C .WhenC is an EB channel, C is a separable state, and X we have 0 ≤ q ≤ 1. Thus ΛðΦCÞ is a separable Choi state that corresponds to an 1 s 1 0 ; Λ Nþ M¼ piðsi þ 1ÞM 2 EB (8) EB channel, which means that is a free transformation and concludes s þ 1 s 1 s 1 i þ þ i the proof. ÀÁ fi P P P therefore by de nition we have R ipiN i ¼ RðN Þ s ¼ ipisi ¼ ipiRðN iÞ. Simulating observational statistics Single-shot memory synthesis Observe that the general simulation strategy is to find a set of free Here we study a more general scenario, imperfect memory synthesis, memories fMigEBP such that the target memory can be linearly which allows a small error between the synthesised memory and the expanded as N¼ c M ; c 2 R. By using M and measuring target memory. The resource cost for this task is defined as the minimal i i i i i Tr½OMiðρÞ,P we can obtain the target statistics as dimension required for the ideal qudit memory I d, ρ ρ ÈÉTr½ONð Þ ¼ iciTr½OMið Þ. Thus, compared with having access to N ε R ðN Þ ¼ min d : 9Λ 2O; k ΛðI dÞÀNk ε ; (9) and directly measuring O, the classical simulation introducesÀÁP an extra syn Å 2 2 sampling overhead with a multiplicative factor k c k1 ¼ ijcij .In where kÁkÅ denotes the diamond norm, which describes the distance of particular, suppose we aim to estimate Tr½ONðρÞ to an additive error ε ε two channels. We also include a smooth parameter of the cost which with failure probability δ. Due to Hoeffding’s inequality27, when having tolerates an arbitrary amount of error in the synthesis protocol. The case 2 −1 access to N we need T0 ∝ 1/ε log(δ ) samples to achieve this estimate to with ε = 0 corresponds to the case with exact synthesis. When considering desired precision, and when onlyP having access to free resources in a the ideal qubit memory I 2 as the unit optimal resource, the minimal fi 2=ε2 δÀ1 speci c decomposition N¼ iciMi, we need T /k c k1 logð Þ n 2 number of ideal qubit memories I 2 required for memory synthesis is = ε samples. The simulation overhead is thus given by k c k1 / T T 0. given by n ¼dlog2ðRsynðN ÞÞe By minimising the simulation overhead over all possible expansions, we fi Correspondingly, we de ne a smoothed version of the robustness obtain the optimal simulation cost measure by minimising over a small neighbourhood of quantum channels, ; : ε 0 : CminðN Þ ¼ minP Cðfci MigÞ R ðN Þ ¼ min RðN Þ N¼ c M (16) 0 ε (10) i i i kN ÀNkÅ We prove that the smoothed robustness measure exactly quantifies the Our second result shows that this optimal cost is quantified by the resource cost for imperfect single-shot memory synthesis. robustness measure. Formal statement of Theorem 1. For any quantum channel N and any Formal statement of Theorem 2. For any quantum channel N , the 0 ≤ ε < 1, the resource cost for single-shot memory synthesis satisfies optimal cost for the observational simulation of N using EB channels is given by ε ε : RsynðN Þ ¼ 1 þdRðN Þe (11) 2: ε = CminðN Þ ¼ ð1 þ 2RðN ÞÞ (17) Note that by setting 0 we recover the result for perfect memory P synthesis stated in the main text. Proof ε ε For any linear expansion N¼ iciMi, denote the positive and Proof We start by proving R ðN Þ 1 þdRðN Þe. The first step is to fi þ À syn negative coef cients of ci by ci and ci , respectively. Then we have show that the robustness of the identity channel is RðI dÞ¼d À 1. The X X proof of this fact is omitted here. Next we show that the desired inequality þ À ; ; N¼ jci jMi À jci jMi Mi 2 EB (18) can be proven using the monotonicity property. For an arbitrary memory i:ci 0 i:ci < 0 synthesis protocol ΛðI Þ¼N0 where kN0 ÀNk ε, we have P P d Å with c cþ cÀ . As the channel is trace preserving, k k¼ i:ci 0j i jþ i:ci < P0j i j P ε 0 þ À 1 þRðN Þ ¼ 1 þ min RðN Þ taking trace on both sides we get : jc jÀ : jc j¼1. Denote 0 ε P i ci 0 i iPci < 0 i kN ÀNkÅ s cÀ , hence with ∥c∥ = 2s + 1, cÀ =s, and ¼ iP:ci < 0j i j 1 M¼ i:ci < 0j i jMi 1 0 0 cþ = 1 s , we have þ RðN Þ M ¼ i:ci 0j i jMi ð þ Þ Λ (12) ¼ 1 þRð ðI dÞÞ N¼ðs þ 1ÞM0 À sM; (19) 1 þ RðI dÞ where by convexity of EB, we have M; M0 2 EB. Therefore finding the d: ¼ optimal expansion is equivalent to finding the smallest s such that Eq. (19) Here the second line follows by definition and the fourth line follows from holds, which by definition equals to the robustness, i.e. smin ¼ RðN Þ. 2 monotonicity. As the above inequality holds for all memory synthesis Then we conclude that CminðN Þ ¼ ð1 þ 2RðN ÞÞ . protocols, it also holds for the optimal protocol. Also notice that ε ε dimensions are integers. Thus we derive that R ðN Þ 1 þdRðN Þe. ε ε syn To prove the other side RsynðN Þ 1 þdRðN Þe, suppose the channel Non-local games 0 0 achieves the mimum of Eq. (10)isN , and let dc ¼ 1 þ dRðN Þe.To Consider a quantum game G defined by the tuple G¼ðfαijg; fσig; fOjgÞ, prove the desired inequality, it suffices to show that 9Λ 2O such that σ 0 where i are input states, {Oj} is a positive observable valued measures at ΛðI d Þ¼N. Indeed such a Λ is a protocol that achieves the required c ε the output, and α 2 R are the coefficients which define the particular accuracy using resource 1 þdRðN Þe, thus the optimal protocol should i game. The maximal performance in the gamePG enabled by a channel N is only use less resource. fi ; α σ Λ quanti ed by the payoff function PðN GÞ ¼ ij ijTr½OjNð iÞ. Theorem 3 Next we explicitly construct such a free transformation , which transforms establishes the connection between the advantage of a quantum channel a quantum channel to another channel. As there is a one-to-one in the game scenario over all EB channels and the robustness measure. To correspondence between Choi states and quantum channels, we give this ensure that the optimisation problem is well-defined and bounded, we will construction based on transformation of the Choi state: optimise over games which give a non-negative payoff for classical Λ Φ ϕþΦ Φ ϕþ Φ Φ ; ð CÞ¼TrðÞC N 0 þ TrðÞðI À Þ C M (13) memories, which include standard state discrimination tasks. Formal statement of Theorem 3. Let 0 denote games such that all EB Φ ϕ+ G where denotes the Choi state of the subscript channel and is the channels achieve a non-negative payoff, that is, maximally entangled state. In the full proof we show that Λ is a valid physical transformation, i.e. a quantum super-channel. PðM; G0Þ0 8M 2 EB: (20) Λ 0 Λ As it is easy to verify that ðI dc Þ¼N, it only remains to show that is a free transformation, which maps EB channels to EB channels. To do this, Then the maximal advantage of a quantum channel N over all EB 0 0 first notice that as dc 1 þ RðN Þ, there exists M; M 2 EB such that channels, maximised over all such games, is given by the robustness: 0 1 0 dc À 1 PðN; G Þ 0: (14) : N þ M¼M max 0 ¼ 1 þ RðN Þ (21) dc dc G0 max PðM; G Þ M2EB Then we can rewrite Eq. (13)as Proof The proof is based on duality in conic optimisation (see ref. 17 and Λ Φ Φ Φ ; ð CÞ¼q M0 þð1 À qÞ M (15) references there in). First we write the robustness as an optimisation

npj Quantum Information (2021) 108 Published in partnership with The University of New South Wales X. Yuan et al. 7 problem Experiment details

The processor has five qubits with T1 and T2 ranging from 25 to 110 μs, RðN Þ þ 1 ¼ min Tr½x − − 1 single-qubit gate error 3.3–6.5 × 10 4, two-qubit gate error 1.0–1.5 × 10 2, : : Φ ; −2 s t x1 À x2 ¼ N (22) and read-out error 1.9–4.5 × 10 . Our experiments are run on the first fi x1; x2 2 coneðChoiðEBÞÞ; three qubits, which have the highest gate delities, and the circuits are implemented with Qiskit63. Φ where N is the Choi state of N , Choi(EB) denotes the Choi states of EB channels, i.e. bipartite separable Choi states, and cone(⋅) represents the unnormalised version. This can be written in the standard form of conic DATA AVAILABILITY programming, based on which we can write the dual form of this optimisation problem. The dual form can be simplified as The authors declare that all data supporting this study are contained within the article and its supplementary files. Φ OPT ¼ max Tr½ N W s:t: W ¼ Wy; (23) 0 CODE AVAILABILITY Tr½Φ 0 W2½0; 1; 8M 2 EB: M The code that supports the findings of this study are available from the We can verify that these primal and dual forms satisfy the condition for corresponding author upon reasonable request. strong duality, therefore OPT ¼ 1 þ RðN Þ, and it remains to show that OPT equals the maximal advantage in games. Received: 8 July 2020; Accepted: 9 June 2021; As the constraints in the dual form (Eq. (23)) are linear, without loss of generality, we can rescale the optimisation so that we only need to consider games G0 that satisfy PðM; G0Þ2½0; 1 (24) for any M2EB. We can then write REFERENCES P 0 PðN; G Þ¼αijTr½OjNðσiÞ; 1. Duan, L.-M., Lukin, M., Cirac, J. I. & Zoller, P. Long-distance quantum commu- ; Pi j nication with atomic ensembles and linear optics. Nature 414, 413 (2001). α Φ σT ; 2. Zhong, M. et al. 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