Universal and operational benchmarking of quantum memories

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.

Memories are essential for information processing, from communication to sensing and computation.In the context of quantum technologies, such memories must also faithfully preserve the uniquely quantum properties that enable quantum advantages, including quantum correlations and coherent superpositions [1].This has motivated extensive work in experimental realisations across numerous physical platforms [2,3], and presents a pressing need to find operationally meaningful means to compare quantum memories across diverse physical and functional settings.In contrast, present approaches towards detecting and benchmarking the quantum properties of memories are often ad-hoc, involving experimentally taxing process tomography, or only furnishing binary measures of performance based on tests of entanglement and coherence preservation [4][5][6][7][8][9][10][11].
Our work addresses these issues by envisioning memory as a physical resource.We provide a means to quantify this resource by asking: how much noise can a quantum memory sustain before it is unable to preserve uniquely quantum aspects of information?Defining this as the robustness of a quantum memory (RQM), we demonstrate that the quantifier has diverse operational relevance in benchmarking the quantum advantages enabled by a memory -from speed-up in statistical sampling to nonlocal quantum games (see Fig. 1).We prove that RQM behaves like a physical resource measure, representing the number of copies of a pure idealised qubit memory that are required to synthesise the target memory.We show the measure to be exactly computable for many relevant cases, and introduce efficient general bounds through experi-mental and numerical methods.The quantifier is, in particular, experimentally accessible without full tomography, enabling immediate applications in benchmarking different memory platforms and error sources, as well as providing a witness for non-Markovianity.We experimentally test our benchmark on the five-qubit IBM Q hardware for different types of error, demonstrating its versatility.In addition, the generality of our methods within the broad physical framework of quantum resource theories [12][13][14] ensures that many of our operational interpretations of the RQM can also extend to the study of more general quantum processes [15][16][17][18][19][20], including general resource theories of quantum channels, gate-based quantum circuits, and dynamics of many-body physics.Our work thus presents an operationally meaningful, accessible, and practical performance-based measure for benchmarking quantum processors that is immediately relevant in today's laboratories.
Framework.Any quantum memory can be viewed as a channel in time -mapping an input state we wish to encode into a state we will eventually retrieve in the future.An ideal memory preserves all information, such that proper post-processing operations on the output state can always undo the effects of the channel.Such channels preserve all state overlaps, in the sense that any pair of distinguishable input states remain distinguishable at output.In contrast, this is not possible with classical memories that store only classical data.To distinguish orthogonal states in some basis |k , we are forced to measure in this basis and record only the classical measurement outcome k.Such a measure-and-prepare process will never distinguish |0 + |1 from |0 − |1 .In fact, this procedure exactly encompasses the class of all entanglement-breaking (EB) channels [21,22]: if we store one part of an entangled bipartite state within classical memory, the output is always separable.As such, classical memories are mathematically synonymous with EB channels.
To systematically characterise how well a general memory preserves quantum information, we consider how robust it is against noise.We define robustness of quantum memories (RQM) as the minimal amount of a classical memory that needs to be mixed with the target memory N such that the resultant probabilistic mixture is also classical: where the minimisation is over the set of all entanglementbreaking channels EB.We explicitly prove that the robustness measure is a bona fide resource measure of quantum memories, satisfying all necessary operational properties.Crucially, we show that the robustness satisfies monotonicity -a memory's RQM can never increase under any resource non-generating (RNG) transformation, that is, any physical transformation of quantum channels that maps EB channels only to EB channels.We thus refer to such transformations as free within the resource theory of quantum memories.Commonly encountered free transformations include pre-or post-processing with an arbitrary channel or, more generally, the class of so-called classically correlated transformations [10].
Operational interpretations.We illustrate the operational relevance of RQM in three distinct settings.The first is memory synthesis.From the perspective of physical resources, one important task is to synthesise a target resource by expending a number of ideal resources, which can be thought of as the "currency" in this process.Intuitively, a more resourceful object would be harder to synthesise and hence require more ideal resources, allowing us to understand the required number of ideal memories as the resource cost of a given memory.
In entanglement theory, an analogous concept involves deter-mining the minimum number of Bell pairs that are required to engineer a particular entangled state using free operations (entanglement cost) [23,24].For quantum memories, we consider an ideal qubit memory I 2 as the identity channel that perfectly preserves any qubit state.The task of single-shot memory synthesis is then to convert n copies of ideal qubit memories I ⊗n 2 to the target memory N via a free transformation.We show that the robustness measure lower bounds the number n of the requisite ideal memories, i.e., n ≥ log 2 ( R(N ) + 1) .Therefore, a larger robustness indicates that the memory requires more ideal resources to synthesise.Furthermore, we show that there always exists an optimal RNG transformation that saturates this lower bound, and thus the robustness tightly captures the optimal resource cost for this task.We summarise our first result as follows.
Theorem 1.The minimal number of ideal qubit memories required to synthesise a memory N is n = log 2 ( R(N ) +1) .
In Methods, we further consider imperfect memory synthesis by allowing an error ε and show that the optimal resource cost is characterised by a smoothed robustness measure with smoothing parameter ε.Theorem 1 thus corresponds to the special case of ε = 0.
In the second task, we consider classical simulation of quantum memories.The motivation here is analogous to computational speed-up -the observational statistics of any quantum algorithm can be simulated on a classical computer, albeit at an exponential overhead.Similarly, one strategy for simulating quantum memories is to perform full tomography of the input state and store the resulting classical density matrix.Then, at the output of the memory, the input state ρ is reconstructed and any observational statistics on ρ can be directly obtained.This method clearly requires an exponential amount of input samples for an n-qubit memory -and thus results in an exponential overhead in resources and speed.
Formally, the functional behaviour of any memory is fully described by how its observational statistics vary as a function of input, i.e., the set of expectation values Tr(ON (ρ)), for each possible observable O and input state ρ.In order to simulate a quantum channel using only classical memories, we aim to estimate this quantity to some fixed additive error with at most some fixed admissible failure probability by taking samplesinputting ρ in a classical memory, measuring O, and repeating to get expectation value estimates.Intuitively, the more non-classical a memory is, the more classical samples will be required to simulate its statistics effectively.We thus define the simulation overhead C as the increase in the number of samples required when using only classical memories, versus having access to N itself.We then prove that the optimal overhead is given exactly by the RQM of the quantum memory.

Theorem 2. The minimal overhead -in terms of extra runs or input samples needed -to simulate the observation statistics of a quantum memory N is given by
For entanglement-breaking channels M, the robustness R(M) vanishes and hence C min (M) = 1, aligning with the intuition that classical memories require no extra simulation cost.For n ideal qubit memories, R(I ⊗n 2 ) = 2 n −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 nonlocal quantum games.Related games of this type have previously been employed in understanding features of Bell nonlocality [25] and detecting quantum memories [10].Consider then a set of states {σ i }, 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 {σ i } was encoded by performing a measurement {O j }.The probability that Bob guesses σ j when the input state is σ i is given by Tr[N (σ i )O j ].Thus, by associating with each such guess a coefficient α i j ∈ 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 G = {{α i j }, {σ i }, {O j }} is then evaluated using the average payoff function, Such games can be considered as a generalisation of the task of quantum state discrimination, as can be seen by taking α i j = δ i j p i for some probability distribution {p i }.We see that the players' maximum achievable performance is limited by Bob's capacity to 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 benchmark for the resourcefulness of a given memory, we can then compute the best advantage it can provide in the same game G over all classical memories.To make such a problem well-defined, we will constrain ourselves to games for which the payoff P(M, G) is non-negative.In the Methods we then show that the maximal capabilities of a quantum memory in this setting are exactly measured by the robustness.

Theorem 3. The advantage that a quantum memory N can provide over classical memories in all nonlocal quantum games is given by
We will shortly see that such games, in addition to showcasing another operational aspect of the robustness, allow us to efficiently bound R(N ) in many relevant cases.
Computability and measurability.We can efficiently detect and bound the robustness of a memory through the performance of the memory in game scenarios.Specifically, consider games G such that all classical memories achieve a pay-off in the range [0, 1].By Thm. 3 we know that any such game G provides a lower bound P(N, G) − 1 on R(N ), akin to an entanglement witness quantitatively bounding measures of entanglement [26].This provides a physically accessible way of bounding the robustness measure by performing measurements on a chosen ensemble of states, and in particular there always exists a choice of a quantum game G such that P(N, G) − 1 is exactly equal to R(N ).This approach makes the measure accessible also in experimental settings, avoiding costly full process tomography.We use this method to explicitly compute the robustness of some typical quantum memories in Fig. 4(a), with detailed construction of the quantum games deferred to the Supplemental Materials.
In addition to the above linear witness method, we also give non-linear witnesses of a memory N based on the moments of its Choi state.Consider channels N with input dimension d and k = 0, 1, . . ., ∞, in Supplementary Materials, we prove and equality holds when We stress that this provides an exact and easily computable expression for the robustness for low-dimensional channels.This contrasts with related measures of entanglement of quantum states such as the robustness of entanglement [27], for which no general expression exists even in 2 × 2-dimensional systems.
Here X, Y, Z are the Pauli matrices.We consider the evolution with time t from 0 to π.To decouple the interaction, we apply X operations on the memory at a constant rate.We show that the memory robustness can be enhanced via dynamical decoupling (DD).Furthermore, as the memory robustness can increase with time, we calculate the non-Markovianity using the robustness derived measure as defined in Eq. (5).
Given a full description of the memory, we can also provide efficiently computable numerical bounds on the robustness via a semi-definite program, which we show to be tight in many relevant cases.We leave the detailed discussion to Supplementary Materials.
Applications.The robustness of quantum memories, being information theoretical in nature, applies across all physical and operational settings.This enables its immediate applica-bility to many present studies of quantum memory.For example, non-Markovianity and mitigation of errors resulting from non-Markovianity are widely studied problems in the context of quantum memories.RQM can be used both to identify the former, and measure the efficacy of the latter.
In particular, considering a memory N t that stores states from time 0 to t ≥ 0, we can quantify its non-Markovianity as For any Markovian process N t , the robustness measure R(N t ) is a decreasing function of time owing to monotonicity of R (see Methods).Thus I(T) = 0 for any Markovian process N t , and nonzero values of I(T) directly quantify the memory's non-Markovianity in a similar way to Ref. [28].Meanwhile, the goal of any error-mitigation procedure is to preserve encoded qubits.Thus, the characterisation of an increase in the RQM of relevant encoded sub-spaces provides a universal measure of the efficacy for any such behaviour.In Fig. 4(b), we illustrate these ideas using a single-qubit memory subject to unwanted coupling from a qubit bath.The robustness of quantum memories degrades over time (yellowstarred line) -but has a revival around t = 1, indicating non-Markovianity.Indeed, plotting I(t), we see clear signatures of non-Markovian effects arise at this moment (cyan-crossed line).Meanwhile, the green-dotted line quantifies how dynamical decoupling improves this memory through increased RQM.This improvement has a direct operational interpretation.For example, the approximate 4-fold increase in robustness around t = 0.8 indicates that a quantum protocol that runs on a dynamically decoupled quantum memory could be much harder to simulate than its counterpart.
Experiment.We experimentally verify our benchmarking method on the 'ibmq-ourense' processor on the IBM Q cloud.We first consider a proof-of-principle verification of the scheme by estimating the RQM of three types of single-qubit noise channels -the dephasing channel, stochastic damping channel, and erasure channels.We synthesise the noise channels by entangling the target state with ancillary qubits.For example, the dephasing channel ∆ p (ρ A ) = pρ A + (1 − p)Z ρ A Z can be realised by the circuit in the dashed box of Fig. 3(a), where we input an ancillary state |0 E , rotate it with R Y θ = exp(−iθY /2), and apply a controlled-Z.Here θ = 2 arccos( √ p) and Y is Pauli-Y matrix.We exploit the quantum game approach to estimate the RQM of the three types of noise channels.We choose a normalised quantum game G with the maximal payoff for EB channels of max M ∈EB P(M, G) = 1, so that the robustness of memory N can be lower bounded by R(N ) ≥ P(N, G) − 1.For each input-output setting (σ i , O j ), we measure the probability p( j |i) = Tr[O j N (σ i )] with 8192 experimental runs.The payoff is obtained as a linear combination of the probabilities P(N, G) = i, j α i, j p( j |i) with real coefficients α i, j .As shown in Fig. 3(b), the experimental data (circles, upper and lower triangles) aligns well with the theoretical result (solid lines), with a deviation of less than 0.13.The deviation mostly results from the inherent noise in the hardware, especially the notable two-qubit gate error and the readout error.Benchmarking IBM Q hardware via the RQM of sequential controlled-X (CX) gates.We interchange the control and target qubit so that two sequential CX gates will not cancel out.For example, denote C X 0 1 to be the CX gate with control qubit 0 and target qubit 1; the three CX gates is the swap gate C X 0 1 C X 1 0 C X 0 1 ≡ SW AP and the six controlled-X gates becomes the identity gate C The error bar is three times the standard deviation for both plots.
Next, we show that the RQM can be applied to benchmark quantum gates and quantum circuits.Conventional quantum process benchmarking approaches, such as randomised benchmarking [29,30], generally focus on characterising the similarity between the noisy circuit and the target circuit.In contrast, our method is concerned with the capability of the noisy quantum processor in preserving quantum information, which can be thus regarded as an alternative operational approach for benchmarking processes.In the experiment, we focus on the two-qubit controlled-X (CX) gate, a standard gate used for entangling qubits.We sequentially apply n (up to six) CX gates with interchanged control and target qubits for two adjacent gates.For example, one, three, and six CX gates correspond to the CX gate, the swap gate, and the identity gate, respectively.
Assuming that the dominant error is due to depolarising or dephasing effects, we estimate the RQM of each circuit via the correspondingly designed quantum game.As shown in Fig. 3(c), we can see that although the robustness with one CX is 2.667 ± 0.106, it only slowly decreases to 2.497 ± 0.115 for six CX gates.Our results thus indicate that while the CX gate is imperfect (with an average 0.0340 decrease of robustness for each CX gate), the dominant noise of the two qubit circuit may instead stem from imperfect state preparation and measurement (roughly leading to a 0.3 decrease in robustness).We also note that the large robustness loss of a single CX gate might also be due to the existence of other errors, which would imply that the choice of the quantum game could be further optimised.However, whenever the quantum game gives a large lower bound for the robustness, this is sufficient to ensure that the quantum process performs well in preserving quantum information.To demonstrate this, we consider the circuit C X 0 2 • C X 0 1 for preparing the three-qubit GHZ state.We lower bound the robustness as 5.837 ± 0.548, verifying that the three-qubit noisy circuit can preserve more quantum infor-mation than all two-qubit circuits, whose robustness is upper bounded by 3. We leave the detailed experimental results and analysis to the Supplementary Materials.
Discussion.In this work, we introduced an operationally meaningful, practically measurable, and platform-independent benchmarking method for quantum memories.We defined the robustness of quantum memories and showed it to be an operational measure of the quality of a memory in three different practical settings.The greater the robustness of a memory, the more ideal qubit memories are needed to synthesise the memory; the more classical resources are required to simulate its observational statistics; and the better the memory is at two-player nonlocal quantum games based on state discrimination.The measure can be evaluated exactly in lowdimensional systems, and efficiently approximated both numerically by semi-definite programming and experimentally through measuring suitable observables.This thus constitutes a promising means to quantify the quantum mechanical aspects of information storage, and provides practical tools for benchmarking quantum memories across different experimental platforms and operational settings.The theory is applicable across different physical platforms exhibiting any known type of error source, as we experimentally confirm on the five-qubit IBM Q hardware.With the development of near-term noisy intermediate-scale quantum technologies [31,32], we anticipate that our quantifier can become an industry standard for benchmarking quantum devices.
From a theoretical perspective, our work also constitutes a significant development in the resource theory of quantum memories.The only previously known general measure of this resource involved a performance optimisation over a large class of possible quantum games [10], thus making it difficult to evaluate, experimentally inaccessible, and obscuring a direct quantitative connection to tasks of practical relevance -the robustness explicitly addresses all of these issues.Furthermore, the generality of the resource-theoretic framework ensures that the tools developed here for quantum memories can be naturally extended to other settings, including purity, coherence, entanglement of channels [18,[33][34][35][36][37][38][39], and the magic of operations [40,41], etc.Another direction is to consider infinite-dimensional quantum systems, such as the optical modes of light.Finally, memories are essentially a question of reversibility, and thus have a natural connection to heat dissipation in thermodynamics [42,43].Indeed, recent results show connections between free energy and information encoding [44], and thus present a natural direction towards understanding what thermodynamic consequences quantum memory quantifiers may have.

Methods
Here we present properties of the robustness measure, formal statements of Theorems 1-4 and sketch their proofs.Full version of the proofs and details on the numerical simulations can be found in Supplementary Materials.

Properties of RQM. Recall the definition of RQM
where our chosen set of free channels are the entanglement-breaking (EB) channels.Define free transformation O as the set of physical transformations on quantum channels (super-channels) that map EB channels to EB channels, i.e.O = {Λ : Λ(M) ∈ EB, ∀M ∈ EB}.This class includes, for instance, the family of classically correlated transformations, which were considered in [10] as a physicallymotivated class of free transformations under which quantum memories can be manipulated.In particular, transformations Λ(N ) = M 1 • N • M 2 with arbitrary pre-and post-processing channels M 1 , M 2 are free.We show that RQM satisfies the following properties.
Non-negativity.R(N ) ≥ 0 with equality if and only if N ∈ EB.Monotonicity.R does not increase under any free transformation, R(Λ(N )) ≤ R(N ) for arbitrary N and Λ ∈ O. Convexity.R does not increase by mixing channels, R ( Additional properties such as bounds under tensor product of channels are presented in Supplementary Materials. Proof of properties.Non-negativity follows directly from the definition.For monotonicity, suppose s = R(N ) with the minimisation achieved by M such that Apply an arbitrary free transformation Λ on both sides and using linearity, we obtain 1 s+1 Λ(N ) + s s+1 Λ(M) = Λ (M ) ∈ EB.Therefore by definition R(Λ(N )) ≤ s = R(N ).For convexity, suppose s i = R(N i ) with the minimisation achieved by M i for each i and let therefore by definition we have R ( Single-shot memory synthesis.Here we study a more general scenario, imperfect memory synthesis, which allows a small error between the synthesised memory and the target memory.The resource cost for this task is defined as the minimal dimension required for the ideal qudit memory where • denotes the diamond norm, which describes the distance of two channels.We also include a smooth parameter ε of the cost which tolerates an arbitrary amount of error in the synthesis protocol. The case with ε = 0 corresponds to the case with exact synthesis.When considering the ideal qubit memory I 2 as the unit optimal resource, the minimal number of ideal qubit memories I ⊗n 2 required for memory synthesis is given by n = log 2 (R ε syn (N )) Correspondingly we define a smoothed version of the robustness measure by minimising over a small neighbourhood of quantum channels, We prove that the smoothed robustness measure exactly quantifies the resource cost for imperfect single-shot memory synthesis.
Formal statement of Theorem 1.For any quantum channel N and any 0 ≤ ε < 1, the resource cost for single-shot memory synthesis satisfies Note that by setting ε = 0 we recover the result for perfect memory synthesis stated in the main text.
Proof.We start by proving The proof of this fact is omitted here.Next we show that the desired inequality can be proven using the monotonicity property.For an arbitrary memory synthesis protocol Λ(I d ) = N where N −N ≤ ε, we have Here the second line follows by definition and the fourth line follows from monotonicity.As the above inequality holds for all memory synthesis protocols, it also holds for the optimal protocol.Also notice that dimensions are integers.Thus we derive that To prove the other side R ε syn (N ) ≤ 1 + R ε (N ) , suppose the channel achieves the mimum of Eq. ( 10) is N , and let d c = 1 + R(N ) .To prove the desired inequality, it suffices to show that ∃Λ ∈ O such that Λ(I d c ) = N .Indeed such a Λ is a protocol that achieves the required accuracy using resource 1 + R ε (N ) , thus the optimal protocol should only use less resource.
Next we explicitly construct such a free transformation Λ, which transforms a quantum channel to another channel.As there is a oneto-one correspondence between Choi states and quantum channels, we give this construction based on transformation of the Choi state: where Φ denotes the Choi state of the subscript channel and φ + is the maximally entangled state.In the full proof we show that Λ is a valid physical transformation, i.e. a quantum super-channel.
As it is easy to verify that Λ(I d c ) = N , it only remains to show that Λ is a free transformation, which maps EB channels to EB channels.To do this, first notice that as Then we can rewrite Eq. (79) as with q = d c Tr φ + Φ C .When C is an EB channel, Φ C is a separable state, and we have 0 ≤ q ≤ 1.Thus Λ(Φ C ) is a separable Choi state that corresponds to an EB channel, which means that Λ is a free transformation and concludes the proof.

Simulating observational statistics.
Observe that the general simulation strategy is to find a set of free memories {M i } ⊆ EB such that the target memory can be linearly expanded as In particular, suppose we aim to estimate Tr[ON (ρ)] to an additive error ε with failure probability δ.Due to Hoeffding's inequality [45], when having access to N we need T 0 ∝ 1/ε 2 log(δ −1 ) samples to achieve this estimate to desired precision, and when only having access to free resources in a specific decomposition ) samples.The simulation overhead is thus given by c 2 1 ∝ T/T 0 .By minimising the simulation overhead over all possible expansions, we obtain the optimal simulation cost Our second result shows that this optimal cost is quantified by the robustness measure.
Formal statement of Theorem 2. For any quantum channel N , the optimal cost for the observational simulation of N using EB channels is given by Proof.For any linear expansion N = i c i M i , denote the positive and negative coefficients of c i by c + i and c − i , respectively.Then we have As the channel is trace preserving, taking trace on both sides we get i: where by convexity of EB we have M, M ∈ EB.Therefore finding the optimal expansion is equivalent to finding the smallest s such that Eq. ( 19) holds, which by definition equals to the robustness, i.e.
Nonlocal games.Consider a quantum game G defined by the tuple G = ({α i j }, {σ i }, {O j }), where σ i are input states, {O j } is a positive observable valued measures at the output, and α i ∈ R are the coefficients which define the particular game.The maximal performance in the game G enabled by a channel N is quantified by the payoff function P(N, G) = i j α i j Tr[O j N (σ i )].Theorem 3 establishes the connection between the advantage of a quantum channel in the game scenario over all EB channels and the robustness measure.To ensure that the optimisation problem is well-defined and bounded, we will optimise over games which give a non-negative payoff for classical memories, which include standard state discrimination tasks.
Formal statement of Theorem 3. Let G denote games such that all EB channels achieve a non-negative payoff, that is, Then the maximal advantage of a quantum channel N over all EB channels, maximised over all such games, is given by the robustness: Proof.The proof is based on duality in conic optimisation (see Ref. [17] and references therein).First we write the robustness as an optimisation problem where Φ 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 standard form of conic programming, based on which we can write the dual form of this optimisation problem.The dual form can be simplified as We can verify that these primal and dual forms satisfy the condition for strong duality, therefore OPT = 1 + R(N ), and it remains to show that OPT equals the maximal advantage in games.
As the constraints in the dual form (23) are linear, without loss of generality we can rescale the optimisation so that we only need to consider games G that satisfy for any M ∈ EB.We can then write where d is the input dimension of N and W = d i, j α i j σ T i ⊗ O j .Using this representation, the maximal advantage can be written as an optimisation problem equivalent to (23).In particular, since any Hermitian matrix can be expressed in the form of W for some real coefficients {α i j }, any witness W in (23) can be used to construct a corresponding game G , and conversely any game G satisfying the optimisation constraints gives rise to a valid witness W in (23).We thus have max concluding the proof.

Computability and bounds.
It is known that the description of the set of separable states is NP-hard in the dimension of the system [46], and indeed this property extends to the set of entanglement-breaking channels [47], making it intractable to describe in general.Nonetheless, we can solve the problem of quantifying the RQM in relevant cases, as well as establish universally applicable bounds.As described in the main text, suitably constructing nonlocal games G can provide such lower bounds, which can indeed be tight.More generally, one can employ the positive partial transpose (PPT) criterion [48] to provide an efficiently computable semidefinite programming relaxation of the problem, often providing non-trivial and useful bounds on the value of the RQM.We leave a detailed discussion of these methods to the Supplemental Material.In the case of low-dimensional channels, which is of particular relevance in many near-term technological applications, we can go further than numerical bounds and establish an analytical description of the RQM.
Formal statement of Theorem 4. For any channel N with input dimension d A and output dimension d B , its RQM satisfies and equality holds when Proof.The idea behind the proof is to employ the reduction criterion for separability [49,50], which can be used to show that any entanglement-breaking channel M : Therefore, the set of channels satisfying this criterion provides a relaxation of the set of EB channels, and we can define a bound on the RQM by computing the minimal robustness with respect to this set.A suitable decomposition of a channel N can then be used to show that, in fact, this bound is given exactly by the larger of d A max eig(Φ N ) − 1 and 0. In the case of d A ≤ 3 and d B = 2, the reduction criterion is also a sufficient condition for separability, which ensures that the robustness R(N ) matches the lower bound.

Supplemental Information: Universal and Operational Benchmarking of Quantum Memories I. RESOURCE FRAMEWORK
We first review the framework of resource theory of memories introduced in Ref. [10].

A. Resource theories of memories
Focusing on two chronologically ordered systems A and B, a quantum memory is described by a channel N A→B that maps system A to B, i.e., a completely positive trace-preserving (CPTP) linear map from D(H A ) to D(H B ). Here, H represents the Hilbert space and D(H ) represents the set of states.The resource theory of quantum memories C = (F , O, R) is a tuple with the free memory set F , free transformations O and resource measure R. The resource theory provides a framework to systematically study the properties of quantum memories.In this work, we mainly focus on the capability of preserving quantum information of memories.In this section, we introduce definitions of the free memory set F and free transformations O, and leave the discussion of resource measures R in the next section.
1. Free memories F : entanglement breaking (EB) or equivalently measure-and-prepare channels, Here The reason that we choose EB channels to be free is because only classical information is stored and forwarded from system A to system B. The channels that have the maximal resource are isometric channels as they are reversible.
We consider the normalised Choi state of a channel N , with maximally entangled state Here d is the dimension of the input system A .For a linear CPTP map N , the corresponding Choi state is a normalised quantum state satisfying Tr B [Φ AB N ] = I A /d. Conversely, for any normalised bipartite quantum state Φ AB which satisfies Tr B [Φ AB ] = I A /d, there is a unique quantum channel N whose Choi state equals to Φ AB and can be espressed as We have the following properties for the free memory set F : (a) The set of entanglement-breaking channels is convex.If a channel M is of the form with ψ i pure and {P i } mutually orthogonal rank one projections, then it is an extreme element of the set of entanglement-breaking channels [21].(b) A quantum channel is EB if and only if its Choi state is a separable state.Equivalently, a free channel admits a Kraus decomposition as M(ρ) = i K i ρK † i where each K i is rank one [21].2. Free super-operations O: any super-operation that only transmits classical information is free, Here U A →AE and V BE→B are arbitrary quantum channels and ∆ E is a dephasing channel that enforces system E to be classical.
For mathematical simplicity, we can also consider free operations as resource non-generating super-operations Λ, which map EB channels to EB channels, Meanwhile, it is not hard to see that free super-operations are also convex.
Quantum channels can be represented with Choi states and transformations of quantum channels, i.e., super-operations, can be similarly regarded as special linear transformations of Choi states.Given the Choi state Φ AB N of channel N A→B , a super-operation Λ(N ) can be equivalently described by a linear map acting on the Choi state.That is, suppose the Choi state of Λ(N ) is Φ AB Λ(N) , we have where φ Λ can be understood as a linear map from state Φ AB N to state Φ A B Λ(N) .

II. ROBUSTNESS OF MEMORY
In this section, we introduce robustness measures of memories and study their properties.We also define the generalised robustness and study its properties.

A. Definition
The robustness of memories is defined as with a minimisation over all possible EB channels.We also define the generalised robustness as with a minimisation over all channels.Note that the robustness measures can be equivalently defined based on the Choi state of channels, Note also that related measures have appeared in general resource theories of states [27,[52][53][54] as well as channels [17,19,55], where their properties were studied.We also define the logarithmic robustness as and the smoothed logarithmic robustness as Similarly, the max-entropy of a memory can be defined as and the smoothed version as Note that, as the smoothing is defined based on the diamond norm of channels, the smoothed measures cannot be obtained by smoothing the Choi state, i.e.,

B. Properties
Here, we focus on properties of the robustness measures.We prove it for R(N ) and the related measures.Unless otherwise mentioned, the same proof for R also holds for R G .
1. Non-negativity.For any channel N , we have More specifically, we also have the following: This follows directly from the definition.
2. Monotonicity.For a resource non-generating super-operation Λ, we have Proof.We first prove R(Λ(N )) ≤ R(N ).Suppose the minimisation of s = R(N ) is achieved with channel M, we have Applying the resource non-generating super-operation Φ to both sides of the above equation, we have Since Λ(M) ∈ F , we conclude that R(Λ(N )) ≤ s = R(N ).As LR(N ) = log 2 (1 + R(N )), we also have LR(Λ(N )) ≤ LR(N ).
As Λ(N ) − Λ(N ) ≤ N − N ≤ ε, we have With monotonicity, we also have that the measures are invariant under reversible transformations.
As a special case of the monotonicity property, we have that the robustness measure of sequentially connected memories is upper bounded by the minimal robustness of each memory.That is, This also holds for the logarithmic robustness and the smoothed logarithmic robustness.

Convexity.
For a set of memories {N i } with probability distribution {p i } satisfying p i ≥ 0 and i p i = 1, the averaged resource measure cannot be increased via mixing memories, i.e., Proof.For all i, suppose the minimisation of s i = R(N i ) is achieved with M i , that is, Since M ∈ F , we have 4. Relation under tensor product.For two channels N 1 and N 2 , we have For the logarithmic robustness and max-entropy, we also have or the tighter relation Proof.For i = 1, 2, suppose the minimisation of s i = R(N i ) is achieved with M i , that is, Then Let s = (s 1 + 1)(s 2 + 1) + s 1 s 2 − 1, then N 1 ⊗ N 2 can be expressed as where and In conclusion, For the generalised robustness, we have To prove the lower bound, we define the partial trace of a quantum channel as By definition (28), the partial trace of a bipartite EB channel is also an EB channel.Suppose the minimisation of where superscript denotes input systems.Now, take partial trace on system B, we get thus R(N 1 ) ≤ s.Symmetrically, we also have R(N 2 ) ≤ s, so the lower bound max{R(N 1 ), R(N 2 )} ≤ R(N 1 ⊗ N 2 ) is obtained.

Stability. For any channel N and any
Proof.As appending a free channel is a resource non-generating operation, we have R(N ⊗ M) ≤ R(N ).Then, from property 4 we also have R(N ⊗ M) ≥ R(N ), thus the equality is obtained.

A. Preliminary
Before proving the main results, we first obtain some preliminary Lemmas which are useful for the following discussions.
Using the same technique as in [27], we show that Φ + AB can be expressed as separable states where . It is shown in [27] that these are both separable states.It remains to show that these correspond to Choi states of EB channels.This is true since By definition of robustness of memory, we have R(I d ) ≤ d − 1.Also, the lower bound R(I d ) ≥ d − 1 is trivial, because if R(I d ) < d − 1, then using the same construction, we can show that the robustness of entanglement of Φ + AB is less than d − 1, violating the known result in [27].
Lemma 2. For any pure state |ψ and quantum channels N 1 , N 2 in the same space, the linear map is a quantum super-channel represented by operation on Choi states.Here, C is an input quantum channel, Φ C is its Choi state, and Λ(Φ C ) is the Choi state of the output channel.
Proof.For an input quantum state ω, the output channel acts on ω as In order to show that Λ is a quantum super-channel, we only need to show that Λ(C) can be decomposed into three steps: pre-processing, action with ancillary system, and post-processing, which are constructed as follows: • pre-processing: the input state is appended with a maximally entangled state • action with ancillary system: the channel C acts on system B while identity acts on systems A and C • post-processing: suppose the channels N 1 , N 2 have Kraus operators {K j } j , {T k } k , respectively.We construct a postprocessing channel with Kraus operators To see that this is a valid quantum channel, we have Also, we can see that the output of this channel is

B. Single-shot memory dilution
We consider the problem of single-shot dilution or channel simulation under resource non-generating super-operations, which is defined as Note that compared to the definition in Methods of the main text, we used the logarithm of the dimension to represent the number of qubit memories required for the task.
Theorem 1.For any quantum channel N , its single-shot memory dilution rate is Proof.First we prove the left hand side LR ε (N ) ≤ R 1,ε c (N ).Suppose there exists a memory dilution protocol Λ such that With the monotonicity of the logarithmic robustness, we have Here, the last line follows from Lemma 1.Since the above equation holds for any dilution protocol, we conclude that LR ε (N ) ≤ R 1,ε c (N ).
Next, we prove the right hand side We construct a linear map as where systems C and D have dimension d c .By Lemma 2, we know that Λ is a super-channel.
Next, we verify that Λ is a free resource non-generating super-operation.We first rewrite Eq. ( 79) as C is a separable state, and we have 0 ≤ q ≤ 1.Thus Λ(Φ CD C ) is a separable Choi state that corresponds to an EB channel.
Lastly, we verify that when inputting the identity channel, the output channel Remark 1. Define the smooth robustness of quantum memory as Since dimensions are integers, the single-shot memory dilution rate can be exactly characterised as The above can be compared with the one-shot characterisation of dilution in resource theories of states [19,24,56], which yields related results but is not applicable to the study of manipulation of quantum channels.

IV. SIMULATING QUANTUM MEMORIES WITH CLASSICAL RESOURCES
The task is to simulate a target quantum memory N with free EB memories and free super-operations.For an unknown input state ρ, the output state is N (ρ).Suppose after another operation U is applied to the output state, we read out the system by the measuring the average value The simulation scheme works as follows.
1.As we require the simulation scheme to be independent of ρ, U, and O, it is equivalent to have N as a linear expansion of EB channels M i , i.e., with coefficients c i ∈ R possibly negative.This linear decomposition always exists as the set of EB channels forms a complete basis for the space of quantum channels.
2. To obtain the averaged value O U•N(ρ) , we first re-express it as an overhead c 1 = i |c i |, and a normalised probability distribution We can obtain O U•N(ρ) by averaging sign(c i ) O U•M i (ρ) with probability p i , which is described as follows: (a) We randomly generate i according to the probability distribution {p i }.
(b) As each M i is EB, we have M i (ρ) = j Tr[ρM Suppose we aim to estimate O N(ρ) to an additive error ε with probability δ, we need the number of samples to be according to the Hoeffding inequality.Meanwhile, given the channel N itself, the number of samples needed is T 0 ∝ 1/ε 2 log(δ −1 ).
Thus the simulation cost can be quantified by the overhead We can further minimise the simulation cost over all possible decomposition strategies of Eq. ( 84).Denote the positive and negative coefficients of c i by c + i and c − i , respectively.Then we have As the channel is trace preserving, we also have i: Optimising over all possible decomposition is equivalent to optimising over all M, M ∈ F , which coincides with the definition of the robustness of memories.Therefore we have min Therefore, the robustness of quantum memory quantifies the cost of simulating the memory with free EB memories.The result of this section can be regarded as an extension of the framework of negativity-based simulators of [45,57], which were recently adapted to the study of general resource theories of states [58], and applied in other settings to investigate the properties of specific channel resources [40,41].

V. QUANTUM GAME FOR TESTING THE POWER OF A MEMORY
In this section, we discuss how to use quantum games to test the power of a memory and how the optimal strategy is related to the robustness of memories.We consider quantum games similar to Ref. [10] by firstly inputting a general set of input states {σ i } to the channel.In Ref. [10], the output state is measured together with an ancillary set of states, so that the test can be independent of whether the measurement is faithfully implemented.In this work, instead of requiring such a measurement-device-independent feature, we directly measure the output states with a general set of observables {O j } by assuming that the measurement is trusted.We can define a general pay-off function as with real coefficients α i, j , where we use G = ({σ i }, {O j }, {α i, j }) to denote a particular game.When the coefficients α i, j are selected randomly, the pay-off function can be arbitrary, and in particular does not have to be non-negative.Instead, we first constrain ourselves to the case where the coefficients are selected such that the pay-off function is non-negative for any channel, P(N, G) ≥ 0. Then the maximal pay-off of the game is max where the maximisation is over all games G ∈ S G with S G = {G : To prove it, we first briefly review the duality in conic optimisation.We follow the description of Ref. [17] and refer to the references therein for more details.Given real complete normed vector spaces W and W , a conic optimisation problem is defined as where A ∈ W * , y ∈ W , Λ : W → W is a linear function, and K ⊆ W is a closed and convex cone.The dual form of the optimisation is given by where The primal and dual problems are equivalent if Slater's condition is satisfied: that is there exists a feasible solution x such that x is in the (relative) interior of K. Now we prove Eq. (92).
Proof.We first write R G (N ) in terms of its Choi state as This can be equivalently recast as where we use V (F ) to represent bipartite (separable) Choi states, while cone(F ) and cone(V) represent their unnormalised versions.Now define can be represented as the form of Eq. (93).Note that W * is the set of Hermitian operators.The dual form of the optimisation gives where Note that the above condition is further equivalent to This is because when we have Tr Similarly, we set x 1 = 0 and get Tr[W x 2 ] ≥ 0, ∀x 2 ∈ cone(V).On the other hand, it is straightfward to verify that Eq. (100) implies Eq. (99).Therefore, the dual form can be written as which can be expressed with quantum channels Slater's condition holds as we can choose, say, W = I/2.Thus we have strong duality, R d G (N ) = R G (N ).Now we show that the dual form is equivalent to the maximal pay-off of the quantum game.We first write the pay-off function in terms of the Choi state of the channel as where σ T i is the transpose of σ i and W is a Hermitian operator Denote P max (N ) = max G ∈S G P(N, G) with S G = {G : P(N, G) ≥ 0, P(M, G) ≤ 1, ∀N ∈ CPTP, M ∈ F }, the optimisation over all games in S G is equivalent to optimise over all Hermitian operators W that satisfies Tr[Φ + M W] ≥ 0 for all channels M and Tr[Φ + M W] ≤ 1 for entanglement breaking channels.Therefore, we have The robustness R(N ) can be understood very similarly in this context.The maximal pay-off of the game is max where the maximisation is over all games G ∈ S with S = {G : Proof.We follow the proof for the generalised robustness.We first write the robustness measure in the standard form of the primal optimisation problem as where we define components in the standard form Eq. (93) as The dual form of the optimisation gives where Note that the above condition is further equivalent to We can express it with quantum channels as which is exactly the maximal pay-off function P max (N ) = max G ∈S P(N, G).Since Slater's condition also holds in this case as we can choose W = I/2, we conclude that strong duality holds, thus By considering games G ∈ S with S = {G : P(M, G) ≥ 0, ∀M ∈ F }, we thus get the result presented in the main text, max The characterisation of R G in terms of performance in nonlocal games can be compared with [17,59], where this quantity was related to the advantage in discrimination tasks in general resource theories of states and channels.The setting considered in that work is different, however, most importantly since the measurements there are performed on the joint Choi state of channels rather than their output states, making them significantly more difficult to perform in practice.

A. Relaxing to PPT-inducing channels
The robustness and generalised robustness measures of memories are both convex optimisation problems due to the convexity property.They can be numerically calculated by focusing on the Choi states as and Instead of directly solving this optimisation problem, we reduce the restrictions that Φ + M , Φ + M are separable Choi states to ones that they are positive partial transpose (PPT) Choi states.We denote the robustness measures against PPT-inducing channels as R * (N ) and R * G (N ), defined as R * (N ) = min s s.t.s ≥ 0, As the set of PPT Choi states is larger than the set of separable Choi states, the optimisation always give lower bounds The bound is tight for channels with qubit inputs and outputs or ones with qubit inputs and qutrit outputs or vice versa.In general cases, the optimisation with respect to PPT Choi states can be efficiently solved as semidefinite programs.In the case of qubit outputs, we will in fact establish an analytical form of the measures. and In Fig. 4(a) we show several the robustness of examples channels.Meanwhile the expression for the robustness of quantum memories N is similar to the robustness of entanglement of Choi matrix Φ + N , which is defined as and Note that Φ + M , Φ + M belong to the subset of separable states whose partial trace is identity due to the property of Choi states.Thus the robustness of entanglement of the Choi state lower bounds the robustness of quantum memory, RE(Φ + N ) ≤ R(N ) and RE G (Φ + N ) ≤ R G (N ).Similarly, we can also relax the separable state set with PPT to efficiently calculate RE * (Φ + N ) and RE * G (Φ + N ), so that we have . By randomly choosing quantum channels, one can find that the quantities RE * (Φ + N ) and R * (N ) are different, as shown in Fig. 4(b) and (d).Note that for the numerical examples in the main text, we focus on quantum channels with input dimension d in and output dimension d out such that d in × d out ≤ 6, in which case the SDP relaxation provided here is actually tight, due to the well-known correspondence between SEP and PPT in low dimensions [48].

B. Examples of quantum game
The pay-off of a quantum game G = ({σ i }, {O j }, {α i, j }) can be expressed as where Φ + N = I ⊗ N (Φ + ) is the Choi state of N , Φ + = 1/d i j |ii j j | is the maximally entangled state, d is the dimension of the input system, and the game operator is For the robustness of quantum memories, we consider games that satisfy For the depolarising channel ∆ p (ρ) = pρ + (1 − p)I/2, the corresponding Choi state is the Werner state, The robustness of ∆ p (ρ) can be obtained by applying the entanglement witness considered in our previous work [60].Here, we construct the game G = ({σ i }, {O j }, {α i, j }) such that Because 2Tr[Φ + σ] ≤ 1 for all bipartite qubit separable states σ, we have P(M, G) ∈ [0, 1], ∀M ∈ F .We can also show that W is the optimal witness of R G (∆ p ) by proving This is equivalent to finding a separable Choi state Φ + M that satisfies This is satisfied by choosing Φ + M = 1 3 Φ + + 1 6 I. Now we show how to decompose the witness into input states and measurements.Suppose we choose {σ i }, {O j } as with O i = σ T i , the corresponding coefficients are For the qubit depolarising channel ∆ p (ρ) = pρ + (1 − p)I/2, the pay-off of this game is This provides a lower bound The equal sign is always achieved, as verified from the numerical calculation in Fig. 4(a).
Now we choose the game with a witness which is consistent with the numerical calculation, We can also prove that the estimate from the game witness is tight.That is, we need to find a separable Choi state Φ + M that satisfies Here, we choose To satisfy the above inequality, we choose q = 2p p+1 when p ≥ 1/3 and q = 1−p p+1 when p ≤ 1/3.The decomposition of the witness is similar to the one for the depolarising channel.The only difference is that we need to consider an additional measurement to take into account of the term I ⊗ |2 2|.That is, with the same inputs given in Eq. ( 129), we consider the game with input states measurements and coefficients Note that the robustness of quantum memories of the two above examples are equal to the robustness of entanglement of the corresponding Choi states.For the stochastic damping channel, we found them different as shown in numerical examples of Fig. 4(c) of the main text.
The Choi state of the stochastic damping channels Suppose the spectral decomposition of this matrix is where λ 0 ≥ λ 1 ≥ λ 2 ≥ λ 3 and ψ i is the density matrix of eigenstate.We can compute that as well as |ψ 0 = α |00 + β |11 with real coefficients α and β depending on p, Now construct the game with a witness W defined by We prove it by considering a stronger scenario for general EB channels with input dimension d and we also consider an optimisation over all possible pure states ψ 0 , max This is equivalent to the following Lemma.
Proof.The maximal eigenvalue of σ AB can be obtained by considering its Schatten ∞-norm.For an EB channel, its Choi state can always be expressed as with i p i ψ i = I/d, i p i = 1, and p i ≥ 0. Therefore we have Here the inequality follows from Tr φ i 1 φ i 2 . . .φ i n ≤ 1.
The decomposition of ψ 0 can be expressed as Thus we can choose and And finally we obtain the lower bound

C. Tight lower bound via moments of Choi states
Consider the Choi state Φ + N of a channel N , we can either lower bound the robustness via a witness measurement or via purity measurement.We prove Lemma 1 in the main text.Lemma 4. For any EB channel M with input dimension d and k = 0, 1, . . ., ∞, we have For any channel N with input dimension d, its RQM can be lower bounded by Proof.The first half of this Lemma, Eq. ( 156), can be proven by following the proof of Lemma 3.
For the second half, we show that R Suppose the optimal decomposition is and we obtain a lower bound for the robustness The comparison of the lower bounds with different moments of Choi states is shown in Fig. 4(d).
We will show that the bound obtained by taking k → ∞ is in fact tight for low-dimensional channels, and furthermore exactly characterises the PPT robustness R * for all channels with qubit output.

Theorem 2. Consider a channel
More generally, for any channel such that d B = 2, the PPT robustness satisfies Proof.We will employ the reduction criterion for separability [49,50], which states that a bipartite state ρ AB with d A ≤ 3 and d B = 2 is separable if and only if Notice that when ρ AB = Φ + N is the Choi state of a channel N : A → B, this reduces to that is, max eig Φ + N ≤ 1 d A .The SDPs for the robustness measures then become The inequality R Noting that µd A ≥ 1 and thus µd , which means that the above constitutes a valid feasible solution for the robustness R(N ) with λ = µd A .We conclude that R G (N ) ≤ R(N ) ≤ µd A = d A max eig Φ + N , and so the quantities must all be equal.
The second part of the Proposition follows since the action of the positive map ρ → (Trρ)I − ρ, on which the reduction criterion is based, is unitarily equivalent to the transpose map when acting on a 2-dimensional system [49,50].This means that the reduction criterion and the PPT criterion are equivalent when d B = 2.
Altogether, the results establish Theorem 4 in the main text.As an immediate consequence of the results, we notice that the multiplicativity of the lower bound coupled with the submultiplicativity of R G (see Sec. II B) yields exact multiplicativity for R G or, equivalently, additivity for D max in the case of low-dimension channels.We implement the dephasing channel, erasure channel, and the stochastic damping channel in Fig. 5, 6, and 7, respectively.For these three different channels, we choose four different noise levels as p = 1, 3/4, 1/2, 1/4 and implement it on the IBM cloud.The game for the dephasing channel is the same one for the depolarising channel defined in Eq. (129) and Eq. ( 130).The game for the erasure channel is defined in Eq. (139) and Eq.(141).We use two qubits to encode |0 , |1 and |2 , via |0 ⊗ |0 → |0 , |0 ⊗ |1 → |1 , |1 ⊗ I → |2 .The game for the stochastic damping channel is defined in Eq. (153) and Eq.(154).The first qubit is used to choose the original state or |0 .We use the second qubit to present the input state and use the third one to replace the original state with |0 .We simulate this channel via collecting the post-processing statistics.When the outcome of first qubit is |0 , we only care about the outcomes of second qubit.Otherwise, we focus on the outcomes of third one.The experiment results of the three channels are shown in Table I, II, and III.

B. Robustness of two qubit and three qubit gates
We also measured the robustness of two qubit and three qubit gates on the IBM quantum cloud.Denote the two qubit controlled-X (CX) gate as C X 0 1 with 0 denoting the controlled qubit and 1 denoting the target qubit.We consider a sequence of CX gates with interchanged control and target qubits for two adjacent gates as Note that the gate U 3 is the swap gate and U 6 is the identity gate.We also considered the three qubit gate that prepares the GHZ state For each gate U, the corresponding Choi state is |Φ + U = U |Φ + with |Φ + = 1/ √ d i |ii being the maximally entangled state and d being the dimension.We can define a witness as W = dΦ + U to lower bound the robustness.Such a witness corresponds to the case where we assume the noise is deploarising or dephasing.We can accordingly change the witness if we estimate that the gate noise is other types such as erasure or stochastic damping.According to Lemma 4, the robustness of a noisy gate Ũ is lower bounded as Note that Tr[Φ + U Φ + Ũ ] corresponds to the gate fidelity between the noisy gate Ũ and the target gate U. To measure Tr[Φ + U Φ + Ũ ], we decompose Φ + U into a linear sum of local Pauli operators.Since we are considering small gates, the decomposition only has a small number terms so that they are directly measured in the experiment.When considering large quantum gates, we can make use of the technique introduced in Ref. [61] to efficiently measure the gate fidelity quantity.This requires to measure expectation values of a constant number of local Pauli operators, which are selected at random according to a weighting determined by Φ + U .We leave the implementation of random measurement of large quantum gates for future works.

FIG. 2 .
FIG. 2. Numerical evaluation of the exact value of robustness of quantum memories.(a) Robustness of memories with qubit inputs and computational basis {|0 , |1 } for dephasing channels ∆ p (ρ) = pρ + (1 − p)Z ρZ, stochastic damping channels D p (ρ) = pρ + (1 − p)|0 0|, and erasure channels E p (ρ) = pρ + (1 − p)|2 2| with |2 orthogonal to {|0 , |1 }.(b) Memory robustness under dynamical decoupling (DD) and its quantification of non-Markovianity.We consider a qubit memory (M) coupled to a qubit bath (B) with an initial state ρ B (0) = 0.4|0 0| + 0.6|1 1| and an interaction HamiltonianH = 0.2(X M ⊗ X B + Y M ⊗ Y B ) + Z M ⊗ Z B .Here X, Y, Z are the Pauli matrices.We consider the evolution with time t from 0 to π.To decouple the interaction, we apply X operations on the memory at a constant rate.We show that the memory robustness can be enhanced via dynamical decoupling (DD).Furthermore, as the memory robustness can increase with time, we calculate the non-Markovianity using the robustness derived measure as defined in Eq. (5).

FIG. 3 .
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 ∆ p (ρ) = pρ + (1 − p)Z ρZ, stochastic damping channels D p (ρ) = pρ + (1 − p)|0 0|, and erasure channels E p (ρ) = pρ + (1 − p)|2 2| with |2 orthogonal to the basis {|0 , |1 }.We synthesise the noise channels by interacting the target system with up to two ancillary qubits.We measure the payoff of quantum games P(N, G) which lower bounds the RQM as R(N ) ≥ P(N, G) − 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 sequential CX gates will not cancel out.For example, denote C X 0 1 to be the CX gate with control qubit 0 and target qubit 1; the three CX gates is the swap gate C X 0 1 C X 1 0 C X 0 1 ≡ SW AP and the six controlled-X gates becomes the identity gate CX 1 0 C X 0 1 C X 1 0 C X 0 1 C X 1 0 C X 0 1 ≡ I 4 .The error bar is three times the standard deviation for both plots.

Lemma 1 .
The logarithmic robustness of an ideal d-dimensional quantum memory I d is LR(I d ) = log 2 d. (68) Proof.It is equivalent to show that R(I d ) = d − 1.We first prove the upper bound R(I d ) ≤ d − 1.Notice that the Choi state of I d is the maximally entangled state |Φ To get O U•M i (ρ) , we first measure ρ with the POVM {M j i ≥ 0}, obtaining outcome j with probability Tr[ρM j i ].Then we prepare σ j i , apply U, and measure the observable O to have Tr[U(σ j i )O].The value O U•M i (ρ) can be evaluated by averaging the measurement results Tr[U(σ j i )O] over all outcomes j.(d) Finally, by multiplying sign(c i ) to O U•M i (ρ) and averaging over all M i with probability p i , we recover the normalised average expectation value O U•N(ρ) .3. The target averaged value O U•N(ρ) is obtained by multiplying the constant overhead c 1 to O U•N(ρ) .

FIG. 4 .
FIG. 4. Numerical calculation of robustness measures of memories and entanglement.(a) Robustness of memories with qubit inputs and computational basis {|0 , |1 } for depolarising channels ∆ p (ρ) = pρ + (1 − p)I/2, stochastic damping channels D p (ρ) = pρ + (1 − p)|0 0|, and erasure channels E p (ρ) = pρ + (1 − p)|2 2| with |2 orthogonal to {|0 , |1 }.(b) Comparison between the robustness of memory and the robustness of entanglement of stochastic damping channels.Here the entanglement of a channel refers to the entanglement of the corresponding Choi state.(c) Comparison between the robustness of memories (horizontal axis) and the robustness of entanglement (vertical axis) of random channels with qubit input and output.(d)Estimation of the RQM via different moments of the Choi state given in Eq. (157) for stochastic damping channels.In particular, the estimation is tight with the infinite moment which corresponds to the maximal eigenvalue of the Choi state.

Lemma 3 .
For any EB channel with input dimension d, the maximal eigenvalue of its Choi state σ AB is upper bounded by 1/d, max eig(σ AB ) ≤ 1/d.
for any feasible λ and Φ + M .On the other hand, let µ denote the largest eigenvalue ofΦ + N .If µ ≤ 1 d A , then Φ + N itselfis a feasible solution and the equality R(N ) = R G (N ) = 0 is trivial, so assume that µ ∈ ( 1 d A , 1].Denoting by ∆ the completely depolarising channel with Φ + ∆ = I/d A d B , we define the channels M ± as the convex combinations