Continuous Groups of Transversal Gates for Quantum Error Correcting Codes from ﬁnite Clock Reference Frames

Continuous groups of transversal gates for quantum error correcting codes from ﬁnite clock reference frames

Mischa P. Woods1 and Alvaro´ M. Alhambra2

1Institute for Theoretical Physics, ETH Zurich, Switzerland 2Perimeter Institute for Theoretical Physics, Waterloo, Canada

Following the introduction of the task of ref- ply logical gates transversally, which one can imple- erence frame error correction [1], we show ment while still being able to correct for local errors. how, by using reference frame alignment with The framework for error correction is based on con- clocks, one can add a continuous Abelian sidering three spaces — a logical HL, physical HP 1 group of transversal logical gates to any error- and a code HCo ⊆ HP space. Logical states ρL correcting code. With this we further explore containing quantum information are encoded via an a way of circumventing the no-go theorem of encoding map E : B(HL) → B(HCo) onto the code Eastin and Knill, which states that if local er- space, which is a subspace of some larger physical rors are correctable, the group of transversal space where errors — represented via error maps gates must be of ﬁnite order. We are able to {Ej}j : B(HCo) → B(HP) — can occur. Decoding do this by introducing a small error on the de- maps {Dj}j : B(HP) → B(HL) can then retrieve the coding procedure that decreases with the di- information while correcting for errors; outputting the mension of the frames used. Furthermore, we logical state ρL. That is: show that there is a direct relationship between how small this error can be and how ρL E Ej Dj ρL, (1) accurate quantum clocks can be: the more ac- for all j and for all states ρ ∈ S (H ). Depending curate the clock, the smaller the error; and the L L on the error model, the index j indicating which error no-go theorem would be violated if time could occurred may or may not be known. If it is unknown, be measured perfectly in quantum mechanics. the decoding map D cannot depend on j. We say The asymptotic scaling of the error is stud- j that a logical gate V can be applied transversally if ied under a number of scenarios of reference L for any state ρ, the encoder E is such that frames and error models. The scheme is also † ⊗K †⊗K extended to errors at unknown locations, and E(VLρVL ) = VCo E(ρ)VCo , (2) we show how to achieve this by simple majority voting related error correction schemes on where the tensor product structure “⊗K” represents the reference frames. In the Outlook, we dis- the division of the code into diﬀerent subsystems or cuss our results in relation to the AdS/CFT “blocks” in which errors can be independently cor- correspondence and the Page-Wooters mecha- rected. This condition means that the action of the nism. encoding map commutes with the action of the logical ⊗K gate VL, which is represented by VCo in the physical space. An interesting case to consider is that of codes E and groups G for which all group elements 1 Introduction and Overview of Re- (indexed by g) can be applied transversally using a sults unitary group representation UL(g),

† ⊗K †⊗K E(UL(g)ρUL(g) ) = UCo(g) E(ρ)UCo(g) ∀g ∈ G. arXiv:1902.07725v4 [quant-ph] 19 Mar 2020 In order to build a functional universal quantum com- (3) puter, full fault-tolerance must be achieved. The idea We will refer to codes whose encoding has this prop- behind fault-tolerance is that the errors that occur erty as covariant codes. There are a number of results at particular points during the computation do not that restrict the existence of such codes, in particu- propagate or amplify along the whole computation to lar for stabilizer codes [2–6]. Most notably, the no-go the point of being uncorrectable. Due to fundamental theorem of Eastin and Knill [7] states that in any physical constraints such as no-cloning, achieving this ﬁnite-dimensional code (not necessarily stabilizer) in is a notoriously challenging task, with a number of dif- which local errors can be corrected, the groups of log- ferent requirements on how to prepare, manipulate, ical gates that can be applied transversally must be and protect the quantum states with error-correcting codes. One of the most desirable features of the codes 1Some authors use the convention of not considering the used in fault-tolerant computation is the ability to ap- code space explicitly, in which case one sets HCo = HP.

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 1 finite. This thus excludes dense sets of logical gates, cannot exist as they require infinite energy.2 These as well as any continuous Lie subgroup of U(d). are known as Idealised clocks. However, this does not Since this no-go result imposes a fundamental limi- rule out the existence of approximate time operators tˆ tation on the possible transversal gates, it is interest- and initial clock states |ψi which serve as a reference ing to find ways of circumventing it, via schemes that frame for the observable t,[19–21] either in the form do not satisfy all of the assumptions. A number of of a stopwatch [22] or ticking clock [23–25] alternatives have been thoroughly explored, includ- Here we explore how certain finite-sized reference ing the protocols of magic state distillation [8], and frames (which we call “clocks”) can be used to build other more specific schemes (see for instance [9–14]), imperfect covariant codes, and we give upper bounds many of which propose a relaxation of the transver- to the errors induced by their finite size. We use the sality condition in some fault-tolerant way. construction of Quasi-Ideal clocks [21] and Salecker- Recently, a new kind of circumvention was put for- Wigner-Peres (SWP) clocks [19, 20], to provide sim- ward in [1], where they show examples of codes with ple encoding and decoding protocols based on the task physical spaces of infinite dimension that allow for of reference frame alignment. We show using Quasi- the transversal implementation of Lie groups. The Ideal clock states entangled over L subsystems, that need for infinite dimensions (and seemingly infinite the worst-case entanglement fidelity between the in- energy too, as we will soon discuss) limits their prac- put ρL and decoded output Dj(Ej (Ecov(ρL))) denoted tical relevance, but the idea motivates the following fworst, and defined in Section 2.2, in our protocol sat- question: do there exist large (but finite) covariant isfies (up to log factors) the lower bound 1 − fworst ≤ 2 codes in which approximate error correction can be O(1/(LdC) ) where L is the number of entangled performed? In other words, can we circumvent the Quasi-Ideal clocks and dC is their dimension. In [26] 2 no-go results by allowing for small errors that decrease the generic upper bound 1 − fworst ≥ O(1/(LdC) ) is with the size of the code? derived for all covariant encoding maps generated by Here we explore this question using the notion of isometries. A direct consequence of the combination reference frames and clocks. We construct imperfect of our results with those of [26] adapted to our setting, codes in which Abelian U(1) groups can be imple- is that all error correcting codes {E, {Ej, Dj}j}, can mented transversally. To that aim, we use a simple fi- be made covariant w.r.t. the U(1) groups considered nite dimensional version of the encoding map from [1], here with an optimal fidelity fworst (largest possible which shows that perfect covariant codes are possible value) satisfying provided one has access to a perfect reference frame. 6 A perfect reference frame [15] is defined as a quan- 1 ln (LdC) O 2 ≤ 1 − fworst ≤ O 2 , (6) tum system that encodes, without error, information (LdC) (LdC) about a particular group element. That is, given a group representation U(g), such that U(0) = 1, the for a large LdC. state |ψi is a perfect reference frame iff ∀ g In order to study the role of different resources in our protocols, we define t-incoherent clock states ρC as those for which there exits g ∈ G such that their hψ| U(g) |ψi = δ0,g, (4) † group evolved initial state, UC(g)ρCUC(g), commutes with the projective measurements used in our protocol where δ is a Kronecker delta for finite groups and 0,g to measure them.3 As we later explain, these states Dirac delta in the case of Lie groups. Hence, each require minimal coherent resources to be created in point of the orbit |ψ(g)i = U(g) |ψi is orthogonal to comparison with other clock states. We find that all all the others (and thus perfectly distinguishable). codes which use t-incoherent clock states satisfy the This connects with the work by Pauli on quan- upper bound 1 − f ≥ O(1/(Ld )). We then con- tum clocks in the case of an Abelian U(1) groups. worst C sider the more commonly used SWP clocks (see for If the group {U(g)} is a one-parameter compact Lie g instance [19, 20, 27–30] and references therein), which group generated by solving the Schrödingerequation, belong to this class, and show that they yield an er- namely U(g) = e−igHˆ for some Hamiltonian Hˆ , where ror of order 1−fworst = O(1/(LdC)) hence saturating g = t is the time transcribed, then the constraint Eq. the bound for t-incoherent clocks. However, we also (4) is analogous to requiring the existence of a perfect show that while coherent clock states are necessary to time operator tˆ. Indeed, defining achieve 1 − fworst ≤ O(1/(LdC)), they are not suffi- Z cient, since finite dimensional analogues of coherent tˆ= dg g |ψ(g)ihψ(g)|, (5) G 2The infinite energy manifests itself either with unbounded from below Hamiltonians (when t belongs to an unbounded in- where the integral is over the Haar measure, one finds terval) or infinitely strong potentials producing strict confine- tˆ|ψ(g)i = g |ψ(g)i for all g ∈ G. Pauli [16, 17] already ment of the wave function (for when t belongs to an bounded interval) [18]. concluded that such time operators require quantum systems that — while mathematically well defined — 3See text around Eq. (17) for full definition.

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 2 states can only achieve the same scaling as the SWP about which transversal gate is applied to it is en- clocks. coded in the reference frames, hence without knowl- The results discussed so far consider codes in which edge from the reference frames during the decoding, the error model on the clock is just erasure at known it would not be possible to discern which gates had locations. However, what if one cannot discern the been applied. If the reference frames are damaged via location at which the error occurred? Our final result an error (which is a reasonable assumption since they is to prove that one can also correct local unknown are now part of the extended code space), this will phase errors which occur at an unknown location in inhibit the ability to decode correctly resulting in a the clocks. For this case, we are able to achieve 1 − worse decoding of the logical information. fworst ≤ O(1/(LdC)) up to log factors. The protection of the information to be encoded re- lies on the channel E, which is arbitrary. On the other hand, the protection for the reference frame in the en- 2 Covariant codes based on reference conding of Eq. (8) will come merely from having a few frames copies of them which may or may not be entangled. In (M) ⊗M the case of no entanglement, ρF = ρF , the states We now outline the generic error correction scheme we are only classically correlated due to the twirling over consider, which is based on a generalisation of the ref- G in Eq. (4), and thus bears strong analogies with a erence frame-based scheme in [1]. Let {E, {Ej, Dj}j} classical repetition code. be any perfect error correcting code satisfying Eq. (1) The encoding-error-decoding procedure consists of (not necessarily covariant). the following steps: (M) Now, let ρF be a reference frame for a group 1) An arbitrary state ρL is encoded as Ecov(ρL). isomorphic to {UL(g)}g and {UCo(g)}g, with unitary ⊗M group representation {UF (g)}g. We define the fol- 2) Errors occur: The error maps considered here are lowing for all g ∈ G now of the form Ej,q = Ej ⊗ξq, with ξq acting on the reference frames and the index q — as with ⊗K † †⊗K Eg(·) := UCo(g) E(UL(g)(·)UL(g))UCo(g) . (7) j — may or may not be known depending on the error model. This may include the loss of up to The covariant code is then: M − 1 frames, or in the case of Theorem4, an Z unknown phase error at an unknown location. ⊗M (M) †⊗M Ecov(·) := dg Eg(·) ⊗ UF(g) ρF UF(g) , (8) G 3) Frames are measured: In the case of erasure er- where dg is the uniform or Haar measure over the rors at known locations of the reference frames, 4 only the N non erased clocks are measured. The group. Note that Ecov is from HL to HP ⊗ HF while usually the encoding map takes logical states to states remaining N frames are measured projectively in the physical space. As such, it is convenient to in some basis, and after tracing out the reference think of the reference frames as an extension of the frames, we write code space to H := HCo ⊗HF. Moreover, the chan- ˆ00 Coe trF[Ej,q (Ecov(ρL))] = Ej (Eg0 (ρL)) + Eg0 , (9) nel Ecov is now covariant in the sense of Eq. (3) if the symmetry group on the r.h.s. is defined in the ex- ˆ00 0 where Eg0 represents the error term and g is cho- ⊗K ⊗M tended physical space, namely UCo(g) ⊗ UF(g) , sen based on the measurement outcomes, the ini- so that UL(g) can be implemented transversally. This tial clock states and the error maps ξq. If the way, the notion of covariance defined in Eq. (3) is now 00 reference frames are good, the error Eˆ 0 will be over H rather than H alone. We note, however, g Co Co small since the measurements will be able to dis- that unlikee the code subspace, none of the logical in- tinguish approximately the group elements. In- formation is encoded directly into frames F, and they deed, in the case of perfect reference frames Eq. merely serve as a reference for g, for which both the (4) or equivalently the Idealised clock Eq. (5), ⊗M (M) group {U }g and initial frame state ρ can be 0 00 F F the optimal choice of g leads to no error, Eˆ 0 = 0. chosen at our convenience to optimise our protocol. g The purpose of the error maps Ej is that they take in- 4) Finally, we apply the decoding map D¯j,g0 defined formation from the code space and “mix” it with the as rest of physical space resulting in errors. On the other ¯ hand, the reference frames play the following role in Dj,g(·) = (10) the encoding map Ecov : the logical information is still −1 †⊗K ⊗K † UL(g)E U (g)E (Dj(·)) U (g) U (g), encoded into the code space, but now the information Co Co L −1 −1 4 where Dj (Ej(·)) = E (·), with E being the It is noteworthy that all our results hold for all K — the number of gates being applied transversally. In fact, one can inverse channel for the encoder E (that is, the choose K = 1 when considering a physical space in which the channel that undoes the encoding when no errors gates are not applied locally. occur).

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 3 This achieves have fixed gaps but arbitrary degeneracy, so that h ∈ {0, 1, 2,..., ∆h } with N 3 ∆h ≤ d . As ¯ ˆ S,n S S S Dj,g0 trF[Ej,q (Ecov(ρL))] = ρL + Eg0 , (11) we will see, the performance improves as the range ∆hCo decreases, so in fact the best choice is a trivial where Eˆg0 is the final error term. generator HˆCo ∝ 1, for which we can set ∆hCo = 0. The circuit diagram is as follows. As motivated by our discussion of Pauli’s findings, in this context, one can think of the group elements UF(g) as providing the dynamics of quantum clocks encoding error Decoding ρF where g = t is time. We will therefore refer to ρL the reference frame as a “clock” and use the labels F 0 ⊗K ¯ ⊗K−1 E Ej UCo(g ) Dj,g and C interchangeably in this context. The clock is |0i ρL +Eˆg0 chosen as the compact non-degenerate representation, −itHˆC PdC−1 namely UC(t) = e with HˆC = ωr|rihr|C | i Ecov r=0 where {|riC} is an arbitrary orthonormal basis. Phys- | i ically, energy gaps which are all integer multiples of ω are required to create degeneracies between systems | i L, Co, C. Technically, it means that the group representations on L, Co, C are compact, and that the ir- This scheme is closely related to the task of reps of U (g) contain those of U (g) and U (g). The reference frame alignment, in which the group of C L Co compactness allows replacement of the integration “transversal gates” corresponds to the unknown ro- range in Eq. (8) with [0,T ] where T = 2π/ω is the tation that happens between two parties, here the 0 0 recurrence time and the measure becomes dg = dt/T . encoder and decoder. This is such that the chan- 0 The main clocks we use in Eq. (8) are based upon nel connecting them is effectively a decoherence chan- the Quasi-Ideal clocks [21] and the SWP clocks [20]. nel. The unknown rotation is inferred by measuring They both share the same Hamiltonian Hˆ . The def- the frame (see [15, 31] for further details). Refer- C inition of SWP clock states are simple, they are any ence frame alignment can in fact be thought of as the one of the pure states of the Fourier-transformed basis particular case where the encoding, error and decod- of eigenbasis of Hˆ , namely ρ = |θ ihθ | with ing are identity channels. We discuss this connection C SWP k k further in Supplementary materialI, where we show 1 dC−1 that the error probability using the Quasi-Ideal clock X −i2πnj/dC |θki = √ e |jiC , (12) 3 2 dC is (ln dC) /dC — a quadratic improvement over the j=0 SWP clock, which only achieves ∝ 1/dC as shown in [31]. where |θki = |θk mod d.i, k ∈ Z; and is referred to as Another question of relevance is what happens the time basis. On the other hand, the Quasi-Ideal 0 0 if the original encoding E already had a group clock states ρQI = |Ψ(k1)ihΨ(k1)| are defined as a of transversal gates {VL,VCo} satisfying Eq. (2). coherent superposition of SWP clocks, If [V ,U (g)] = 0 and [V ,U (g)] = 0 for L L Co Co π 0 2 0 0 X − 2 (k−k1 ) i2πn0(k−k1 )/dC all g ∈ G then these gates are also transver- |Ψ(k1)i = Ae σ e |θki , 0 sal for Ecov. Moreover, while these gates are k∈SdC (k1 ) not transversal for Ecov in general, the gates 0 † 0 0 † 0 where A is a normalization constant, S (k0) is the {UL(g )VLUL(g ),UCo(g )VCoUCo(g )}, which can be dC 1 0 R viewed as merely a change of basis on the original set of dC consecutive integers centred about k1 ∈ 00 and ωn ∈ (0, ωd ) is approximately the mean energy gates, are — up to a small error Eg0 — pairs of 0 C 0 transversal gates on Ecov after the reference frames of the clock. k1 can be thought of as an approximate 0 have been measured and a value g has been discerned initial time marked by the clock. The parameters n0, 0 in step 3); which is performed before decoding. See k1 and σ ∈ (0, dC) can be tuned to our convenience. Supplementary materialA for details. Both the Quasi-Ideal and SWP clocks are usually associated with the same projective operators in the 2.1 Finite-sized clocks “time” basis, defined as We now detail the unitary group representations to dC−1 ˆ X T0 which our results apply to. For the logical and phys- tC = k |θkihθk|. (13) dC ical spaces, we consider all compact representations k=0 of the Abelian U(1) group. These can be written in When measuring individual clocks, we will do so in −itHˆL ⊗K −itHˆCo the form UL(t) = e , UCo (t) = e respec- this time basis. When the clocks are entangled, we tively, where using the shorthand S ∈ {L, Co}, the will use a time operator of this form but with the ˆ PdS−1 generators are HS = n=0 ωhS,n|nihn|S for a dS di- projectors |θkihθk| replaced with non-local projectors mensional space, for some frequency ω > 0. They over the multiple entangled clocks, as explained later.

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 4 For a detailed description of the measurement proto- {E, {Ej, Dj}j}, there exists decoding channels D˜j,q for cols, see Supplementary materialB. Ecov such that the worst-case entanglement fidelity of The Quasi-Ideal clock states have been shown to the covariant encoding satisfies yield a good performance in the context of quantum √ 3 2 control [21] and measurement of time [25]. On the 3π dLdP ln (dC) 2 fworst ≥ 1 − (∆hL + ∆hCo) other hand, in [21, 25] the SWP clocks appeared to 4 dC be suboptimal. For the task at hand, we will prove 3/2 ! p ln (dC) that this difference of performance still occurs. + O dLdP 3 . (16) dC 2.2 Entanglement fidelity For the proof, we need to write the encoding- error-decoding scheme in the form described in The figure of merit we will use for quantifying the supplementary materialB when we have a single clock performance of codes which cannot decode perfectly unaffected by errors. After working out the decod- is the worst-case entanglement fidelity. Given a se- ing protocol explicitly, we find a bound that, to- quence of encoding, error and decoding which we la- gether with the results on the entanglement fidelity bel as the channel K : HL → HL, the entanglement of supplementary materialC allows us to derive the fidelity is defined as Theorem in supplementary materialD, after choos- 3 ing σ = ln (dC) for the single clock we measure. fworst(K) = min hφ| K ⊗ I(|φi hφ|) |φi , (14) φ In order to study the role that the different resources play in the current scheme, consider the fol- where I is an identity channel acting on a copy of HL, lowing definitions. We call a clock state ρinc ∈ S(HL), and the optimization is over all bipartite states on with time operator tˆC and unitary group representa- 5 which the map K ⊗ I acts. Having a perfect code as tion {UC(t)}t∈R, t-incoherent if there exists a state those satisfying Eq. (1), is equivalent to fworst(K) = in its periodic orbit which commutes with the time 1, which is achieved with perfect reference frames or operator, namely if there exists t0 ∈ R such that Idealised clocks. For a justification of why this is a † ˆ good measure of approximate codes, see for instance [UC(t0)ρincUC(t0), tC] = 0, (17) [32, 33]. where tˆC is the operator used in step 3) to measure The channel K that we consider here is for the error 6 ˜ the clock. Conversely, clock states which are not t- correction codes {Ecov, {Ej,q, Dj,q}j,q} as described in incoherent are called t-coherent. section2, in which we use clocks that are not idealised. Observe that if one can construct the code E and (M) decoders Dj, then for a given initial clock state ρC , 3 Results in order to construct the covariant code Ecov in Eq. (8), one needs to be able to apply the transversal gates For simplicity, we will only state our bounds to lead- to the clock and code E, and create classical correla- ing order in d , d , d ,L. Here d ≥ d is the di- tions (a separable state) between them. For the de- C P L P Co ˜ mension of the physical space and L is introduced coders Dj, the only additional resource required to later. Except for Theorem4, explicit bounds — apply them is the ability to measure the clocks pro- not just the leading order terms — can be found in jectively in the time basis. The ability to perform such the supplementary material. Furthermore, in all the- operations is always required for the construction of ˜ orems and corollaries in which the Quasi-Ideal clock is any error correction code {Ecov, {Ej,q, Dj,q}j,q}. In involved, the width σ scales logarithmically with dC. the case of t-incoherent clocks, the initial clock state (M) The exact value can be found in the proofs. ρC can then easily be constructed by simply measur- Our first result covers the case in which at least one ing any state on the clock Hilbert space and applying of the clocks is left untouched by erasure errors. the appropriate traversal gate. However, in the case (M) that the initial clock state ρC is t-coherent (such Theorem 1. Consider the covariant code Ecov in Eq. as in the case of Quasi-Ideal clock states in Theorem (8), with M Quasi-Ideal clocks, namely 1), one cannot constructed it with any of the above mentioned operations since they do not allow for the 1 Z T0 E (·) = dt E (·) ⊗ U (t)⊗M ρ⊗M U (t)†⊗M , creation of the necessary coherence (that is, with re- cov T t C QI C 0 0 spect to the basis of tˆ ). Therefore, from a resource- (15) C theoretic standpoint, it is interesting to characterise with error channels {E = E ⊗ ξ } , where {ξ } j,q j q j,q q q how good codes can be if they only use t-incoherent are erasure at known locations of at most M −1 clocks clocks: and {Ej}j are the error channels for the code E. Then + for all M ∈ N , and for all error correcting codes 6Note that in the case of time operators which are not projective POVMs, t-incoherent clocks can still be defined by re- 5 Note that it is sufficient to take an ancillary space of size placing tˆC by the 1st moment operator of the POVM. We do dL. not need to consider such generalisations here however.

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 5 Theorem 2. Consider the covariant code Ecov in Eq. to leading order in dC for all Quasi-Ideal clock states. (8), with one dC dimensional t-incoherent clock ρinc, They are thus approximately minimum√ uncertainty with time operator tˆC defined in Eq. (13), states. By changing the value of σ/ dC from small to large one can achieve the limiting cases of a time Z T0 1 † eigenstate ∆E ∆t, and an energy eigenstate ∆E Ecov(·) = dt Et(·) ⊗ UC(t) ρinc UC(t). (18) T0 0 ∆t. Eq. (16) in Theorem1 corresponds to the opti- 3 mal value of σ = ln (dC), which corresponds to states Allow for no errors on the clock, {Ej,q = Ej ⊗ which are time squeezed (∆E ≥ ∆t), but not by too ξq = Ej ⊗ I}j,q. Then for all error correcting codes much — the states do not become (t-incoherent) time {E, {Ej, Dj}j}, the entanglement fidelity that can be eigenstates. On the other hand, Quasi-Ideal clock achieved is upper bounded by states which are energy squeezed (∆E ≤ ∆t), yield C an entanglement fidelity fworst which scales with dC fworst ≤ 1 − , (19) even worse than t-incoherent states. In-between, one dC finds the non squeezed (∆E = ∆t) Quasi-Ideal states where C > 0 is independent of d . Moreover, −1 C which have the same dC scaling as the t-incoherent there exists a scheme using the Salecker-Wigner-Peres SWP clock. clock, ρinc = ρSWP that achieves The latter Quasi-Ideal clock states behave anal- C∗ ogously to “classical” coherent states — their ex- fworst = 1 − , (20) pectation values in the time and energy bases oscil- d C late like simple harmonic oscillators, and they min- ∗ where C ≥ C is also independent of dC. imize the Heisenberg uncertainty with equal uncertainty in each basis. However, the analogy ends here. The full proof is shown in supplementary material The time squeezed Quasi-Ideal clock states remain E and it goes as follows: we write explicitly the state time squeezed under the application of UC(t) for all of the code after the clock has been measured, and t ∈ R, while squeezed coherent states in one quadra- we apply an arbitrary decoder. Then we write the ture (e.g. position) become squeezed in the comple- entanglement fidelity explicitly and show that the er- mentary quadrature (e.g. momentum) under evolu- ror term decays at best linearly with the dimension tion of its Hamiltonian — broadening in the initial of the clock. The performance achieved by the SWP quadrature basis. Intuitively, this is expected to be clock is calculated by looking at the scaling of the an important difference between squeezed coherent errors in the decoding scheme of Section2. states and squeezed Quasi-Ideal clock states, at least The setup in the above Theorem is the same as in regarding the present task. This is because good de- Theorem1 in the special case that M = 1 and one ex- coding maps would require the states to be squeezed changes the Quasi-Ideal clock for a t-incoherent one. during the entire periodic orbit in the basis in which So by comparing Eqs. (16) and (19) we see that there they are measured in step 3) of the decoding protocol. is essentially a quadratic improvement in the scaling We also find a significant improvement to the en- w.r.t. the clock dimension d . This demonstrates the C tanglement fidelity when one has access to a larger necessity of t-coherent clock states to achieve the im- number of clocks. To achieve it, we embed a large proved scaling in our protocols. Furthermore, in all entangled clock within the Hilbert space of L ∈ N+ our protocols, the final state of the clock after apply- smaller ones, effectively creating a clock of dimension ing a decoder D˜ is t-incoherent and thus in protocols j d(L) := L(d − 1) + 1. in which the initial clock state was t-coherent, the C coherence was “used up” in the process. Theorem 3. Given a covariant error correction code Interestingly, we have observed the same effec- {Ecov, {Ej,q, D˜j,q}j,q} as described in section2, with tive quadratic advantage by using the Quasi-Ideal 0 entanglement fidelity f(dC) ≤ fworst ≤ f (dC) in clock rather than the SWP clock for the related + which a single dC ∈ SN ⊆ N dimensional clock task of reference frame alignment (See Section2 and is used, there exists another covariant error correc- Supplementary materialI) and ticking clocks [25]. tion code with the same error channels {E } in While t-coherent clock states are necessary to j,q j,q which L clocks of dimension dC are entangled, such achieve the improved scaling, they are clearly not suf- 0 that f(L(dC − 1) + 1) ≤ fworst ≤ f (L(dC − 1) + 1), ficient. A case in point are clock states which are for all L, d ∈ N+, s.t. L(d − 1) + 1 ∈ S . incoherent in the energy basis. Indeed, by inverting C C N Eq. (12) one observes that they are t-coherent yet The embedding needed for this theorem leads natu- since they do not evolve in time are useless for this rally to an entangled discrete Fourier transform basis task. on the Hilbert space of L clocks Changing the width σ of the Quasi-Ideal clock state allows one to understand these differences better. The d(L) uncertainty in both the energy and time basis, de- 1 X −i2πnk/d(L) |θk(L)i = p e |En,1i , (21) noted ∆E and ∆t respectively, satisfy ∆E∆t = 1/2 d(L) n=0

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 6 where |En,1i are a non-degenerate set of L For the details about how this inequality follows from LL ˆ 7 the results of [26] see Supplementary materialG. 8 eigenvectors of the generator i=1 HC (see supplementary materialF for details). If we keep the code parameters ∆hCo, ∆hL, dP, dL When this embedding is applied to the SWP clock fixed and scale up the clock size and the number of and Quasi-Ideal clock, it gives rise to a L-site SWP clocks, the combination of lower and upper bounds Entangled clock state, ρSWPE,L = |θk(L)ihθk(L)| Eqs. 23, 25 prove that our construction achieves an and an L-site Quasi-Ideal Entangled clock, ρQIE,L = optimal scaling with both the dimension of the clock L 0 L 0 d and the number of them L, up to the logarithmic |Ψ (k1)ihΨ (k1)|, C factors. Furthermore, this proves that the bound is (k−k0) π 0 2 1 L 0 X − 2 (k−k1 ) i2πn0 tight for all choices of {E, {Ej, Dj}j}. |Ψ (k )i = Ae σ e d(L) |θk(L)i , 1 So far we have only considered erasure errors at k∈S (k0) d(L) 1 known locations on the clock. However, what if one with the new time operator, cannot detect where the error occurred without dam- aging the code? This is the case in the most elemen- dC−1 X T0 tary error correcting codes. If one has many clocks tˆC(L) = k |θk(L)ihθk(L)|. (22) dC and the error occurs with an approximately uniform k=0 error probability distribution over the clock locations, The combination of Theorem3 with Theorems1, one simple approach would be to choose one of the 2, yields an important corollary: clocks, erase the other clocks, and perform error correction with this clock. If the probability of the error Corollary 1. Consider the setup in Theorem1, but M occurring on this clock is low, then this would work with ML-site Quasi-Ideal Entangled clocks ρ⊗ , QIE,L well on average. However, this approach is wasteful rather than M Quasi-Ideal clocks and the erasure now since it does not use the encoded information in the being on at most M − 1 of the L-site Quasi-Ideal En- other clocks and requires a low probability of error on tangled clocks. The worst-case entanglement fidelity a particular clock. Now we will investigate how well of the covariant encoding now satisfies we can recover in the case of unknown phase errors at √ 2 3π d d ln3(L d ) unknown locations which also works well for a small f ≥ 1 − L P C (∆h + ∆h )2 worst 4 L d L Co number of clocks. Consider the case in which one C clock (or an entangled block of L clocks in the sense 3/2 ! p ln (L dC) of Theorem3) whose location is unknown has a ran- + O dLdP 3 . (23) (L dC) dom phase applied to it (i.e. an un-known group ele- ⊗L R ment UC (tph), tph ∈ ). We call this a 2-unknown Similarly, consider the setup in Theorem2, but with a N phase error and denote it ξph,q(tph), where q ∈ and dC dimensional t-incoherent clock state ρinc,L, with a t ∈ R are the unknown site location and phase re- ˆ ph time operator tC(L), rather than a SWP clock with spectively. The following result shows that such errors ˆ time operator tC. The worst-case entanglement fi- are correctable up to an arbitrarily small error. delity of the covariant encoding now satisfies Theorem 4. Consider the covariant code Ecov in C Eq. (8), with 3 blocks of L-site Quasi-Ideal Entan- fworst ≤ 1 − , (24) L dC gled clocks, namely T where C is independent of dC, L; and equality is ob- 1 Z 0 E (·) = dt E (·)⊗U (t)⊗3L ρ⊗3 U (t)†⊗3L, tained for the L-site SWP Entangled clock ρSWPE,L. cov t C QIE,L C T0 0 If one compares the scaling with the number of (26) copies L in Eqs. (23),(24) one finds effectively a with error channels {Ej,q = Ej ⊗ ξph,q(tph)}j,q, q ∈ quadratic advantage in the case of the t-coherent 1, 2, 3 where {Ej}j are the error channels for the code states. The difference in scaling between the two cases E, and ξph,q is a 2-unknown phase error acting on is the same as that found in metrology when compar- one of the three L-site Quasi-Ideal Entangled clocks, N+ R ing the classical shot noise scaling with the quantum ρQIE,L. Then for all L ∈ , q ∈ {1, 2, 3}, tph ∈ Heisenberg scaling [34]. and error correcting codes {E, {Ej, Dj}j}, there exists ˜ The bound Eq. (23) in Corollary1 effectively satu- a decoding channel Dj for Ecov, which is independent rates the upper bound derived in [26], which is proven of the unknown block q and phase tph, such that to hold for all covariant error correction codes gener- 7 p 10π(∆hL + ∆hCo) ln (L dC) ated by isometries. Applying it to our constructions, fworst ≥ 1− dLdP √ 3 L dC it takes the form 11 ln (L dC) 2 p ∆h + O dLdP 2 . (27) L (L dC) fworst ≤ 1 − 2 . (25) 16(∆hCo + LdC) 8 Note that the extra additive factor ∆hP in Eq. (25) is to 7The symbol L denotes the Kronecker sum. be expected since the code E could itself contain a clock.

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 7 ment of the qth clock, is thus incorrect. Since the value of q is also unknown, one cannot simply ignore the corresponding measurement outcome, so our protocol is to order the three measured elapsed times in ascending order and apply a unitary corresponding to the elapsed time which is neither the smallest nor the largest out of the three. There are two possibilities: 1) there was no phase error on this clock, in which case the marked elapsed time is approximately correct. In this case, the phase error could have been large. 2) the phase error occurred on this clock. In this case, the phase error must have been small, since the (incorrect) measured elapsed time is upper and lower bounded by that of the other two clocks (which must both be correct, since there is at most one 2- Figure 1: Illustration of 3 clocks superimposed with a unknown phase error). This corresponds to the case 2-unknown phase error. All clocks are set to have the of Fig.1. 0 0 0 same initial time, k1 = k2 = k3 (red glowing hand). This processing of the measurement outcomes is The three measurement outcomes of the three clocks (black very closely related to majority voting used in a clas- hands) attain similar values with high probability. The yel- sical repetition code. Here the main difference is that low shaded region represents possible values of kα for this 0 0 the outcomes of the clocks are not binary and are un- example, where kα = g (dC − 1)/(2πT0), and g is described in step 3) of the decoding protocol in Section2. An un- likely to agree exactly. known phase error occurs on an unknown clock (for the Note how our protocol does not rely on any assump- purpose of illustration, this is the 2nd clock). It has the tions about the probability distributions over blocks nd effect of shifting the initial time of the 2 clock by an un- q = 1, 2, 3 and phases tph ∈ R over which the 2- known amount kph. The apparent elapsed times are ∆t1 = unknown phase error occurs. However, if one does 0 0 2π(k1 −k1)T0/(dC −1), ∆t2 = 2π(k2 −k2 −kph)T0/(dC −1), assume that the probability of the 2-unknown phase 0 ∆t3 = 2π(k3 − k3)T0/(dC − 1), however, due to the 2- error ξph,q(tph) is small and allows for an arbitrary unknown phase error, ∆t2 will give the incorrect prediction. number of 2-unknown phase errors to occur in an in- Nevertheless, in this example, the phase error is small and no dependent and identically distributed manner, then correction is needed. the above protocol will also work with high fidelity since the probability of two or more phase errors occurring on two or three clocks will be very small. This result extends trivially to the more general Finally, what about infinite dimensional covariant case in which one has M blocks (rather than 3) of codes of finite energy? We leave the motivation to the L-site Quasi-Ideal Entangled clocks in the covari- Outlook (Section4) and state our conclusions here. ant code and has erasure errors at up to M − 3 One can embed our finite dimensional clock into an known locations, and the 2-unknown phase error on infinite dimensional space, and express d in terms one of the 3 blocks of the remaining copies. See C of the mean energy of the clock, denoted hHˆ i. For supplementary materialH for proof. C all the clocks considered here, this corresponds to the To gain an intuitive picture of how the protocol mapping works, it is best to consider the case L = 1 (the gen- d 7→ 2hHˆ i/ω + 1, (28) eral L case is then a direct consequence of Theorem C C 3). For L = 1 the protocol is similar to our previous (up to additive higher order corrections in hHˆCi, ones: we measure the 3 clocks locally in the time basis, for the case of Quasi-Ideal clocks). Our theorems and based upon the information from the 3 measure- and corollary; when framed in this context, provide ment outcomes, we calculate a time g0 = t0 [step 3)] bounds as a function of energy for how well the errors and apply a corresponding decoding map D¯ j,t0 [step on covariant codes can be corrected. In this context, 4)] on the physical space. Due to the classical corre- due to the infinite dimensions of the physical space, lations between the code and clocks, the outcomes of the Eastin-Knill no-go theorem does not apply, so in the 3 clocks are correlated. This is such that, if there principle one could have perfect covariant error cor- were no 2-unknown phase errors, the clocks would all recting codes. However, as we have seen in Section2, indicate approximately the same “elapsed time” and due to the insights of Pauli, covariant error correcting one could correct the code up to an error of order codes based on reference frames can only be decodable Eq. (27) with the knowledge of only one of the 3 without errors (i.e. fworst = 1) iff they use Idealised measurement outcomes. However, when a phase er- clocks which necessitate infinite energy. We conjec- th 0 ror occurs in the q clock, its initial time kq shifts by ture that this is true in general. Specifically, that an unknown phase, making it an unknown variable all error correcting codes which are covariant w.r.t. and the reported elapsed time by the clock measure- any faithful representation of a non-trivial Lie group,

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 8 necessitate infinite energy. This would be an exten- ment fidelity fworst for finite dC have been derived in sion of the Eastin-Knill no-go theorem since all finite the supplementary material which can be evaluated dimensional systems have finite energy but not vice- numerically for experimentally feasible parameters. versa. Were this the case, we would hope that the reference frames used here might be useful in some near-term applications of quantum technologies in which error 4 Outlook correction has started playing a role, such as quantum metrology [37–39]. In particular, given Eq. (25), Given any error correcting code, we have seen how the Quasi-Ideal clock states appear to be almost op- to construct simple classes of approximate codes in timally distiguishable along a whole U(1) orbit. This which compact U(1) groups can be implemented suggests that a setting like that of Theorem4 could transversally. We study a specific scheme using Quasi- be of use in the construction of a metrological scheme Ideal clocks [21] that saturates optimal bounds, ex- aiming to determine unknown parameters (such as the plore the performance of alternative schemes, and also time t of the operator UC (t)). extend to an error model beyond errors at known lo- Covariant infinite dimensional error correcting cations. The present codes are based on the task codes are also of interest. Perhaps most prominently of reference frame alignment, which has been stud- is the example of the hypothesized AdS/CFT dual- ied quite extensively [15] and also formalized within ity between quantum gravity in asymptotically AdS the resource theory of asymmetry [35, 36], which we space (known as the bulk) and a conformal field the- hope will be of further use in the context of error cor- ory on the boundary, where the theory on the bulk rection [1, 26]. Furthermore, since the clocks need to and boundary are related via quantum error correct- be measured projectively in our protocols, a preferred ing codes [40]. Specifically, the bulk constitutes the basis naturally arises and necessary conditions w.r.t. logical space while the boundary is the physical/code this basis for a quadratic advantage are identified. space. Global symmetries in the bulk should cor- An obvious extension of our results is to groups be- respond to symmetries on the boundary which con- yond U(1). This requires the construction of more serve the local structure of the theory — in a simi- involved reference frames, such as finite-sized quan- lar spirit to how transversal gates act globally in the tum “gyroscopes” (for the case of SU(2)). We hope logical space while locally in the physical space [c.f. that our techniques may serve as a starting point for Eq. (3)]. It is argued that variants of the Eastin- generalising the results found here in this direction Knill no-go theorem have important interpretations — perhaps together with the construction of approx- in AdS/CFT: all global symmetries in the bulk (both imate frames for general groups from [31]. continuous and discrete) are ruled out, since their An important question is to understand and char- existence would be in contradiction with the struc- acterize the sort of error models and implementations ture of the correspondence9 [41, 42]. A related is- for which the present scheme is adequate. We have sue is time dynamics which is given by a U(1) sym- mostly explored the case of erasure errors at known ˆ ˆ metry with representations e−itHblk and e−itHbdy for locations, and a type of dephasing errors on the clocks Hamiltonians Hˆblk, Hˆbdy on the bulk and boundary at unknown locations. While these are quite natural ˆ ˆ error models, it may well be that with more involved respectively. Often Hblk, Hbdy are quasi-local and schemes of codes involving clocks, other types of er- thus the corresponding U(1) group action has to also rors can be dealt with. preserve this local structure. An approximate U(1) Also, the results of [26] show tight bounds that can- covariant code with these properties for finite dimen- not be overcome only for erasure errors at known lo- sional Hamiltonians has been developed in [43] us- cations, and it would be interesting to know if their ing techniques from [44]. By choosing the number bounds are also tight for other error models. We of code blocks to be one i.e. K = 1, our protocols suspect that the error scaling in Theorem4 is op- allow for such bulk-boundary time dynamics covari- timal, even though it is only as good as that of the t- ance for arbitrary (e.g. quasi-local) finite dimensional incoherent clock states when the error model was that Hamiltonians on HL and HCo. An open question is of erasure at know locations. Intuitively, the limiting whether further work may allow for clocks with inter- factor in this case is that the 3 blocks of clocks are acting quasi-local Hamiltonians too. only classically correlated (separable states). While Since both theories of the duality are infinite- one can easily construct entangled counterparts, it is dimensional but of finite energy locally, further quan- not clear how this can help when the errors occur over titative variants of the Eastin-Knill no-go theorem for the unknown blocks. infinite dimensional physical spaces, seem more ap- One key point to be determined is whether the er- 9 ror bounds shown here, even if they are essentially Local CFT operators acting on a spatial region R in the boundary, can only access information in a limited region in optimal, are not too large to be useful in practice for the bulk via entanglement wedge reconstruction. This is in- some type of architecture. In the case of Theorem compatible with the existence of charged localised objects in 1 and Corollary1, explicit bounds on the entangle- the bulk.

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 9 propriate (for instance, by taking into account the Inf. Theory, 57(9):6272–6284, 2011. DOI: “average energy density” of the code space). Our re- 10.1109/TIT.2011.2161917. sults suggest that given sufficient energy, the preser- [3] Sergey Bravyi and Robert König.Classification vation of local symmetries in the boundary is at least of topologically protected gates for local stabi- approximately possible. lizer codes. Phys. Rev. Lett., 110(17):170503, Finally, it is worth noting that the extension of 2013. DOI: 10.1103/PhysRevLett.110.170503. a physical Hilbert space by including clocks has [4] Fernando Pastawski and Beni Yoshida. Fault- been proposed in a different context before. The tolerant logical gates in quantum error-correcting aim of the Wheeler-DeWit equation is to describe codes. Phys. Rev. A, 91(1):012305, 2015. DOI: a time-static theory of quantum gravity in which 10.1103/PhysRevA.91.012305. locally one finds dynamics [45]. This motivated [5] Tomas Jochym-O’Connor, Aleksander Kubica, the Page-Wooters mechanism for describing how a and Theodore J Yoder. Disjointness of stabi- quantum clock can allow for time evolution to follow lizer codes and limitations on fault-tolerant logi- from a static universe using Idealised clocks [46]. In cal gates. Phys. Rev. X, 8(2):021047, 2018. DOI: supplementary materialJ we show that when the 10.1103/PhysRevX.8.021047. logical space has the trivial group representation of [6] Benjamin J Brown, Daniel Loss, Jiannis K the U(1) symmetry, our formalism is an example Pachos, Chris N Self, and James R Woot- of an approximate Page-Wooters mechanism where ton. Quantum memories at finite temperature. the approximation comes from using non Idealised Rev. Mod. Phys., 88(4):045005, 2016. DOI: clocks. 10.1103/RevModPhys.88.045005. [7] Bryan Eastin and Emanuel Knill. Restric- tions on transversal encoded quantum gate sets. Note on Related Work: Around the time this Phys. Rev. Lett., 102(11):110502, 2009. DOI: work was being developed, it was realized by the au- 10.1103/PhysRevLett.102.110502. thors that a complementary approach to characteriz- [8] Sergey Bravyi and Alexei Kitaev. Universal ing how well the Eastin-Knill no-go theorem can be quantum computation with ideal clifford gates circumvented by allowing for a small error, was being and noisy ancillas. Phys. Rev. A, 71(2):022316, developed independently by the authors of [26]. We 2005. DOI: 10.1103/PhysRevA.71.022316. thank them for their community spirit. [9] Emanuel Knill, Raymond Laflamme, and W Zurek. Threshold accuracy for quantum computation. ArXiv:quant-ph/9610011, 1996. URL Acknowledgements https://arxiv.org/abs/quant-ph/9610011. [10] Héctor Bomb´ın and Miguel Angel´ Mart´ın- We thank useful conversations with Benjamin Brown, Delgado. Topological computation without Philippe Faist, Daniel Harlow, Tamara Kohler and braiding. Physical review letters, 98(16):160502, Rob Spekkens. This work was initiated at QIP 2018 2007. DOI: 10.1103/PhysRevLett.98.160502. hosted by QuTech at Delft University of Technol- ogy. M.P.W. acknowledges funding from the Swiss [11] Adam Paetznick and Ben W Reichardt. Uni- National Science Foundation (AMBIZIONE Fellow- versal fault-tolerant quantum computation with ship PZ00P2 179914) in addition to the NCCR QSIT. only transversal gates and error correction. This research was supported in part by Perimeter In- Phys. Rev. Lett., 111(9):090505, 2013. DOI: stitute for Theoretical Physics. Research at Perimeter 10.1103/PhysRevLett.111.090505. Institute is supported by the Government of Canada [12] Tomas Jochym-O’Connor and Raymond through the Department of Innovation, Science and Laflamme. Using concatenated quantum Economic Development and by the Province of On- codes for universal fault-tolerant quantum gates. tario through the Ministry of Research, Innovation Phys. Rev. Lett., 112(1):010505, 2014. DOI: and Science. 10.1103/PhysRevLett.112.010505. [13] Héctor Bomb´ın. Gauge color codes: optimal transversal gates and gauge fixing in topologi- References cal stabilizer codes. New J. Phys., 17(8):083002, 2015. DOI: 10.1088/1367-2630/17/8/083002. [1] Patrick Hayden, Sepehr Nezami, Sandu Popescu, [14] Theodore J Yoder, Ryuji Takagi, and Isaac L and Grant Salton. Error correction of quantum Chuang. Universal fault-tolerant gates on reference frame information. ArXiv:1709.04471, concatenated stabilizer codes. Phys. Rev. 2017. URL https://arxiv.org/abs/1709. X, 6(3):031039, 2016. DOI: 10.1103/Phys- 04471. RevX.6.031039. [2] Bei Zeng, Andrew Cross, and Isaac L [15] Stephen D Bartlett, Terry Rudolph, and Chuang. Transversality versus universality Robert W Spekkens. Reference frames, su- for additive quantum codes. IEEE Trans. perselection rules, and quantum information.

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A Compatibility with the transversal gates of E

In our schemes we start with a code E(·) which is in principle arbitrary. We can ask what happens if this code already has a group of transversal or covariant gates (possibly of ﬁnite order as granted by [7]). Does that covariance remain after the reference frame is added? Let {VL,VCo,VF} be an element of a representation of the group of transversal gates of E on the logical, physical and reference frame spaces. Thus:

† ⊗K †⊗K E VL(·)VL = VCo E(·)VCo , (29) where ⊗K represents the tensor product structure of the physical space of E. Does the same symmetry hold for E ? Let us write cov Z † † ⊗M (M) †⊗M Ecov(VL(·)VL ) = dg Eg(VL(·)VL ) ⊗ UF(g) ρC UF(g) , (30) G whereas we have, on the other hand

Z ⊗K †⊗K † ⊗K †⊗K ⊗M (M) †⊗M † VCo ⊗ VF Ecov(·) VCo ⊗ VF = dg VCo Eg(·)VCo ⊗ VF(UF(g) ρC UF(g) )VF , (31) G

⊗K It is clear that Eqs. (30) and (31) coincide when i) both commutations [VCo,UCo(g) ] = 0 and [VL,UL(g)] = 0 hold for all g ∈ G and ii) VF acts trivially on the clocks. Otherwise, the covariance/transversality is lost. While the latter can be obtained by choosing VF to act trivially on the clocks, the former seems much more restrictive. However, there is a sense in which we might still have transversal gates in the code, if we are able to set the ⊗K representation UCo(g) on the physical space to the trivial case. After measuring the clocks and obtaining 0 ⊗K some outcome g , one can apply the gates VP ⊗ VF. Then, applying the decoder, the resulting state is

† 0 0 † VLUL(g )ρLUL(g ))VL . (32)

If [VL,UL(g)] = 0 we may just apply UL(g) and end up with the desired state to which the gate has been applied † 0 transversally. Otherwise, what has happened is that we have applied the gate VLUL(g ). We have knowledge

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 12 0 † 0 † of g , and so to end up with VLρLVL we may just apply the unitary VLUL(g )VL . This assumes that one can apply the logical gate both at the logical and the physical level. This discussion gives a further example, together with the errors of the main results, of why it is advantageous to choose the generator HˆCo to be trivial HˆCo ∝ 1P, if possible.

B General encoding-decoding error with ﬁnite clocks

Here we explain the form for the encoding-error-decoding protocol for all the schemes of the present work. We refer to this discussion repeatedly in the diﬀerent proofs. We will here assume that the M clocks are of product form. We will see later that this construction is general enough for our purposes even when the clock are entangled, by considering further divisions of the local sites considered here.

We deﬁne the dimensions of the input and output of the error correcting code E to be dL, dP. The dimension of the ith clock is di. If all clocks have the same dimension we use the notation dC := d1 = d2 = .... If we do not write the range over a summation, it will be over the full range.

−itHˆC For our case, the unitary representation is UC(t) = e where HˆC is deﬁned in Section 2.1. The integral measure over the group is dg = dt/T0 on the interval [0,T0]. The encoded state in which M clocks in states ρC,i with i ∈ {1, ..M} are used has the form of Eq. (8)

M 1 Z T0 O E (ρ ) = dt E (ρ ) U (t)ρ U † (t). (33) cov L T t L C C,i C 0 0 i=1

Let us ﬁrst calculate the integral over t by writing the unitary operators UC(t), UL(t), UCo(t) in their eigenbasis:

dL−1 † X −iωt(hL,n−hL,n0 ) 0 0 E UL(t)ρLUL(t) = e E (L hn | ρL |niL |n ihn|) , (34) n,n0=0 and thus,

dP−1 dL−1 X X −iωt(hCo,q −h 0 +hL,n−h 0 ) 0 Et (ρL) = e Co,q L,n Eq,q0,n,n0 (ρL) |qihq |P, (35) q,q0=0 n,n0=0 with 0 0 0 Eq,q0,n,n0 (ρL) := P hq| E (L hn | ρL |niL |n ihn|) |q iP . (36)

For simplicity, let us deﬁne Q := hCo,q − hCo,q0 + hL,n − hL,n0 . Therefore

1 Z T0 Ecov(ρL) = dt Et (·) ⊗ ρC,1(t) ⊗ ρC,2(t) ⊗ ... ⊗ ρC,M(t), (37) T0 0

dP−1 dL−1 T0 1 Z 0 0 X X X −iωt(Q+r1−r +...+rM −r ) 0 = dt e 1 M Eq,q0,n,n0 (ρL) |qihq | (38) T0 q,q0=0 n,n0=0 0 0 0 ~rM ,~rM

⊗ hr1| ρC,1 |r1i ⊗ ... ⊗ hrM | ρC,M |rM i (39)

dP−1 dL−1 X X X 0 = δ 0 0 E 0 0 (ρ ) |qihq | (40) Q+r1−r1+...+rM −rM ,0 q,q ,n,n L q,q0=0 n,n0=0 0 ~rM ,~rM 0 0 0 0 ⊗ hr1| ρC,1 |r1i |r1ihr1| ⊗ ... ⊗ hrM | ρC,M |rM i |rM ihrM |, (41)

ˆ where {|rii} is the eigenbasis of the generator HC of the i-th clock, δ(·,·) is the Kronecker-Delta function and

d −1 d −1 X X1 XM = ... . (42) 0 0 0 ~rM ,~rM r1,r1,=0 rM ,rM ,=0

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 13 Since by assumption the last N + 1,...,M clocks may be lost due to erasure errors, we can trace them out:

dP−1 dL−1 X X X 0 tr [E (ρ )] = δ 0 0 E 0 0 (ρ ) |qihq | (43) CN+1...CM cov L Q+r1−r1+...+rN −rN ,0 q,q ,n,n L q,q0=0 n,n0=0 0 ~rM ,~rM 0 0 0 0 ⊗ hr1| ρC,1 |r1i|r1ihr1| ⊗ ... ⊗ hrN | ρC,N |rN i |rN ihrN | (hrN+1| ρC,N+1 |rN+1i ... hrM | ρC,M |rM i) (44)

dP−1 dL−1 X X X 0 = δ 0 0 E 0 0 (ρ ) |qihq | (45) Q+r1−r1+...+rN −rN ,0 q,q ,n,n L q,q0=0 n,n0=0 0 ~rN ,~rN 0 0 0 0 ⊗ hr1| ρC,1 |r1i|r1ihr1| ⊗ ... ⊗ hrN | ρC,N |rN i |rN ihrN |. (46) So by comparing Eqs. (46), (41), we observe that after tracing out the additional clocks, the resultant channel is of the same form as the original channel, when evaluated for the renaming number of clocks. We now assume that an error due to the environment occurs. This means that we apply a CPTP error map Ej : HP → HP corresponding to an error j. We thus denote

j 0 0 0 Eq,q0,n,n0 (ρL) := P hq| Ej (E (L hn | ρL |niL |n ihn|)) |q iP . (47) We now measure the remaining clocks, performing projective measurements in the “time” basis

( d −1 )dm 1 m X −i2πnk/dm |θkim = √ e |nim (48) dm n=0 k=0 on the mth clock. Let us drop the m subindex for simplicity of notation. The state after the outcome ~k := k1, k2, . . . , kN is:

Thus, after postselecting on a particular outcome represented with the vector ~k, we obtain the following state on the Hilbert space of the output of EjE(·)

Z T0+t0 ~ 1 −iωQt R Z FQ(k) := dt e hθk1 |ρC,1(t)|θk1 i ... hθkN |ρC,N(t)|θkN i , ∀ t0 ∈ ,Q ∈ , (58) T0 t0 where for simplicity we will assume that all the clock dimensions are equal d1 = d2 = ... = dN =: dC. This deﬁnition has the following properties:

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 14 ~ 1) Since the integrand has period T0, FQ(k) is independent of t0 and we can set it to any preferred real number to help perform the calculations. 2) We can perform the change of variable t = t0 + a, a = −l 2π , l ∈ Z, to show ωdC

2π i2πlQ/d Z T0+t0−l ωd ~ e C 0 −iωQt FQ(k) = dt e hθk1+l|ρC,1(t)|θk1+li ... hθkN +l|ρC,N(t)|θkN +li (59) 2π T0 t0−l ωdC i2πlQ/d ~ = e FQ(k + l), (60) ~ ~ ~ where the jth vector component of k+l is deﬁned by [k+l]j := [k]j +l for all vector component j. Note how ~ ~ this expression is invariant under l 7→ l + jdC, j ∈ Z. So w.l.o.g. we can exchange k + l with [k + l](mod. d), where the map acts element-wise on vectors and (mod. d) denotes modular d arithmetic.

3) By inspection, we see that FQ is invariant under any pairwise interchanges

kq ↔ kr and ρC,q(t) ↔ ρC,r(t), q, r ∈ 1,...,N, (61) ~ where kq and kr are the qth and pth vector elements of k.

4) By Eq. (57), it follows that FQ encodes the measurement outcome probabilities, ~ p~k = F0(k). (62)

This has important consequences when property 2) is taken into account. For example, in the case of one clock, it implies that all measurement outcomes are equally likely, so 1 p~k = ∀ k1 = 0, . . . , dC − 1. (63) dC

~k Thus using property 5), we can write ρP in Eq. 52 as

It is convenient to introduce an arbitrary phase kα ∈ R which for our purposes has to be deﬁned modulo dC. It will depend on ~k, the measurement outcomes. We will choose it later depending on the particular covariant error correcting code set-up and protocol. We can now write

dP−1 dL−1 ~ ~k X X FQ(k) j 0 ρ = E 0 0 (ρ ) |qihq |. (69) P ~ q,q ,n,n L q,q0=0 n,n0=0 F0(k) dP−1 dL−1 " ~ # ! X X FQ(k) = e2πikαQ/dC − 1 + 1 (70) ~ q,q0=0 n,n0=0 F0(k)

⊗K† j † 0 ⊗K UP (tα)Eq,q0,n,n0 UL(tα)ρLUL(tα) |qihq |UP (tα) (71) =U ⊗K†(t ) Eˆ0 (˜ρ ) + Ej(˜ρ ) U ⊗K (t ), (72) P α ~k L L P α

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 15 where † T0 ρ˜L := UL(tα)ρLUL(tα), tα := kα , (73) dC and dP−1 dL−1 " ~ # FQ(k) ˆ0 X X 2πikαQ/dC j 0 E (·) := e − 1 E 0 0 (·)|qihq | (74) ~k ~ q,q ,n,n q,q0=0 n,n0=0 F0(k) In order to proceed with the decoding, let us deﬁne the function p(Q,~k) as: ~ FQ(k) p(Q,~k) = 1 − e2πikαQ/dC , (75) ~ F0(k) where p(Q,~k) is a complex number for which we expect |p(Q,~k)| 1 (how small will determine the size of the error of the particular sheme). The phase kα determines the group elements we should apply in the decoding procedure, as per Eq. (73). ⊗K 1 Since by assumption the group action only acts non-trivially in the physical space, UP (tα) = UCo(tα)⊕ P/Co, the decoder may take the form

¯ −1 † Dj,tα (·) = UL(tα)E (UCo(tα)E (Dj(·)) UCo(tα)) UL(tα), (76)

−1 where E (·) = Dj (Ej(·)) is the inverse channel for the encoder E and Dj the decoding map of E. ~k With it we see that if we apply the decoder to the state ρP, we obtain

~k ˆ ρP = ρL − E~k(ρL), (77) where we deﬁne

dP−1 dL−1 ˆ X X ~ ¯ j †⊗K 0 ⊗K E~k(ρL) = p(Q, k) Dj,tα Eq,q0,n,n0 (ρL) UP (tα)|qihq |UP (tα) . (78) q,q0=0 n,n0=0

~ ~ Since each of the outcomes k occurred with probability F0(k), the ﬁnal decoded state, averaged over all measurement outcomes, is of the form X ~ ˆ K(ρL) = ρL − F0(k)E~k(ρL). (79) ~k In AppendixC we show how a bound on the entanglement ﬁdelity of the code follows from this expression. The last fact that then needs to be shown is the form of the RHS of Eq. (75) for particular cases of clocks.

C Calculation of the entanglement ﬁdelity

We here give the bounds on the entanglement ﬁdelity which will be used to prove the main results. Again, it is deﬁned as fworst(K) = min hφ| K ⊗ I(|φi hφ|) |φi , (80) φ where the minimization is over all the pure bipartite states |φi. Motivated by the discussion in AppendixB leading to Eq. (79), let us start with the assumption that after the encoding, error and decoding steps, the map has the following form when applied to a state ρ.

K(ρL) = ρL − Eˆ(ρL), (81)

A result from [32] allows us to lower bound the entanglement ﬁdelity in terms of Eˆ(ρL). If for all pure states on a single system on Hdin the following holds hφ| K(|φi hφ|) |φi ≥ 1 − , (82)

3 ˆ ˆ then we have that fworst(K) ≥ 1 − 2 . Given that hφ| E(|φi hφ|) |φi ≤ ||E(|φi hφ|)||1, we can choose = ˆ max|φihφ| ||E(|φi hφ|)||1, where the optimization is over all pure states, thus obtaining 3 fworst(K) ≥ 1 − max ||Eˆ(|φi hφ|)||1. (83) 2 |φihφ|

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 16 The next step is to give an upper bound to the 1-norm of Eˆ that holds for any pure state. First, by the triangle inequality ˆ X ~ ˆ X ~ ˆ ||E||1 = || F0(k)E~k||1 ≤ F0(k)||E~k||1. (84) ~k ~k As seen in AppendixB, in all of the cases considered here, the operator Eˆ is deﬁned as Eˆ = D¯ Eˆ0 where ~k ~k j,tα ~k ¯ Dj,tα is the decoder, and

dout−1 din−1 X X K Eˆ0 = U †⊗ (t )|qihq0|U ⊗K (t ) hq| E E (|ni hn| hn| ρ |n0i) |q0i p(Q,~k), (85) ~k Co α Co α j 0 L q,q0=0 n,n0=0 where p(Q,~k) is in principle constrained by complete positivity and trace preservation (as deﬁned in Eq. (78)). By contractivity of CPTP maps, ||Eˆ || ≤ ||Eˆ0 || . Now, we can write ~k 1 ~k 1

din−1 ˆ0 X X 0 0 0 ~ ||E~ ||1 = max | hq| Ej(|ni hn | hn| ρL |n i) |q i p(Q, k)| (86) k q0 q n,n0=0   din−1 X 0 X 0 0 ~ = max tr |q i hq| Ej(|ni hn | hn| ρL |n i)p(Q, k) (87) q0 q n,n0=0

din−1 X X 0 0 ~ ≤ max ||Ej( |ni hn | hn| ρL |n i p(Q, k))||1 (88) q0 q n,n0=0

din−1 X X 0 0 ~ ≤ ||Ej||1−1 max || |ni hn | hn| ρL |n i p(Q, k)||1 (89) q0 q n,n0=0 where the third line follows from the inequality [47],

† |tr[BA ]| ≤ ||A||p||B||r, (90) with 1 = 1/p + 1/r and choosing p = 1, r = ∞, with B = |q0i hq| and A the rest. The fourth line follows from ||Ej (X)||1 the deﬁnition of the 1 − 1 norm for CPTP maps, which is ||Ej||1−1 = sup , where the optimization is X ||X||1 over operators on the Hilbert space of the input of Ej .√ By contractivity, we have that ||Ej||1−1 ≤ 1. The last step is to bound the left-over 1-norm. By using ||A||1 ≤ din||A||2 we can bound the 2 norm instead, as  2 din−1 din−1 X 0 ~ 2  X 0 ~  || |ni hn| hn| ρL |n i p(Q, k)||2 = tr  |ni hn| hn| ρL |n i p(Q, k)  (91) n,n0=0 n,n0=0

din−1 X 00 00 ~ ~ = hn| ρL |n i hn | ρL |ni p(hCo,q − hCo,q0 + hL,n − hL,n00 , k)p(hCo,q − hCo,q0 + hL,n − hL,n00 , k) (92) n,n00=0

din−1 ~ 2 X 00 00 ~ 2 2 ~ 2 ≤ max |p(Q, k)| hn| ρL |n i hn | ρL |ni = max |p(Q, k)| tr[ρL] = max |p(Q, k)| , Q Q Q n,n00=0 since ρL = |φihφ|. Generically in our examples the optimization is over the range Q ∈ {−(∆hL + † ∆hCo),..., ∆hL + ∆hCo} as deﬁned in Sec. 2.1, but if the representation UCo(t) is trivial (that is, the generator is the identity) then Q ∈ {−∆hL, ∆hL}, which yields the best performance. Finally we have ˆ p X ~ p ~ ||E~ ||1 ≤ din max max |p(Q, k)| = dindout max |p(Q, k)|, (93) k q0 Q Q q ~ ~ which holds for any |φihφ|. Since outcome k occurs with probability p~k = F0(k), we have 3p X ~ ~ fworst(K) ≥ 1 − dindout F0(k) max |p(Q, k)|. (94) 2 Q ~k ~ ~ For particular sets of clocks and schemes, we will give bounds for F0(k) max |p(Q, k)|, and then use Eq. (94) to Q estimate the entanglement ﬁdelity.

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 17 D Proof of Theorem1, a bound for a single remaining clock

Following the discussion of appendicesB andC, here we provide the proof of the bound on the quantity that we use to bound the entanglement ﬁdelity for the case in which, after erasure errors, only a single clock is ensured not to be erased. For simplicity of notation, let us use the shorthand d = dC. The ﬁrst step is to notice that the discussion of AppendicesB andC shows we only have to compute a bound on max |p(Q,~k)|, where Q

~ FQ(k) −p(Q,~k) = e2πikαQ/d − 1, (95) ~ F0(k) as deﬁned in Eq. (75). Then, any upper bound on |p(Q,~k)| ∀ Q,~k will yield a lower bound on the worst-case entanglement ﬁdelity as per Eq. (94). We now compute this ratio explicitly for a single Quasi-Ideal clock, which we do following the notation of AppendixB. Let us recall that

Z T0+t0 ~ 1 −iωQt R Z FQ(k) = dt e hθk1 |ρC,1(t)|θk1 i , ∀ t0 ∈ ,Q ∈ . (96) T0 t0

0 In this case of this proof, we let kα = k1 − k1. After measuring the single clock, and right before applying the decoding, the state is that of Eq. (72), that is

dP−1 dL−1 " ~ # ! ~ X X FQ(k) ρk = e2πikαQ/d − 1 + 1 (97) P ~ q,q0=0 n,n0=0 F0(k)

⊗K† j † 0 ⊗K UCo (tα)Eq,q0,n,n0 UL(tα)ρLUL(tα) |qihq |UCo (tα). (98)

~ Using property 2) of FQ(k) (see Eq. (60)), we observe that the first line of Eq. (98) is invariant under the ~ −i2πlQ/d ~ change of variable k1 7→ k1 + l (Since FQ(k) maps to e FQ(k) while kα to kα + l). Thus the situation is the same for all measurement outcomes k1. As such we will only need to concern ourselves with one of the 0 measurement outcome k1, which we choose for convenience to be k1 = kMa := max{Sd1 (k1)}, where the set 0 Sd(k1) is described in the main text is defined to be n d do S (k0) = k : k ∈ Z, − ≤ k0 − k < , (99) d 1 2 1 2 0 R 0 0 for k1 ∈ . Thus kMa = k1 + d/2 if k1 + d/2 is integer (we will assume this is the case here, but one can find analogous results for when it is half integer)10 We are now interested in bounding the overlap

−itHˆc 0 hθkMa |e |ψnor(k1)i . (100)

This can be bounded with the results in Theorem 8.1 on page 151 of [21]. The theorem tells us that for all t ∈ R d −itHˆc 0 0 e |Ψnor(k1)i = |Ψnor(k1 + t )i + |εi , (101) T0 where 0 X 0 |Ψnor(k1)i = ψnor(k1; k) |θki , (102) 0 k∈Sd(k1 ) with k−k0 2 2 1 π 0 i2πn1 d 0 −( σ ) (k−k1 ) ψnor(k1; k) = Ae e . (103)

The error term εc := |||εi||2 satisﬁes

d d εc(t) < |t| εtotal + |t| + 1 εstep + εnor, (104) T0 T0

10 0 Recall that k1 is the value at which the Gaussian amplitude of the initial Quasi-Ideal clock state is centred.

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 18 where ! ! 2 2 α0 1 1 − πσ α2 1 1 d 1 − πd ε = 2πAd 2σ + + e 4 0 + + + + e 4σ2 total 2 −πσ2α πd πd2 2 2 2πσ 1 − e 0 − 2 − 2σ 2πd 1 − e σ 1 − e σ2 (105) and A is deﬁned in Eq. (124), and α0 ∈ (0, 1] is deﬁned in [21] to be:

Deﬁnition 1. (Distance of the mean energy from the edge of the spectrum) We deﬁne the parameter α0 ∈ (0, 1] as a measure of how close n0 ∈ (0, d − 1) is to the edge of the energy spectrum, namely 2 α = min{n , (d − 1) − n } (106) 0 d − 1 0 0 2 = 1 − 1 − n0 ∈ (0, 1]. (107) d − 1

The maximum value α0 = 1 is obtained for n0 = (d − 1)/2 when the mean energy is at the mid point of the energy spectrum, while α0 → 0 as n0 approaches the edge values 0 or d − 1. Furthermore,

2 − πd εstep < 2Ae 4σ2 (108)

√  πd2 2  − − πσ 40 2 e 2σ2 σ e 2 εnor ≤ + √ (109)  − 2πd −πσ2  3σ 1 − e σ2 2 1 − e where on the r.h.s. of the inequality in Eq. (109) we have assumed σ ≥ 1 and d = 2, 3, 4,... (tighter bounds can be found in Section E.A.2 of [21]). We can now get back to estimating the overlap Eq. (100). We ﬁnd that for all t ∈ [0,T0],

Thus using ω = 2π/T0 and the change of variable x = t/T0 − 1/2, and setting t0 = 0 such that integral is over t ∈ [0,T0], we ﬁnd

Z T0 ~ 1 −iωQt FQ(k) = dt e hθkMa |ρC,1(t)|θkMa i (119) T0 0 T 2 1 Z 0 d 2 t 1 −iωQt 2 −2π( σ ) T − 2 = dt e A e 0 + ε1(t/T0) (120) T0 0 1 Z 2 2 −πiQ −i2πQx 2 −2π( d ) x2 = e dx e A e σ + ε1(x + 1/2) (121) 1 − 2 Z ∞ 2 −πiQ −i2πQx 2 −2π( d ) x2 = e dx e A e σ + ε2 (122) −∞ 2 −πiQ 2 σ − π ( σ ) Q2 = e A √ e 2 d + ε2, (123) 2d

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 19 ~ Recalling that F0(k) = 1/d and that, following Section E.1.1 of [21] we have √ 2 A = 2/σ + εA, (124)

εA is speciﬁed later in Eq. (131), we can write

~ 2 FQ(k) 2πik Q/d 2 σ − π σ Q2 πi(2k −d)Q/d e α = A √ e 2 ( d ) + dε e α (125) ~ 2 F0(k) 2 σ π σ 2 2 − ( ) Q πi(2kα−d)Q/d = (1 + √ εA)e 2 d + dε2 e (126) 2 2 σ − π ( σ ) Q2 = (1 + √ εA)e 2 d + dε2, (127) 2 0 where in the last line we have used kα = k1 − k1 = d/2 in order to get rid of the phase. Finally, we need to ﬁnd bounds for both ε2 and εA. Let us start with ε2, deﬁned as

1 Z 2 π d2 2 − 2 2 ε2 = ε1(x + 1/2)dx + A e σ , (128) 1 − 2 where ε1 is deﬁned in Eq. (118). By inspection, we have 2 ε1(t) ≤ 2εc(t) + εc(t) (129) where εc is reproduced in Eq. (104). Hence going back to the deﬁnition of 2 in Eq. (128), we can then conclude that π d2 2 2 − 2 2 |ε2| ≤ 2 (dεtotal + (d + 1)εstep + εnor) + (dεtotal + (d + 1)εstep + εnor) + A e σ (130)

To bound the error term εA, we use the results from Eq. (477-483) from Section E.1.1 on page 84 of [21]. The bound is

ε¯1 +ε ¯2 |εA| ≤ , (131) √σ √σ − ε¯ − ε¯ 2 2 1 2 where

2 2 − πd − πσ 2e 2σ2 σ 2e 2 ε¯1 := , ε¯2 := √ . (132) − 2πd −πσ2 1 − e σ2 2 1 − e

Thus, the leading contribution to εA comes from ε¯2, and the leading contribution to ε2 comes from the ﬁrst term of εtotal and the last of εnor. It can be seen in Eq. (105) that the ﬁrst term of the sum only decays as 2 √ − πσ α2 σde 4 0 . Thus, we can write

˜ ~ 2 2 2 2 FQ(k) 2πik Q/d − π σ Q2 − πσ −d2 3 − πsσ − πσ −d2 e α = e 2 ( d ) 1 + O e 2 + O(e ) + O(d e 4 ) + O(e 2 ) + O(e ) . (133) ~ F0(k) Going back to the deﬁnition of Eq. (75), this gives the following bound on max |p(Q,~k)|, Q

2 2 2 2 − π σ Q2 − πσ −d2 3 − πsσ − πσ −d2 max |p(Q,~k)| ≤ 1 − min e 2 ( d ) 1 + O e 2 + O(e ) + O(d e 4 ) + O(e 2 ) + O(e ) , Q Q (134)

2 2 2 2 − π σ (∆h +∆h )2 − πσ −d2 3 − πsσ − πσ −d2 ≤ 1 − e 2 ( d ) L Co 1 + O e 2 + O(e ) + O(d e 4 ) + O(e 2 ) + O(e ) , (135) from which, given the discussion of AppendixC, the bound on the entanglement ﬁdelity follows. To ﬁnalise the proof, we need to pick an optimal value of σ. Choosing σ = ln3(d), we obtain the best scaling of the leading error term in d, which gives

2 3 π ln (d) 2 3/2 ! − 2 d Q ln (d) max |p(Q,~k)| ≤ 1 − min e + O (136) Q Q d3 2 ! π ln3(d) ln3/2(d) = (∆h + ∆h )2 + O , (137) 2 d L Co d3

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 20 from which, given the discussion of AppendixC, the bound on the entanglement ﬁdelity follows. The choice σ = ln3(d) corresponds to time squeezed Quasi-Ideal lock state (i.e. ∆t < ∆E, where ∆t and ∆E are the standard deviation in the time and energy bases as described in the main text).

E Proof of bound for clocks diagonal in the time eigenbasis

We here give a proof of Theorem2, which is a bound on the entanglement ﬁdelity of clocks which are, at some point in their periodic orbit, diagonal in the basis in which the are measured — the time eigenbasis {|θki}, conjugate to the energy eigenbasis. That is, we assume that there exits t0 ∈ R such that the initial clock state ρinc satisﬁes dC−1 † X 2 UC(t0)ρincUC(t0) = |Ak| |θkihθk| :=ρ ˜C,1, (138) k=0

dC−1 for some probability amplitudes { |Ak| }k=0 . Therefore,

dC−1 dC−1 2 |Ak| 0 0 X 2 X −i2πk(r1−r1)/dC hr1|ρ˜C,1|r1i = |Ak| hr1|θki hθk|r1i = e . (139) dC k=0 k=0

0 Unlike in the previous case of an optimal clock, the above equation only depends on the difference r1 −r1 rather 0 than r1, r1 individually (c.f. the Quasi-Ideal clock case). Before preceding, we need to write the covariant encoding channel in a way which reflects the form of Eq. (138). Note that Eq. (18) in the theorem has the property 0 Z T0 Z T0+t 1 † 1 † Ecov(·) = dt Et(·) ⊗ UC(t) ρinc UC(t) = dt Et(·) ⊗ UC(t) ρinc UC(t), (140) T0 0 T0 t0 for all t0 ∈ R. This is easily provable via a change of variable and is a consequence of the fact that the integrand 0 is periodic and we are integrating over one period. Hence via the change x = t − t0 and setting t = t0, we find

Z T0 Z T0 1 † 1 † Ecov(ρL) = dt Et(·) ⊗ UC(t) ρinc UC(t) = dt Et+t0 (˜ρL) ⊗ UC(t)ρ ˜C,1 UC(t), (141) T0 0 T0 0

† where ρ˜L = UL(t0)ρLUL(t0) ∈ HL. The above equation shows how to write the covariant encoding channel in terms of the clock state ρ˜C,1 which is diagonal in the time basis. Since our results are stated in terms of the entanglement fidelity which is independent of any particular input, the change in inputs ρL to ρ˜L in the above equation is irrelevant and we will thus hence ignore this difference. The only relevant difference is thus that the encoding map Et is shifted to Et+t0 . This small difference can easily be accounted for at the decoding stage without extra complication.

0 0 From Eq. (35), we see that changing Et for Et+t0 is equivalent to changing Eq,q ,n,n to −iωt0(hCo,q −hCo,q0 +hL,n−hL,n0 ) j e Eq,q0,n,n0 or equivalently, changing Eq,q0,n,n0 to

˜j −iωt0(hCo,q −hCo,q0 +hL,n−hL,n0 ) j Eq,q0,n,n0 := e Eq,q0,n,n0 (142)

j ˜j Hence we can use Eq. 52 by making the replacements Eq,q0,n,n0 (deﬁned in Eq. (47)) with Eq,q0,n,n0 (·) and ρC,1 with ρ˜C,1, followed by plugging in Eq. (139). Hence recalling the short hand notation Q := hCo,q − hCo,q0 + hL,n − hL,n0 , we ﬁnd

dP−1 dL−1 dC−1 1 X X X ˜j 0 = δ 0 E 0 0 (ρ ) |qihq | (145) Q+r1−r1,0 q,q ,n,n L p~k 0 0 0 q,q =0 n,n =0 r1,r1=0

dC−1 2 X |Ak| −i2π(k −k)(r −r0 )/d e 1 1 1 C . (146) d2 k=0 C

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 21 Now applying d−1 d−1 X X f(r − r0) = (d − |x|)f(x) ∀f : R → R (147) r,r0=0 x=−(d−1) to Eq. (146), we have

dP−1 dL−1 dC−1 dC−1−|x1| ~k 1 X X X X ˜j 0 ρP = δQ+x1,0 Eq,q0,n,n0 (ρL) |qihq | (148) p~k 0 0 q,q =0 n,n =0 x1=−(dC−1) p1=0

dC−1 2 X |Ak| e−i2π(k1−k)x1/dC (149) d2 k=0 C

dP−1 dL−1 dC−1 1 X X X ˜j 0 = (dC − |x1|) δQ+x1+...+xN ,0 Eq,q0,n,n0 (ρL) |qihq | (150) p~k 0 0 q,q =0 n,n =0 x1=−(dC−1)

dC−1 2 X |Ak| e−i2π(k1−k)x1/dC . (151) d2 k=0 C

dP−1 dL−1 1 X X ˜j 0 = (d − |Q|) E 0 0 (ρ ) |qihq | (152) p C q,q ,n,n L ~k q,q0=0 n,n0=0

dC−1 2 X |Ak| e−i2π(k1−k)Q/dC . (153) d2 k=0 C

As all the outcomes are equally likely, we have that p~k = 1/dC. Hence

dP−1 dL−1 dC−1 ~k X X |Q| X 2 ˆ 0 0 ρP = 1 − |Ak| Oq,q ,n,n (tk−k1 ), (154) dC q,q0=0 n,n0=0 k=0 where we have deﬁned

⊗K† j † ⊗K ˆ 0 0 ˜ Oq,q ,n,n (tk−k1 ) := UCo (tk−k1 ) Eq,q0,n,n0 UL(tk−k1 )ρLUL(tk−k1 ) UCo (tk−k1 ) (155)

2π with tk−k1 = ω (k − k1). It is convenient to write this as

dC−1 ~k X 2 1 ˆ ρP = |Ak| ρP(tk−k1 ) + δ(tk−k1 ) , (156) dC k=0 where

dP−1 dL−1 X X ˆ ⊗K† † ⊗K ρP(t) := Oq,q0,n,n0 (t) = UCo (t + t0) EjE UL(t)ρLUL(t) UCo (t + t0), (157) q,q0=0 n,n0=0

dP−1 dL−1 ˆ X X δ(t) := − |Q| Oˆq,q0,n,n0 (t). (158) q,q0=0 n,n0=0

Observe that if we can apply the map

† ⊗K ⊗K† UL(tk−k1 )Dj UCo (tk−k1 + t0)(·)UCo (tk−k1 + t0) UL(tk−k1 ) (159)

to ρP(tk−k1 ), we have perfect error correction, i.e. † ⊗K ⊗K† R UL(tk−k1 )Dj UCo (tk−k1 + t0)ρP(tk−k1 )UCo (tk−k1 + t0) UL(tk−k1 ) = ρL ∀ t0, tk−k1 ∈ . (160) as such we deﬁne the error term ˆ † ⊗K ⊗K† E(t) := UL(t)Dj UCo (t + t0)δ(t)UCo (t + t0) UL(t). (161)

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 22 ~k Let us assume we apply an arbitrary decoder D to ρP. The ﬁdelity with an arbitrary initial pure state |ψi hψ| is

dC−1 X 2 1 ˆ 1 ˆ hψ| |Ak| D ρP(tk−k1 ) + δ(tk−k1 ) |ψi ≤ max hψ| D ρP(tk−k1 ) + δ(tk−k1 ) |ψi (162) dC k dC k=0 1 ˆ ≤ 1 − min hψ| D δ(tk−k1 ) |ψi (163) dC k 2 which shows that a the Salecker-Wigner-Peres clock with |Ak| = δk,k0 is the optimal choice. ˆ Crucially, the term in the optimization hψ| D δ(tk−k1 ) |ψi now only depends on parameters of the encoding E and decoding D independently of the clock and its dimension dC. Now, let us assume there exists a sequence of encoding-decoding schemes such that for some fixed dP, ˆ lim min hψ| Dl δl(tk−k ) |ψi = 0. (164) l→∞ k 1 If this were true, one could achieve an arbitrarily large entanglement fidelity for all pure states without increasing the size of the clock. This contradicts the existing no-go results [7, 26], so there must exist a constant C, such ˆ that for all D and δ(tk−k1 ) with fixed dP ˆ min hψ| D δ(tk−k ) |ψi ≥ C > 0, (165) k 1 ˆ which can be chosen such that the inequality is saturated for some D and δ(tk−k1 ). Hence, we have that C fworst(K) ≤ 1 − . (166) dC On the other hand, since the Salecker-Wigner-Peres clock saturates the inequalities (162) and (163) we can † apply the decoder UL(t)DjUL(t) in (159) to obtain C∗ fworst(K) = 1 − , (167) dC ∗ 1 ˆ for some C ≥ C, as the only dependence with the dimension of the clock is in the term δ(tk−k ). dC 1

F Proof of Theorem3

We just need to show that with L dC-dimensional clocks we can construct a clock state with the same properties as a single clock of dimension L(dC −1)+1. Let us take the Hamiltonian of L non-interacting clocks of dimension dC, L L(dC−1) M X X Htot = HC = ωr |Er,αihEr,α| (168) l=1 r=0 α which has energy levels {0,L(dC − 1)}, and α is the degeneracy index, as this new Hamiltonian has a very large degeneracy. Let us choose an arbitrary non-degenerate subspace of maximal dimension d(L) := L(dC − 1) + 1, 11 such that Htot = Hclock ⊕ˆ H⊥. This can be done for instance by choosing all the eigenvectors with a fixed α = 1, in which case d −1 XL Hclock = ωr|Er,1ihEr,1|. (169) r=0 This defines a subspace with dimension d(L) and with a Hamiltonian with equally-spaced energy levels with gap ω. Now given any single clock state ρ = PdC−1 A |r ihr | ∈ S(H ) in a d dimensional space, we can C r1,r2=0 r1,r2 1 2 C C C construct a d(L) dimensional space via the mappings d 7→ d(L) and ρ 7→ ρ0 := Pd(L)−1 A |E ihE | ∈ C C r1,r2=0 r1,r2 r1,1 r2,1 ⊗L S(HC ). This large dimensional clock state will now be a superposition of product states (the energy eigenstates) and will thus be entangled. Thus, in any scheme that uses a clock with dimension dC, such that it achieves a fidelity bounded by f(dC) (either from above or below), one can now use this d(L) dimensional clock, to achieve a fidelity bounded by f(d(L)) (again, either from above or below). This way, we can use a large amount of clocks to vastly improve the performance of the codes.

11Here ⊕ˆ denotes the Direct sum.

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 23 G The converse bound from [26]

Here we show how to adapt the bound from [26] to our setting 12. To understand why the bound cannot be straightforwardly applied to our setting, first recall our definition of the covariant code Ecov(·). For all t ∈ R, ⊗K and a given encoding E(·) and group representations UL(t),UCo(t) , we define

⊗K † †⊗K Et(·) := UCo(t) E(UL(t)(·)UL(t))UCo(t) . (170) The covariant encoding that we use throughout our work is then the CPTP map:

Z T0 1 ⊗M (M) †⊗M Ecov(·) := dt Et(·) ⊗ UC(t) ρC UC(t) . (171) T0 0

This construction is by definition not an isometry, and in general the output Ecov(·) is a mixed state even for pure inputs. However, the theorem of [26] assumes that the encoding map is an isometry, and as such it does not directly apply to our results. However, since in our setup we assume that the encoding of Eq. (171) is covariant, it is possible to construct a Stinespring dilation with an isometry to which the bound of [26] can be directly applied. We can construct the Stinespring dilation with an isometry directly for Eq. (171). However, since in our set-up M − N clocks are lost due to erasure errors, and the erased clocks are inaccessible, we can trace out M −N clocks and work with the resultant effective encoding map instead. Since we also assumed no correlations between the erased and non-erased clocks, this leads to the effective channel

Z T0 1 ⊗N (N) †⊗N trCN+1...CM [Ecov(·)] = dt Et(·) ⊗ UC(t) ρC UC(t) . (172) T0 0 Finally, since in some instances the generator of the unitary group on the clock can be exchanged for an effective generator w.l.o.g. (noticeably Theorem3), we will assume in Eq. (172) that the unitary representation of the ⊗N ⊗N −itH¯C compact U(1) group on the clock UC(t) , takes on the generic form UC(t) = e and specialise it later in this section to specific cases. We can now proceed with the isometry. Notice that E(·) can be dilated to an isometry VL→CoA defined such (N) that the representation of the symmetry group acts trivially on system A. Also, the state ρC may not be a (N) pure state, but it can also be dilated to |ΨCC¯ i hΨCC¯ | , again such that the symmetry group again acts trivially on system C.¯ We thus define 1 Z T0 E¯ (·) := dtV (·) ⊗ U (t)⊗N ⊗ 1⊗N |Ψ ihΨ | U (t)†⊗N ⊗ 1⊗N , (173) cov t C C¯ CC¯ CC¯ C C¯ T0 0 where now we have

⊗K 1 † † †⊗K 1 Vt(·) := UCo(t) ⊗ A VL→CoA(UL(t)(·)UL(t))VL→CoA UCo(t) ⊗ A . (174)

The only reason why E¯cov(·) is not yet an isometry is the twirling over the group, for which we are also able to deﬁne a dilation with the help of the Choi-Jamiolkowski isomorphism. First, let {|kiL} be the eigenbasis ˆ P P of L with HL = k ωhL,k|kihk|L, and let |ΦLL¯ i = k |kiL ⊗ |kiL¯ be an unnormalized maximally entangled P state between the logical space L and a copy L.¯ Moreover, let us deﬁne Hˆ ¯ = −ωhL,k|kihk|¯ such that L k L ˆ 1 1 ˆ ¯ HL ⊗ L¯ + L ⊗ HL¯ |ΦLL¯ i = 0. The Choi-Jamiolkowski representation of channel Ecov is (we now omit writing the trivial representations such as 1 , 1⊗N for simplicity of notation) A C¯

12We thank Philippe Faist for sharing this argument with us and allowing us to reproduce it here.

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 24 ˆ ¯ ˆ x Since the operators HCo, HC, HL¯ act on diﬀerent subsystems, we can decompose ΠCoCL¯ as

Now we can define an extra system B on a Hilbert space spanned by orthonormal basis vectors {|xiB}, and such that Hˆ = P (x)|xihx| , so that every single energy difference h − h appears in the B x∈{hCoC,l−hL,k} B CoC,l L,k sum only once (note that by assumption this set is finite). With this, we can define the following isometry from the logical space labeled by L to BCoACC.¯

X X l k WL→BCoACC¯ = |xiB ⊗ ΠCoC(VL→CoAΠL) ⊗ |ΨCC¯ i . (180) x k,l hCoC,l−hL,k=x

This is indeed an isometry since W † W = 1 . It is covariant, in the sense that by construction L→BCoACC¯ L→BCoACC¯ L

ˆ ˆ ¯ ˆ −itHL −it(HCo⊕HC⊕HB) 1 WL→BCoACC¯ e = e ⊗ AC¯ WL→BCoACC¯ . (181) Moreover, it can be easily computed that h i tr W |Φ ihΦ |W † = E¯ (|Φ ihΦ |) (182) B L→BCoACC¯ LL¯ LL¯ L→BCoACC¯ cov LL¯ LL¯ and subsequently h i tr W |Φ ihΦ |W † = tr [E (|Φ ihΦ |)] , (183) BAC¯ L→BCoACC¯ LL¯ LL¯ L→BCoACC¯ CN+1...CM cov LL¯ LL¯ so WL→BCoACC¯ is a covariant Stinespring dilation of channel trCN+1...CM [Ecov(·)], which we can see as an encoding isometry for which the error is i) ﬁrst the loss to the environment of systems BAC,¯ and then ii) an erasure error channel for Ecov. The result of [26] states that for these isometries, the entanglement ﬁdelity achieved by any decoding scheme is bounded by

2 ∆hL fworst ≤ 1 − 2 2 , (184) 4N ∆hloss where ∆hL is the range of values of the set {hL,k} of HˆL, N is the number of subsystems in the encoding that can be erased independently by the error model, and where ∆hloss is the largest energy difference in the Hamiltonians of all the subsystems that are lost to the environment. We have that BAC¯ are lost and since A, C¯ have trivial generators we just have to look at HˆB, so that ∆hloss = ∆hB. The eigenvalues of HˆB are of the form hCoC,l − hL,k, so that the range of HˆB is ∆hB = ∆hCo + ∆hC (the terms with hL,k always contribute negatively and hence ∆hL does not enter here). We will now specialise the bound to the first case considered in Corollary1. Here we have erasure of up to M − 1 blocks of L entangled clocks so N = 1. Furthermore, the effective clock generator on the remaining block of L entangled clocks is H¯C = Hclock (recall Eq. (169) for an expression for Hclock). This effective clock system has ∆hC = LdC. Therefore ∆hloss = ∆hCo + LdC. To compute N , first notice that BAC¯ counts as a single system as it is always lost to the environment. Moreover, as mentioned above, the M − 1 clocks that are lost to the environment do not appear in the decoding procedure at all, and as detailed above are such that effectively they were not there in the first place. Finally, the error channel on P is later corrected by its own decoding map Dj(·) and is independent of whether we make the code covariant in the first place, and the block of L entangled clocks that is left (which we have taken to be the whole C system) does not get erased, so we can also count both these two as a single subsystem (which gets erased with probability 0). Hence we have N = 2. Putting everything together, we finally obtain

2 ∆hL fworst ≤ 1 − 2 . (185) 16(∆hCo + LdC)

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 25 The appearance of ∆hCo as an additive contribution to the effective clock dimension LdC in the upper bound on the fidelity is to be expected, since in our setup one could have chosen the encoding map E to be a clock whose decoding map Dj measures this clock in the same way that the decoding map D˜j,q measures the clocks on C. This scenario would help to make the encoding channel Ecov increase its decoding fidelity, since its effective clock dimension would be ∆hCo +LdC. Contrarily, if one considers encoding and decoding channels E, Dj which do not help to make the encoding channel Ecov reduce its decoding errors at all, then the value of ∆hCo should not be expected to play a role in the decoding fidelity fworst. Finally, comparing Eq. (185) with the lower bound of Eq. (23), we see that up to logarithmic factors, the L entangled Quasi-Ideal clocks achieve the optimal error scaling with both their dimension dC and number of clocks L.

H Proof of Theorem4

In this section, we will prove Theorem4. We will prove it only for the special case that the three blocks consist of one clock each, i.e. L = 1. Once Theorem4 is proven for this specialised case, the result for larger L is a direct consequence of Theorem3. Again for simplicity of notation, we will label d = dC. For simplicity, we will start assuming, as for a single clock, that d is even (d odd follows analogously), and that σ satisfies d (0, d) 3 σ → ∞ and → ∞ as d → ∞. (186) σ ~ The first goal will be to work out an explicit expression for FQ(k) evaluated for the case of three Quasi-Ideal clocks in which a 2-unknown phase error applied to one of them. To indicate this difference (i.e. that un unknown phase has now been applied), we add a tilde to F . Specifically, we have

F˜Q(k1, k2, k3) := (187) Z T0 1 −iωQt ˜ ˜ ˜ dt e hθk1 |ρC,1(t + tph,1)|θk1 i hθk2 |ρC,2(t + tph,2)|θk2 i hθk3 |ρC,3(t + tph,3)|θk3 i , (188) T0 0 where ( tph if q = r, t˜ph,q = (189) 0 otherwise, and r = 1, 2, 3 denotes the clock to which the phase is applied. The variables r and tph are assumed to be unknown. Applying Theorem 8.1 (Eq. 71) in [21] we ﬁnd ˜ hθkq | ρC,q(t + tph,q) |θkq i (190)

= hθkq |ρC,q(t)|θkq i + εI0, (193) 0 ˜0 kq 7→kq 0 0 where we have used the Quasi-Ideal clock states ρC,q = |ψnor(kq )ihψnor(kq )|q and deﬁned

( 0 ˜0 kq + tphd/T0 if q = r, kq = 0 (194) kq otherwise. In the last line of Eq. (193) we have deﬁned

F˜Q(k1, k2, k3) = FQ(k1, k2, k3) + 7 εI0, (198) 0 ˜0 3 {kq 7→kq }q=1

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 26 so we see that a 2-unknown phase error, up to a small error εc, simply re-maps the initial time of one of the clocks to an unknown value. Given the relation Eq. (198), our 1st task will be to ﬁnd an explicit expression for the integral (58). We do this in the following section.

H.1 An Expression for the Integral in FQ 0 We start by deriving a general expression for hθk|ρC,q(t)|θki for k ∈ Sd(kq ), q ∈ {1, 2, 3} for Quasi-Ideal clock states. For this we need to consider the overlaps

Let tq be deﬁned via the relation 0 kq − tqd/T0 = min{Sd(kq )}. (201) 0 0 d Using Eq. (99) we ﬁnd min{Sd(kq )} = bkq c − 2 + 1, giving d T t = k − bk0c + − 1 0 . (202) q q q 2 d Therefore,

Now consider Eq. (58) with measurement outcomes k1 ≤ k2 ≤ k3, with identical clocks other then their 0 0 0 starting times denoted k1, k2, k3 respectively. We consider this special case w.l.o.g. since the other cases can be reconstructed later using property 3) in SectionB. Thus using Theorem 9.1 in [21] we have

Z T0 1 −iωQt FQ(k1, k2, k3) = dt e hθk1 |ρC,1(t)|θk1 i hθk2 |ρC,2(t)|θk2 i hθk3 |ρC,3(t)|θk3 i (205) T0 0 1 Z T0 2 = dt e−iωQt Aψ (k0; kˆ (t, k ) − td/T ) + hθ |ε (t)i (206) nor 1 1 1 0 k1 c T0 0 2 Aψ (k0; kˆ (t, k ) − td/T ) + hθ |ε (t)i (207) nor 2 2 2 0 k2 c 2 Aψ (k0; kˆ (t, k ) − td/T ) + hθ |ε (t)i (208) nor 3 3 3 0 k3 c Z 1 2 = dx e−i2πxQ Aψ (k0; kˆ (xT , k ) − xd) + hθ |ε (xT )i (209) nor 1 1 0 1 k1 c 0 0 2 Aψ (k0; kˆ (xT , k ) − xd) + hθ |ε (xT )i (210) nor 2 2 0 2 k2 c 0 2 Aψ (k0; kˆ (xT , k ) − xd) + hθ |ε (xT )i , (211) nor 3 3 0 3 k3 c 0 where in the last line we performed the change of variable x = t/T0 and used ω = 2π/T0 and deﬁned tm = 0 d T0 km − bkmc + 2 − 1 d and for m = 1,...,N ( ˆ km if t ∈ [0, tm], km(t, km) = (212) km + d if t ∈ (tm,T0]. From Eq. (211) we have Z 1 6 −i2πxQ 2 0 ˆ FQ(k1, k2, k3) = A dx e ψnor k1; k1(xT0, k1) − xd (213) 0 2 0 ˆ 2 0 ˆ ψnor k2; k2(xT0, k2) − xd ψnor k3; k3(xT0, k3) − xd (214)

+ εI1. (215)

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 27 Using the triangle inequality and the identity |R + C|2 = R2 + ε where |ε| ≤ |C|(2R + |C|) for R ≥ 0, C ∈ C, we can bound the εI1 term, Z 1 6 2 6 |εI1| ≤ A dx 5 max ψ (0; y) εc(T0, d) ≤ 5A εc(T0, d), (216) R nor 0 y∈ where εc(T0, d) is an upper bound to k |εc(t)i k2 evaluated at t = T0. This quantity, (which already appeared in AppendixD) is given in Theorem 8.1 of [21] and satisﬁes

3 2 2 T0 2 1/2 − π σ2α2 d − π d − π d − π σ2 ε (T , d) = O d σ e 4 0 + O 1 + e 4 ( σ ) + O e 4 ( σ ) + O e 4 (217) c 0 3/4 T0 σ

3 2 2 1/2 − π σ2α2 d − π d = O d σ e 4 0 + O e 4 ( σ ) (218) σ3/4 where α0 ∈ (0,√1] is deﬁned in Def. 2 in [21] and explained in Eq. (106). 2 Using A = 2/σ + εA = O(1/σ) from Eq. 483 in [21], and Eq. (216) we conclude

2 3 2 d − π σ2α2 d − π d |ε | =O e 4 0 + O e 4 ( σ ) . (219) I1 σ5/2 σ15/4

Dividing the integral in Eq. (215) into subintervals and substituting for ψnor we ﬁnd

t1/T0 Z 2π 0 2 0 2 0 2 6 −i2πxQ − [(k −k1+xd) +(k −k2+xd) +(k −k3+xd) ] FQ(k1, k2, k3) = A dx e e σ2 1 2 3 (220) 0 t2/T0 Z 2π 0 2 0 2 0 2 6 −i2πxQ − [(k −k1−d+xd) +(k −k2+xd) +(k −k3+xd) ] + A dx e e σ2 1 2 3 (221) t1/T0

t3/T0 Z 2π 0 2 0 2 0 2 6 −i2πxQ − [(k −k1−d+xd) +(k −k2−d+xd) +(k −k3+xd) ] + A dx e e σ2 1 2 3 (222) t2/T0 1 Z 2π 0 2 0 2 0 2 6 −i2πxQ − [(k −k1−d+xd) +(k −k2−d+xd) +(k −k3+xd) ] + A dx e e σ2 1 2 3 (223) t3/T0

+ εI1. (224)

For the rest of the proof, it is convenient to work with F˜Q rather than FQ. Substituting the above into Eq. 0 ˜0 (198) and performing the mapping kq 7→ kq , q = 1, 2, 3, we ﬁnd

F˜Q(k1, k2, k3) = (225)

0 2 0 2 0 2 ¯ 2h ¯ ∆k ¯ ∆k ¯ ∆k i Z k1/d+1/2−1/d −2π( d ) x− k1 + 1 + x− k2 + 2 + x− k3 + 3 A6 dx e−i2πxQ e σ d d d d d d (226) 0 0 2 0 2 0 2 ¯ 2h ¯ ∆k ¯ ∆k ¯ ∆k i Z k2/d+1/2−1/d −2π( d ) x−1− k1 + 1 + x− k2 + 2 + x− k3 + 3 + A6 dx e−i2πxQ e σ d d d d d d (227) k¯1/d+1/2−1/d 0 2 0 2 0 2 ¯ 2h ¯ ∆k ¯ ∆k ¯ ∆k i Z k3/d+1/2−1/d −2π( d ) x−1− k1 + 1 + x−1− k2 + 2 + x− k3 + 3 + A6 dx e−i2πxQ e σ d d d d d d (228) k¯2/d+1/2−1/d 0 2 0 2 0 2 2h ¯ ∆k ¯ ∆k ¯ ∆k i Z 1 −2π( d ) x−1− k1 + 1 + x−1− k2 + 2 + x−1− k3 + 3 + A6 dx e−i2πxQ e σ d d d d d d (229) k¯3/d+1/2−1/d

+ εI1 + 7 εI0. (230) where we have deﬁned initial clock time independent measurement outcomes, d d d k¯ := k − bk˜0c ∈ S (k˜0) − bk˜0c = − + 1, − + 2,..., , p = 1, 2, 3, (231) p p p d p p 2 2 2

0 ˜0 ˜0 and 1 > ∆kp := kp − bkpc ≥ 0, p = 1, 2, 3. Before proceeding further, we write explicitly bounds for the ε terms. Using Eq. (218) we ﬁnd

3 2 2 2 1/2 − π σ2α2 d − π d |ε | ≤ 2ε (t˜ , d) + ε (t˜ , d) = O d σ e 4 0 + O e 4 ( σ ) . (232) I0 c ph c ph σ3/4

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 28 Thus taking into account (219), we conclude

3 2 2 1/2 − π σ2α2 d − π d |ε | + 7|ε | = O d σ e 4 0 + O e 4 ( σ ) . (233) I1 I0 σ3/4

In order to simplify the equations further, we will set k1 to the smallest measurement outcome possible, i.e. ˜0 k1 = bk1c − d/2 + 1. We can easily generate the other cases by employing Eq. (60), which we postpone to Section H.4. Substituting into Eq. (230) gives us

F˜Q(k1, k2, k3) = (234)

0 2 0 2 0 2 ¯ 2h 1−∆k ¯ ∆k ¯ ∆k i Z k2/d+1/2−1/d −2π( d ) x− 1 − 1 + x− k2 + 2 + x− k3 + 3 A6 dx e−i2πxQ e σ 2 d d d d d (235) 0 0 2 0 2 0 2 ¯ 2h 1−∆k ¯ ∆k ¯ ∆k i Z k3/d+1/2−1/d −2π( d ) x− 1 − 1 + x−1− k2 + 2 + x− k3 + 3 + A6 dx e−i2πxQ e σ 2 d d d d d (236) k¯2/d+1/2−1/d 0 2 0 2 0 2 2h 1−∆k ¯ ∆k ¯ ∆k i Z 1 −2π( d ) x− 1 − 1 + x−1− k2 + 2 + x−1− k3 + 3 + A6 dx e−i2πxQ e σ 2 d d d d d (237) k¯3/d+1/2−1/d

+ εI1 + 7 εI0. (238)

The intuition is that each of the three above integrals will be small whenever the the corresponding term in square brackets in the exponent does not pass through zero in the interval over which it is being integrated. We ¯ ¯ now solve for k2, k3 for when this happens. We start with the 1st integral. From line (235), we observe that the term " 2 2# 1 1 − ∆k0 2 k¯ ∆k0 k¯ ∆k0 x − − 1 + x − 2 + 2 + x − 3 + 3 (239) 2 d d d d d

0 ¯ 0 0 ¯ can only be zero iff x = 1/2 + 1/d − ∆k1/d, k2/d = 1/2 + 1/d + ∆k2 − ∆k1 /d, and k3/d = 1/2 + 1/d + 0 0 ¯ 0 0 ∆k3 − ∆k1 /d. In this case, x ∈ [0, k2/d + 1/2 − 1/d] = [0, 1 + + ∆k2 − ∆k1 /d], so x passes through 0 ¯ ¯ ¯ ¯ x = 1/2 + 1/d − ∆k1/d. Observe that this values of k2/d, k3/d are outside the domain of k2/d, k3/d by an 0 0 0 0 additive factor of 1/d + ∆k2 − ∆k1 /d and 1/d + ∆k3 − ∆k1 /d respectively. However, since this quantity tends to zero as d becomes large, the integral on line (235) will provide a significant contribution. For this case, ¯ ¯ we will make the parametrization k2/d = 1 + 1/d − k/d, k3/d = 1 + 1/d − l/d, k, l = 1, . . . , d. Similarly for the integral on line (236), we find that

" 2 2# 1 1 − ∆k0 2 k¯ ∆k0 k¯ ∆k0 x − − 1 + x − 1 − 2 + 2 + x − 3 + 3 (240) 2 d d d d d

0 ¯ 0 0 ¯ can only be zero iﬀ x = 1/2 + 1/d − ∆k1/d, k2/d = −1/2 + 1/d + ∆k2 − ∆k1 /d, and k3/d = 1/2 + 1/d + 0 0 ¯ 0 0 ∆k3 − ∆k1 /d. In this case, x ∈ [0, k2/d + 1/2 − 1/d] = [0, 1 + ∆k2 − ∆k1 /d] For this case, we will make ¯ ¯ the parametrization k2/d = −1/2 + 1/d + m/d, k3/d = 1 + 1/d − l/d, m = 0, . . . , d − 1, l = 1, . . . , d. Similarly for the integral on line (237), we ﬁnd that

" 2 2# 1 1 − ∆k0 2 k¯ ∆k0 k¯ ∆k0 x − − 1 + x − 1 − 2 + 2 + x − 1 − 3 + 3 (241) 2 d d d d d

0 ¯ 0 0 ¯ can only be zero iff x = 1/2 + 1/d − ∆k1/d, k2/d = −1/2 + 1/d + ∆k2 − ∆k1 /d, and k3/d = −1/2 + 1/d + 0 0 ¯ 0 0 ∆k3 − ∆k1 /d with x ∈ [k3/d + 1/2 − 1/d, 1] = [ ∆k3 − ∆k1 /d, 1]. Intuitively, the three cases represent all possibilities (satisfying our assumptions k1 ≤ k2 ≤ k3 ) in which 0 0 0 approximately either all three measured elapsed times (these are all proportional to k1 − k1, k2 − k2, k3 − k3) 0 coincide with the measured elapsed time of the 1st clock (proportional to k1 −k1), or the three measured elapsed times correspond to some combination of either the measured elapsed time of the first clock or approximately the time corresponding to one period T0 later. In all cases, these correspond approximately to the initial time, due to the periodic motion of the clock, i.e. the clock resets back to the initial time after one period of its motion. So in all cases, we should expect that to a good approximation, the outcomes which are most likely are those when all the clocks mark (at least approximately) the same elapsed time. We will now find a good approximation for the last integral (line (237)), the others can be approximated in the same way.

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 29 H.2 Approximating the integral in line (237). ¯ ¯ We will start by substituting the parametrizations k2/d = −1/2 + 1/d + m/d, k3/d = −1/2 + 1/d + p/d, m, p = 0, . . . , d − 1 into the integral on line (237), obtaining

0 2 0 2 0 2 2h ∆k ¯ ∆k ¯ ∆k i Z 1 −2π d x− 1 − 1 + 1 + x−1− k2 + 2 + x−1− k3 + 3 −i2πxQ ( σ ) 2 d d d d d d Int3(m, p) := dx e e (242) k¯3/d+1/2−1/d 0 2 0 2 0 2 2h ∆k ∆k ∆k i Z 1 −2π( d ) x− 1 − 1 + 1 + x− 1 − 1 + 2 − m + x− 1 − 1 + 3 − p = dx e−i2πxQ e σ 2 d d 2 d d d 2 d d d . (243) p/d We now consider the case in which the integral may contain a signiﬁcant contribution to Eq. (238), and write

0 2 0 2 0 2 2h ∆k ∆k ∆k i Z ∞ −2π d x− 1 − 1 + 1 + x− 1 − 1 + 2 − m + x− 1 − 1 + 3 − p −i2πxQ ( σ ) 2 d d 2 d d d 2 d d d Int3(m, p) = dx e e (244) −∞ L R + ε3 (m, p) + ε3 (m, p), (245) where

0 2 0 2 0 2 2h ∆k ∆k ∆k i Z p/d −2π d x− 1 − 1 + 1 + x− 1 − 1 + 2 − m + x− 1 − 1 + 3 − p L −i2πxQ ( σ ) 2 d d 2 d d d 2 d d d ε (m, p) = − dx e e (246) 3 −∞ 0 2 0 2 0 2 2h ∆k ∆k ∆k i Z p/d −2π( d ) x− 1 − 1 + 1 + x− 1 − 1 + 2 − m + x− 1 − 1 + 3 − p ≤ dx e σ 2 d d 2 d d d 2 d d d . (247) −∞ We now employ the relation 1 1 2 (x + A)2 + (x + A − y)2 + (x + A − z)2 = − (y + z)2 + y2 + z2 + 3 x + A − (y + z) (248) 3 3 for x, y, z, A ∈ R followed by introducing the condition (later we will workout a bound for when this condition is not satisﬁed) p 1 1 p + m − ∆k0 − ∆k0 − ∆k0 − − − 1 2 3 ≤ −εL for some εL > 0, (249) d 2 d 3d 3,1 3,1 we ﬁnd

0 0 0 2 0 0 2 0 0 2 2h p+m+2∆k −∆k −∆k p+∆k −∆k m+∆k −∆k i −2π d − 1 1 2 3 + 1 3 + 1 2 L ( σ ) 3 d d d ε3 (m, p) ≤e (250) 0 0 0 0 2 2 ∆k p+m+2∆k −∆k −∆k Z p/d −6π( d ) x− 1 − 1 + 1 − 1 2 3 dx e σ 2 d d 3d (251) −∞ 0 0 0 2 0 0 2 0 0 2 2h p+m+2∆k −∆k −∆k p+∆k −∆k m+∆k −∆k i −2π( d ) − 1 1 2 3 + 1 3 + 1 2 ≤e σ 3 d d d (252)

0 0 0 0 0 0 2 p+m−∆k −∆k −∆k 2 p+m−∆k −∆k −∆k Z p/d x − 1 − 1 − 1 2 3 −6π( d ) x− 1 − 1 − 1 2 3 dx 2 d 3d e σ 2 d 3d (253) p p+m−∆k0−∆k0−∆k0 −∞ 1 1 1 2 3 d − 2 − d − 3d 0 0 0 2 0 0 2 0 0 2 2h p+m+2∆k −∆k −∆k p+∆k −∆k m+∆k −∆k i −2π( d ) − 1 1 2 3 + 1 3 + 1 2 =e σ 3 d d d (254)

0 0 0 2 2 p+m−∆k −∆k −∆k −6π( d ) p − 1 − 1 − 1 2 3 −1 σ 2 e σ d 2 d 3d 0 0 0 (255) 2π6 d p 1 1 p+m−∆k1 −∆k2 −∆k3 d − 2 − d − 3d 0 2 0 2 0 2 2h ∆k ∆k ∆k i −2π( d ) p − 1 − 1 + 1 + p − 1 − 1 + 2 − m + p − 1 − 1 + 3 − p 1 σ 2 e σ d 2 d d d 2 d d d d 2 d d d ≤ L (256) 2π6 d ε3,1 0 2 2 1−∆k −2π( d ) 1 + 3 1 σ 2 e σ 2 d ≤ L (257) 2π6 d ε3,1 2 − π d 1 σ 2 e 2 ( σ ) ≤ L , ∀ m, p = 0, . . . , d − 1. (258) 2π6 d ε3,1

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 30 R where in the penultimate line we have used Eq. (248) and (249). Analogously, we can bound ε3 (m, p). We ﬁnd

0 2 0 2 0 2 2h ∆k ∆k ∆k i Z ∞ −2π d x− 1 − 1 + 1 + x− 1 − 1 + 2 − m + x− 1 − 1 + 3 − p R −i2πxQ ( σ ) 2 d d 2 d d d 2 d d d ε (m, p) = − dx e e (259) 3 1 2 2 − π d 1− 2 1 σ 2 e 2 ( σ ) ( d ) ≤ R , m, p = 0, . . . , d − 1. (260) 2π6 d ε3,1 where we have introduced the constraint 1 1 p + m − ∆k0 − ∆k0 − ∆k0 1 − − − 1 2 3 ≥ εR for some εR > 0. (261) 2 d 3d 3,1 3,1 Thus computing the integral in Eq. (245) and combining the epsilons into a single term, we have

1 σ −iπQ iπ 2 ∆k0 Q/d −iπ 2 (3+m+p)Q/d Int3(m, p) =√ e e 3 T e 3 (262) 6 d

[∆k0−∆k0+m]2+[∆k0−∆k0+p]2−[∆k0−∆k0+m][∆k0−∆k0+p] 4 d 2 1 2 1 3 1 2 1 3 −π ( ) π σ 2 2 3 σ d2 − ( ) Q e e 6 d + ε3(m, p), (263)

1 σ −i2πkQ/d˜ −iπ 2 (m+p)Q/d =√ e e 3 (264) 6 d

[∆k0−∆k0+m]2+[∆k0−∆k0+p]2−[∆k0−∆k0+m][∆k0−∆k0+p] 4 d 2 1 2 1 3 1 2 1 3 −π ( ) π σ 2 2 3 σ d2 − ( ) Q e e 6 d + ε3(m, p), (265) where in the last line we have multiplied by 1 = e2πiQ and deﬁned the quantities

1 1 ∆k0 k˜ := − + − T , (266) 2 d 3d 0 0 0 0 ∆kT := ∆k1 + ∆k2 + ∆k3. (267)

Note that if inequalities (249) and (261) are both satisﬁed, then ! 1 σ 2 1 1 π d 2 2 2 − 2 ( σ ) (1− d ) |ε3(m, p)| ≤ L + R e , ∀ m, p = 0, . . . , d − 1. (268) 2π6 d ε3,1 ε3,1

Whenever inequality (249) and/or inequality (261) are/is not satisﬁed, we can bound the integral by the largest value of the integrand, which is exponentially small in (d/σ)2. In this case since the point at which the Gaussian takes on its maximum value lays outside of the integration region. However, it turns out that (m2 +p2 −mp)/d2 is uniformly bounded away from zero in this case and thus the 1st term in (265) is exponentially small in (d/σ)2. We will prove this result, since it will provide a more useful expression. We start by deﬁning three functions for x ∈ R

1 1 ∆k0 1 C (x) := −p + + − 3 + (p + x) − εL , (269) 3L,p 2 d d 3 3,1 1 1 ∆k0 1 C (x) := 1 − − + 1 − (p + x) − εR (270) 3R,p 2 d d 3 3,1 2 2 F3,p(x) := x + p − x p. (271)

0 0 0 0 With the identiﬁcation m = (∆k1 − ∆k2 + m)/d and p = (∆k1 − ∆k3 + p)/d, we see that the constraints in Eqs. (249), (261) are the conditions C3L,p(m) ≤ 0 and C3R,p(m) ≥ 0 respectively, while F3,p(m) is the exponent in Eq. (265) which we aim to prove is uniformly bounded away from zero whenever C3L,p(m) ≤ 0 and C3R,p(m) ≥ 0 is not satisﬁed, i.e. when

C3L,p(m) > 0 and/or C3R,p(m) < 0. (272)

Given the range of p = 0, . . . , d − 1, it is easy to verify that both Eqs, (272) cannot simultaneously hold since C3L,p(m) > 0 implies C3R,p(m) ≥ 0 and C3R,p(m) < 0 implies C3L,p(m) ≤ 0. Therefore, it suﬃces to consider Eq. (272) with the “or” case only. It is useful to think of these functions as functions of m which are

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 31 d2 parametrized by p, as the chosen notation suggests. Furthermore, dx2 F3,p(x) = 2, and thus F3,p(x) is a convex function for all p. As such, we can lower bound F3,p by any tangent straight line. First consider the case that C3L,p(m) > 0 holds. We will proceed to lower bound F3,p(x) by the tangent line ? ? ? Ap + Bp x which intersects F3,p(x) at x = m , where m is deﬁned by C3L,p(m ) = 0. It follows that 1 1 ∆k0 m? = 3 εL + p − − + 3 − p, (273) 3,1 2 d d from which we ﬁnd

0 d ? L 1 1 ∆k3 Bp = F3p(x) = 2m − p = 6 ε3,1 − − + + 3p (274) dx x=m? 2 d d 1 1 ∆k0 1 ∆k0 ∆k0 ≤ 6 εL − − + 3 + 3 1 − + 1 − 3 . (275) 3,1 2 d d d d d

It is convenient to demand Bp ≤ 0. We thus set

3 1 1 ∆k0 + ∆k0 1 εL := − 1 3 > , (276) 3,1 2 d 2 d 2d so that Bp ≤ 0 for all p = 0, . . . , d − 1. To ﬁnd Ap, we need to solve the equation

? ? Ap + Bpm = F3p(m ), (277) giving ? ? ? 2 2 Ap = F3p(m ) − Bpm = − (m ) + p . (278) Therefore, putting it all together we have for all m, p = 0, 1/d, . . . , 1 − 1/d,

? 2 2 ? F3p(m) ≥ Ap + Bp m = − (m ) + p + (2m − p) m. (279)

? ? Taking into account Bp ≤ 0, we have F3p(m) ≥ Ap + Bp m ≥ Ap + Bp m for all m < m . Furthermore, by ? construction, C3L,p(m) > 0 holds iﬀ m < m . Thus simplifying, we ﬁnd

9 2 9 F (m) ≥ (m?)2 + p2 − p m? = d − 1 + ∆k0 − ∆k0 − d − 1 + ∆k0 − ∆k0 p + 3p2, (280) 3p 4d2 1 3 2d 1 3 if C3L,p(m) > 0 holds. The R.H.S. is a convex function in p, thus calculating its stationary point, it can be lower bounded by 9 1 ∆k0 − ∆k0 2 9 2 2 F (m) ≥ 1 − + 3 1 ≥ 1 − , (281) 3p 16 d d 16 d if C3L,p(m) > 0 holds. Performing the analogous procedure for the case that C3R,p(m) < 0 holds, and deﬁning

1 1 εR := 1 + ∆k0 + ∆k0 ≥ , (282) 3,1 2d 1 3 2d we ﬁnd 9 3 2 F (m) ≥ 1 − . (283) 3p 16 d Thus summarising the two cases, we see that

9 3 2 F (m) ≥ 1 − . (284) 3p 16 d if the statement “Inequalities (249), (261) are both satisﬁed” is false. Thus deﬁning εF by the equation

1 σ −i2πkQ/d˜ −iπ 2 (m+p)Q/d εF := Int3(m, p) − √ e e 3 (285) 6 d 0 0 2 0 0 2 0 0 0 0 2 [∆k −∆k +m] +[∆k −∆k +p] −[∆k −∆k +m][∆k −∆k +p] −π 4 d 1 2 1 3 1 2 1 3 2 3 ( σ ) d2 − π σ Q2 e e 6 ( d ) , (286)

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 32 we have that

|εF | ≤ Int3(m, p) (287) 0 0 2 0 0 2 0 0 0 0 2 [∆k −∆k +m] +[∆k −∆k +p] −[∆k −∆k +m][∆k −∆k +p] −π 4 d 1 2 1 3 1 2 1 3 2 1 σ 3 ( σ ) d2 − π σ Q2 + √ e e 6 ( d ) (288) 6 d 2 2 1 σ −π 3 ( d ) (1− 3 ) ≤ Int3(m, p) + √ e 4 σ d , (289) 6 d if “Inequalities (249), (261) are both satisﬁed” is false. We now bound Int3(m, p) . From Eq. (243) it follows

2 2 0 2 0 2 2h ∆k ∆k ∆k i Z 1 −2π( d ) x− 1 − 1 + 1 + x− 1 − 1 − m + 2 + x− 1 − 1 − p + 3 σ 2 d d 2 d d d 2 d d d Int3(m, p) ≤ dx e (290) p/d 0 0 0 2 0 0 2 0 0 2 2h p+m+2∆k −∆k −∆k p+∆k −∆k m+∆k −∆k i −2π( d ) − 1 1 2 3 + 1 3 + 1 2 ≤ e σ 3 d d d (291)

0 2 2 p+m−∆k Z 1 −6π( d ) x− 1 − 1 − T dx e σ 2 d 3d (292) p/d 0 0 0 2 0 0 2 0 0 2 2h p+m+2∆k −∆k −∆k p+∆k −∆k m+∆k −∆k i −2π( d ) − 1 1 2 3 + 1 3 + 1 2 ≤ e σ 3 d d d (293) 2 p+m−∆k0 d 2 1 1 T p −6π( σ ) minx∈[p/d,1] x− 2 − d − 3d 1 − e . (294) d where we have used Eq. (248). Since the above minimisation is over a parabola, where the smallest value at 0 1 1 p+m−∆kT x = 2 + d + 3d lays outside the interval [p/d, 1] whenever “Inequalities (249), (261) are both satisfied” is false, the solution to the minimization is achieved at one of the boundary points, x = p/d or x = 1. Specifically we find 0 2 0 2 1 1 p + m − ∆kT 1 1 p + m − ∆kT min x − − − = ∆3,p − − − , (295) x∈[p/d,1] 2 d 3d 2 d 3d where ( p d if C3L,p(m) ≤ 0 is false ∆3,p = (296) 1 if C3R,p(m) ≥ 0 is false.

Thus from Eq. (294), it follows

0 0 0 2 0 0 2 0 0 2 2h p+m+2∆k −∆k −∆k p+∆k −∆k m+∆k −∆k i −2π( d ) − 1 1 2 3 + 1 3 + 1 2 σ 3 d d d Int3(m, p) ≤ e (297) 2 p+m−∆k0 d 2 1 1 T −6π( ) ∆3,p− − − e σ 2 d 3d (298) 2 2 2 h ∆k0 ∆k0 ∆k0 i d 2 1 1 1 1 1 3 p 1 1 2 m 1 −2π( σ ) ∆3,p− 2 − d + d + ∆3,p− 2 − d + d − d + ∆3,p− 2 − d + d − d ≤ e (299) d 2 2 2 2 1 −2π d 1 − 2 1 − π d 1− 4 ≤ e ( σ ) ( 2 d ) = e 2 ( σ ) ( d ) , d = 4, 6, 8,..., (300) d d if “Inequalities (249), (261) are both satisfied” is false. Hence, finally, taking into account Eq. (265) and Eqs. (286), (300), we find that for all m, p = 0, . . . , d − 1, such that m ≤ p,13

1 σ −i2πkQ/d˜ −iπ 2 (m+p)Q/d Int3(m, p) =√ e e 3 (301) 6 d 2 2 m+∆k0−∆k0 + p+∆k0−∆k0 − m+∆k0−∆k0 p+∆k0−∆k0 4 d 2 [ 1 2] [ 1 3] [ 1 2][ 1 3] −π 3 ( σ ) 2 2 d − π ( σ ) Q2 e e 6 d + ε3T, (302)

13 This last condition is simply to ensure that k¯2 ≤ k¯3, as has been assumed from the outset.

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 33 where for d = 4, 6, 8 ... n o |ε3T| ≤ max |εF | , |ε3(m, p)| (303)

2 2 2 2 2 2 2 1 σ −π 3 d 1− 3 1 − π d 1− 4 1 σ − π d 1− 2 ≤ max √ e 4 ( σ ) ( d ) + e 2 ( σ ) ( d ) , e 2 ( σ ) ( d ) (304) 6 d d 3π d

2 2 2 σ − π d 1− 4 = O e 2 ( σ ) ( d ) as d → ∞, (305) d

L R where we have used Eqs. (289), (268) and the values of ε3,1, ε3,1 from Eqs. (282), (276).

H.3 Approximating the integrals in lines (235), (236). The steps taken in the previous section for approximating the integral in line (237), can be repeated for the integrals in lines (235), (236). We summarise here the ﬁnal results, starting with the integrals in line (235): For ¯ ¯ the parametrization k2/d = 1/2 + 1/d − k/d, k3/d = 1/2 + 1/d − l/d, k, l = 1, . . . , d.

0 2 0 2 0 2 ¯ 2h 1−∆k ¯ ∆k ¯ ∆k i Z k2/d+1/2−1/d −2π d x− 1 − 1 + x− k2 + 2 + x− k3 + 3 −i2πxQ ( σ ) 2 d d d d d Int1(k, l) := dx e e (306) 0 1 σ −2iπkQ˜ iπ 2 (k+l)Q/d = √ e e 3 (307) 6 d 2 2 k+∆k0−∆k0 + l+∆k0−∆k0 − k+∆k0−∆k0 l+∆k0−∆k0 4 d 2 [ 2 1] [ 3 1] [ 2 1][ 3 1] −π 3 ( σ ) 2 2 d − π ( σ ) Q2 e e 6 d + ε1T, (308) where 2 2 2 σ − π d 1− 4 |ε | = O e 2 ( σ ) ( d ) as d → ∞. (309) 1T d ¯ ¯ In the case of the integral in line Eq. (236), for the parametrization k2/d = −1/2 + 1/d + m/d, k3/d = 1/2 + 1/d − n/d, m = 0, . . . , d − 1, n = 1, . . . , d; we ﬁnd

0 2 0 2 0 2 ¯ 2h 1−∆k ¯ ∆k ¯ ∆k i Z k3/d+1/2−1/d −2π d x− 1 − 1 + x−1− k2 + 2 + x− k3 + 3 −i2πxQ ( σ ) 2 d d d d d Int2(m, n) := dx e e (310) k¯2/d+1/2−1/d

1 σ −i2πkQ/d˜ −iπ 2 (m−n)Q/d = √ e e 3 (311) 6 d 2 2 m−∆k1+∆k0 + n+∆k1−∆k0 + n+∆k1−∆k0 m−∆k1+∆k0 4 d 2 [ 2 1] [ 3 1] [ 3 1][ 2 1] −π 3 ( σ ) 2 2 d − π ( σ ) Q2 e e 6 d + ε2T, (312)

where 2 2 2 σ − π d 1− 4 |ε | = O e 2 ( σ ) ( d ) as d → ∞. (313) 2T d

˜ H.4 Final expression for FQ(k1, k2, k3)

With the results from the previous section, we can ﬁnally formulate a useful approximation for F˜Q(k1, k2, k3). ¯ Plugging Eqs. (308), (312), (302), into Eq. (238) and after some re-parametrization, we ﬁnd for k1/d = ¯ ¯ ¯ ¯ ¯ −1/2 + 1/d, k2/d = −1/2 + 1/d + m/d, k3/d = −1/2 + 1/d + p/d, m, p = 0, . . . , d − 1, with k1 ≤ k2 ≤ k3,

6 2 A σ −i2πkQ/d˜ − π ( σ ) Q2 F˜Q(k1, k2, k3) =√ e e 6 d GQ(d; m, p) + εT, (314) 6 d where

2 4 d 2 2 4 d 2 −iπ (m+p)Q/d −π ( ) H1(d,m,p) −iπ (m+p−d)Q/d −π ( ) H2(d,m,p) GQ(d; m, p) :=e 3 e 3 σ + e 3 e 3 σ (315) 2 −iπ 2 (m+p−2d)Q/d −π 4 d H (d,m,p) + e 3 e 3 ( σ ) 3 , (316)

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 34 and 2 2 [m+∆k0−∆k0] +[p+∆k0−∆k0] −[m+∆k0−∆k0][p+∆k0−∆k0] H (d, m, p) := 1 2 1 3 1 2 1 3 , (317) 1 d2

2 2 [m−∆k0+∆k0] +[−p+d+∆k0−∆k0] +[−p+d+∆k0−∆k0][m−∆k0+∆k0] H (d, m, p) := 2 1 3 1 3 1 2 1 , (318) 2 d2

2 2 [−m+d+∆k0−∆k0] +[−p+d+∆k0−∆k0] −[−m+d+∆k0−∆k0][−p+d+∆k0−∆k0] 2 1 3 1 2 1 3 1 . H3(d, m, p) := d2 (319)

The small error term εT in Eq. (314) is deﬁned via 6 εT = A (ε1T + ε2T + ε3T) + εI1 + 7εI0, (320) and bounded by 6 |εT| = A (ε1T + ε2T + ε3T) + εI1 + 7εI0 (321) √ 3 2 ≤ + |ε | |ε | + |ε | + |ε | + ε + 7ε (322) σ A 1T 2T 3T I1 I0

2 2 1 − π d 1− 4 = O e 2 ( σ ) ( d ) + ε + 7ε as d → ∞, (323) d σ I1 I0 √ 2 where we have used A = 2/σ + εA = O(1/σ) (from page 483 in [21]) in line (322), followed by Eqs. (309), (313). Finally, using Eq. (233) we conclude

2 2 2 1 − π d 1− 4 d − π σ2α2 |ε | = O e 2 ( σ ) ( d ) + O e 4 0 as d → ∞. (324) T d σ σ5/2 0 0 0 We can now easily generalise this result to any measurement outcomes. Deﬁne k1, k2, k3 ∈ Id = {0, 1, . . . , d − 1}, by

 d ˜0  − + 1 + bk1c + l if q = 1,  2 (mod. d) 0 d ˜0 kq := [kq + l](mod. d) = − 2 + 1 + bk2c + l + m (mod. d) if q = 2, (325)  d ˜0  − 2 + 1 + bk3c + l + p (mod. d) if q = 3, for l, m, p = 0, . . . , d − 1. Rather than the previous case where the measurement outcomes kq where chosen ˜0 0 0 0 kq ∈ Sd(kq ), q = 1, 2, 3; in the present case of the measurement outcomes k1, k2, k3, for simplicity we will use 0 0 0 14 0 the convention k1, k2, k3 ∈ Id = {0, 1, . . . , d − 1} , with outcome kq associated with projective measurement |θ 0 ihθ 0 | , q = 1, 2, 3. kq kq q Now employing Eqs. (60), (314) we achieve ˜ 0 0 0 −i2πlQ/d ˜ FQ(k1, k2, k3) = e FQ(k1, k2, k3) (326) 6 A σ ˜ π σ 2 2 −i2πklQ/d − ( ) Q −i2πlQ/d = √ e e 6 d GQ(d; m, p) + εT e , if m ≤ p (327) 6 d Here we have defined the common phase, d ∆k0 k˜ := − + 1 − T + l, (328) l 2 3 0 0 0 0 where recall ∆kT is defined in Eq. (267). Finally, we note that not all values of k1, k2, k3 ∈ Id are achievable due to the constraint m ≤ p. However, we can remedy this by using property 3) [see Eq. (61)]. Since in this case ˜0 ˜0 ˜0 the clocks only differ by the values k1, k2, k3, interchanging ρC,2(t) with ρC,3(t) is equivalent to interchanging ˜0 ˜0 ¯ ˜0 ˜0 ¯ ˜0 ˜0 k2 with k3. Furthermore, recalling k2 = k2 + bk2c = −d/2 + 1 + m + bk2c, k3 = k3 + bk3c = −d/2 + 1 + p + bk3c, ˜0 ˜0 ˜0 ˜0 we see that interchanging k2 with k3 and k2 with k3 is equivalent to interchanging k2 with k3 and m with p. ¯ ¯ Finally, by construction k2 > k3 iff m > p. Thus using Eq. (327) we find that if m > p,

" 6 # A σ ˜ π σ 2 2 ˜ 0 0 0 −i2πklQ/d − 6 ( d ) Q −i2πlQ/d FQ(k1, k2, k3) = √ e e GQ(d; m, p) + εT e (329) 6 d ˜0 ˜0 k2 ↔ k3 m ↔ p 6 A σ ˜ π σ 2 2 h i −i2πklQ/d − ( ) Q = √ e e 6 d GQ(d; p, m) +ε ˜T, (330) ˜0 ˜0 6 d k2 ↔ k3

14Note that this simply corresponds to a shifting the values of all the measurement outcomes by a constant value.

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 35 where |ε˜T| satisﬁes the same upper bound in Eq. (323) that |εT| satisﬁes. Thus, in general, from Eqs. (327) and (330) we conclude

6 A σ ˜ π σ 2 2 0 0 0 −i2πklQ/d − ( ) Q F˜Q(k , k , k ) = √ e e 6 d G˜Q(d; m, p) +ε ¯T, (331) 1 2 3 6 d where  GQ(d; m, p) if m ≤ p G˜Q(d; m, p) := h i (332) GQ(d; p, m) if p < m  ˜0 ˜0 k2 ↔ k3 and |ε¯T| is upper bounded by the r.h.s. of Eq. (323).

ˆ H.5 Bounding kEk1

Now that we have from the previous section an expression for F˜Q for all measurement outcomes of the 3 clocks, the next task is to bound kEˆk1 deﬁned in 72 to be

ˆ p X ˜ 0 0 0 ~ 2 ||E||1 ≤dL dP F0(k1, k2, k3) max |p(Q, k)| (333) Q 0 0 0 k1,k2,k3 0 0 0 F˜Q(k , k , k ) p X ˜ 0 0 0 1 2 3 2πikαQ/d =dL dP F0(k1, k2, k3) max e − 1 (334) Q ˜ 0 0 0 0 0 0 F0(k1, k2, k3) k1,k2,k3

p X ˜ 0 0 0 2πikαQ/d ˜ 0 0 0 =dL dP max FQ(k1, k2, k3) e − F0(k1, k2, k3) (335) Q 0 0 0 k1,k2,k3 p A6 σ =dL dP √ max (336) 6 d Q d−1 2 X − π σ Q2 2πi(k −k˜ )Q/d 6 ( d ) ˜ α l ˜ e GQ(d; m, p)e − G0(d; m, p) +ε ˜T +ε ¯T . (337) l,m,p=0

In all cases, the optimization is over Q ∈ {−(∆hCo + ∆hL),..., (∆hCo + ∆hL)}. In order to proceed further, we need to specify kα for this protocol. If there were no phase errors, it turns ¯0 ¯0 out that defining it equal to either of the three quantities k1 := −d/2 + 1 + l, k2 := −d/2 + 1 + l + m, ¯0 k3 := −d/2 + 1 + l + p, would suffice. Note how doing so would only require information from one clock measurement — the other two clock results would be redundant information. Intuitively, this is because with high probability, all three clocks will have measurement outcomes corresponding to approximately the same elapsed time (if there were no phase errors). For the case of no phase error, it is convenient to prove that the 15 ¯0 protocol works up to the specified error when kα is chosen to be “any angle in-between the three angles k1, ¯0 ¯0 ¯0 ¯0 ¯0 k2, k3” when k1, k2, k3 belong to a to-be specified domain, and “any angle otherwise”. Later in section H.6 we will show how such a result is sufficient in the case of the unknown phase error. Specifically, we define d k = − + 1 + γ (α) (338) α 2 l,m,p where γl,m,p are functions indexed by l, m, p = 0, . . . , d − 1 of the form

γl,m,p(α) = α. (339)

Due to the modular arithmetic, depending on which sector l, m, p belong to, the domain of γl,m,p changes. Speciﬁcally, for R ∈ [1, d/4] we deﬁne the sets

2 S11(R) := (m, p) ∈ Z | 0 ≤ m, p ≤ R − 1 , (340) 2 S31(R) := {(m, p) ∈ Z | 0 ≤ m ≤ R − 1 & d − R ≤ p ≤ d − 1}, (341) 2 S13(R) := {(m, p) ∈ Z | d − R ≤ m ≤ d − 1 & 0 ≤ p ≤ R − 1}, (342) 2 S33(R) := {(m, p) ∈ Z | d − R ≤ m, p ≤ d − 1}. (343)

15Throughout this proof, we call “angle” to real numbers which only need to be speciﬁed up to modulo d. These angles can be converted to radian by multiplying them by 2π/d.

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 36 Figure 2: Dom(γl,m,p) for (m, p) in diﬀerent intervals and l = 0, 1, . . . , d − 1. Case 1): Yellow highlighted interval represents Dom(γl,m,p) for (m, p) ∈ S11(R). Case 2): Yellow highlighted interval represents Dom(γl,m,p) for (m, p) ∈ S31(R). Case 3): Yellow highlighted interval represents Dom(γl,m,p) for (m, p) ∈ S33(R). The case Dom(γl,m,p) for (m, p) ∈ S13(R) is the same as case 2) above under the interchange m ↔ p. In all four cases, CoDom(2πγl,m,p/d) is “any angle between angles l, m, p.”

Now we can deﬁne Dom(γl,m,p):  l, l + 1, . . . , l + max{m, p} if (m, p) ∈ S11(R)  l + p, l + p + 1, . . . , l + m + d if (m, p) ∈ S (R)  31 Dom(γl,m,p) := l + m, l + m + 1, . . . , l + p + d if (m, p) ∈ S13(R) (344)  l + min{m, p}, l + min{m, p} + 1, . . . , l + d if (m, p) ∈ S33(R)  l, l + 1, . . . , l + d − 1 otherwise

Observer that l, (l + m)(mod. d), (l + p)(mod. d) ∈ Dom(γl,m,p)(mod. d) always holds. In particular, the function γm,p(α) covers up to (but not more than), all angles 0, 1, . . . , d−1 which are between the three angles l, l+m, l+p whenever (m, p) ∈ S11(R) ∪ S31(R) ∪ S13(R) ∪ S33(R) and l ∈ Id. See Fig.2. Writing Eq. (337) in terms of this new notation, we have

p A6 σ ||Eˆ||1 ≤dL dP √ max (345) 6 d Q∈{−(∆hCo+∆hL),...,(∆hCo+∆hL)}   d−1 X π σ 2 2 2πiγ (α)+ ∆kT Q/d − 6 ( d ) Q ˜ m,p 3 ˜  e GQ(d; m, p)e − G0(d; m, p) + 2d |εˆT| (346)

l,m,p=0

 d−1  p A6 σ X =dL dP √ max  ∆F (d; l, m, p) + 2d |εˆT| , (347) 6 d Q∈{−(∆hCo+∆hL),...,(∆hCo+∆hL)} l,m,p=0

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 37 where |εˆ | = max 0 |ε˜ +ε ¯ |/2 is upper bounded by the r.h.s. of Eq. (323). It is now convenient to bound the T k1 T T summand ∆F (d; l, m, p). Using Eqs. (316) and (331), we ﬁnd

π σ 2 2 ∆kT − Q 2πi γm,p(α)−l+ Q/d 6 ( d ) ˜ 3 ˜ ∆F (d; l, m, p) := e GQ(d; m, p)e − G0(d; m, p) (348)

∆k π σ 2 2 2 T 4 d 2 ˜ − ( ) Q −iπ (m+p)Q/d 2πi γm,p(α)−l+ 3 Q/d −π ( ) H1(d,m,p) ≤ e 6 d e 3 e − 1 e 3 σ (349)

∆k π σ 2 2 2 T 4 d 2 ˜ − ( ) Q −iπ (m+p−d)Q/d 2πi γm,p(α)−l+ 3 Q/d −π ( ) H2(d,m,p) + e 6 d e 3 e − 1 e 3 σ (350)

∆k π σ 2 2 2 T 4 d 2 ˜ − ( ) Q −iπ (m+p−2d)Q/d 2πi γm,p(α)−l+ 3 Q/d −π ( ) H3(d,m,p) + e 6 d e 3 e − 1 e 3 σ , (351) where for q = 1, 2, 3, we have deﬁned  Hq(d, m, p) if m ≤ p H˜q(d, m, p) := h i (352) Hq(d, p, m) if p < m  0 0 k2 ↔ k3 The evaluation of the summations over m, p for ∆F (d; l, m, p) is more complicated than the summation over l and it is convenient to perform the summation separately over subsets of the domain of m, p. With this in mind, we deﬁne for R ∈ [1, d/4] the following sets.

Stot = S11(R) ∪ S¯12(R) ∪ S13(R) ∪ S¯21(R) ∪ S¯22(R) ∪ S¯23(R) ∪ S31(R) ∪ S¯32(R) ∪ S33(R). (353) 2 where Stot = {(m, p) ∈ Z | 0 ≤ m, p ≤ d − 1} is the set m and p run over in the summation in Eq. (347) and S11(R), S31(R), S11(R), S33(R), are the sets which when R is chosen to scale with d appropriately, will include 0 0 all measurement outcomes k2, k3 which, as a whole, occur with high probability. On the other hand, the sets 2 S¯12(R) := {(m, p) ∈ Z | R ≤ m ≤ d − 1 − R & 0 ≤ p ≤ R − 1} (354) 2 S¯21(R) := {(m, p) ∈ Z | 0 ≤ m ≤ R − 1 & R ≤ p ≤ d − 1 − R} (355) 2 S¯22(R) := {(m, p) ∈ Z | R ≤ m, p ≤ d − 1 − R} (356) 2 S¯23(R) := {(m, p) ∈ Z | d − R ≤ m ≤ d − 1 & R ≤ p ≤ d − 1 − R} (357) 2 S¯32(R) := {(m, p) ∈ Z | R ≤ m ≤ d − 1 − R & d − R ≤ p ≤ d − 1}, (358) 0 0 correspond to all measurement outcomes k2, k3 which, as a whole, occur with very low probability. See graphical depiction Fig.3. In addition to the limits Eqs. (186) assumed thought, we now demand that R satisfies the following limits. The exact choice of R will be determined later. R R lim = +∞, lim Qmax = 0, (359) d→+∞ σ d→+∞ d 16 where Qmax := maxQ∈{−(∆hP+∆hL),...,(∆hP+∆hL)} |Q| = ∆hL+∆hP. We now want to bound the ∆F (d, l, m, p) in Eq. (348) for measurement outcomes in sets Eqs. (340) to (343). We start with the set S12. For all (m, p) ∈ S12(R), we have that limd→∞ H˜2(d, m, p) = limd→∞ H˜3(d, m, p) = 1, so the terms in lines Eqs. 2 (350),(351) are exponentially small in (d/σ) . On the other hand, limd→∞ H˜2(d, m, p) converges to zero and the common factor in line (349) is significant. For the term in line Eq. (349), it is the terms within the absolute value which are small. Specifically, for all αl,m,p ∈ Dom(γl,m,p) and for all (m, p) ∈ S12(R), l ∈ Id,

∆k π σ 2 2 2 T − Q −iπ (m+p)Q/d 2πi γl,m,p(αl,m,p)−l+ Q/d 6 ( d ) 3 3 e e e − 1 (360)

2 π σ 2 2 ∆kT = − Q − iπ (m + p)Q/d + 2πi γl,m,p(αl,m,p) − l + Q/d (361) 6 d 3 3 π σ 2 2 ∆k 2 + O − Q2 − iπ (m + p)Q/d + 2πi γ (α ) − l + T Q/d (362) 6 d 3 l,m,p l,m,p 3 2 4 R R 4 R R ≤ π Qmax + 2π Qmax + O π Qmax + 2π Qmax (363) 3 d d 3 d d 10 R 10 R 2 =π Q + O π Q , (364) 3 d max 3 d max

16 Note that Qmax ≥ 1 uniformly in d and thus it follows from Eq. (359) that limd→∞ R/d = 0.

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 38 푝 푑 − 1

풮31(푅) 풮32ҧ (푅) 풮33(푅) 푑 − 푅 푑 − 1 − 푅

ҧ ҧ 풮21(푅) 풮22ҧ (푅) 풮23(푅)

푅 푅 − 1

풮11(푅) 풮12ҧ (푅) 풮13(푅) 푚 0 푅 − 1 푑 − 푅 푑 − 1 푅 푑 − 1 − 푅

Figure 3: Diagrammatic view of the nine distinct sets used in the proof. Note that all pairs have the empty set as their intersection and the union of all sets is the set Stot given by Eq. (353).

where we have used that for (m, p) ∈ S11(R), Dom(γl,m,p) = l, . . . , l + max{m, p}. Thus we conclude that for all αl,m,p ∈ Dom(γl,m,p) and for all (m, p) ∈ S12(R), l ∈ Id

2 2 10 R 10 R −π 4 d ∆F (d; l, m, p) ≤ π Q + O π Q + O 2e 3 ( σ ) . (365) 3 d max 3 d max

Now consider the set S33. For all (m, p) ∈ S33(R), we have that limd→∞ H˜1(d, m, p) = limd→∞ H˜2(d, m, p) = 1, so the terms in lines Eqs. (349),(350) are exponentially small in (d/σ)2. For the other term, i.e. line Eq. (351), we proceed by bounding the part within absolute values. Speciﬁcally, for all αl,m,p ∈ Dom(γl,m,p) and for all (m, p) ∈ S33(R), l ∈ Id,

∆k π σ 2 2 2 T − ( ) Q −iπ (m+p−2d)Q/d 2πi γl,m,p(αl,m,p)−l+ 3 Q/d e 6 d e 3 e − 1 (366)

∆k π σ 2 2 2 T − ( ) Q −iπ (m+p−2d)Q/d 2πi γl,m,p(αl,m,p)−l−d+ 3 Q/d = e 6 d e 3 e − 1 (367)

2 4 R R 4 R R ≤ π Qmax + 2π Qmax + O π Qmax + 2π Qmax (368) 3 d d 3 d d 10 R 10 R 2 =π Q + O π Q , (369) 3 d max 3 d max where we have used that for (m, p) ∈ S33(R), CoDom(γl,m,p(αl,m,p) − l − d) = min{m, p} − d, . . . , 0 and −R ≤ min{m, p} − d, giving us |γl,m,p(αl,m,p) − l − d| ≤ R. Thus we conclude that for all αl,m,p ∈ Dom(γl,m,p) and for all (m, p) ∈ S33(R), l ∈ Id

2 2 10 R 10 R −π 4 d ∆F (d; l, m, p) ≤ π Q + O π Q + O 2e 3 ( σ ) . (370) 3 d max 3 d max

In general, for every set Sxy(R), x, y ∈ {1, 3}, two out of the three terms H˜1(d; m, p), H˜2(d; m, p), H˜3(d; m, p) converge to unity as d tends to inﬁnity, resulting in two of the lines Eqs. (349), (350), (351) being exponentially small in (d/σ)2. The terms in the remaining line can be expended in a power expansion similarly to Eq. (369), resulting in an order QmaxR/d contribution. We therefore conclude that the following is true for all x, y ∈ {1, 3}: For all αl,m,p ∈ Dom(γl,m,p) and for all (m, p) ∈ Sxy(R), l ∈ Id,

2 2 10 R 10 R −π 4 d ∆F (d; l, m, p) ≤ π Q + O π Q + O 2e 3 ( σ ) . (371) 3 d max 3 d max

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 39 2 From the deﬁnition of the sets x, y ∈ {1, 3} with m, p ∈ Sxy(R) we see that they all have cardinality R . Thus from Eq. (371), it follows that for all αl,m,p ∈ Dom(γl,m,p) and for all (m, p) ∈ S11(R)∪S13(R)∪S31(R)∪S33(R), l ∈ Id, X X ∆F (d; l, m, p) (372)

l∈Id (m,p)∈S11(R)∪S13(R)∪S31(R)∪S33(R) 2 ! 2 2 10 R 10 R −π 4 d ≤ dR π Q + O π Q + O 2e 3 ( σ ) . (373) 3 d max 3 d max

We will now bound ∆F (d; l, m, p) for (m, p) belonging to the sets Eqs. (354) to (354). We start with the set S¯12. For all (m, p) ∈ S¯12(R), we have for suﬃciently large d

d 2 R2 H˜ (d, m, p) ≥ , (374) σ 1 σ d 2 R2 H˜ (d, m, p) ≥ , (375) σ 2 σ d 2 d 2 H˜ (d, m, p) ≥ . (376) σ 3 σ

Thus from Eq. (348) we conclude that for all αl,m,p ∈ Dom(γl,m,p) and for all (m, p) ∈ S¯12(R), l ∈ Id,

2 2 −π 4 d −π 4 R ∆F (d; l, m, p) ≤ O 2 e 3 ( σ ) + O 4 e 3 ( σ ) . (377)

Similarly, one ﬁnds that Eq. (377) holds for all αl,m,p ∈ Dom(γl,m,p) and for all (m, p) ∈ S¯xy(R), l ∈ Id, for xy = 21, 23, 32. In the case of the set S¯22(R) we ﬁnd that for all αl,m,p ∈ Dom(γl,m,p) and for all (m, p) ∈ S¯22(R), l ∈ Id, 2 −π 4 R ∆F (d; l, m, p) ≤ O 6 e 3 ( σ ) . (378)

2 Finally, since the cardinality of the sets S¯12(R), S¯21(R), S¯22(R), S¯23(R), S¯32(R) are all upper bounded by d , we have that for all αl,m,p ∈ Dom(γl,m,p), and for all (m, p) ∈ S¯12(R) ∪ S¯21(R) ∪ S¯22(R) ∪ S¯23(R) ∪ S¯32(R), l ∈ Id, X X ∆F (d; l, m, p) (379)

l∈Id (m,p)∈S¯12(R)∪S¯21(R)∪S¯22(R)∪S¯23(R)∪S¯32(R) 2 2 3 −π 4 d 3 −π 4 R ≤ O d e 3 ( σ ) + O d e 3 ( σ ) . (380)

Hence, combining Eqs. (373) and (380), and plugging into Eq. (347) we ﬁnd for all αl,m,p ∈ Dom(γl,m,p) and for all (m, p) ∈ Stot, l ∈ Id

" 2 ! 6 2 p A σ 2 10 R R −π 4 ( d ) ||Eˆ||1 ≤ dLdP √ R π Qmax + O Qmax + O e 3 σ (381) 6 3 d d # 2 2 2 −π 4 ( d ) 2 −π 4 ( R ) 2 + O d e 3 σ + O d e 3 σ + 2d |εˆT| (382)

6 " 2! p A σ 2 10 R R = dLdP √ R π Qmax + O Qmax (383) 6 3 d d # 2 2 −π 4 ( R ) 2 + O d e 3 σ + 2d |εˆT| , (384) where, to ﬁnd the higher order terms, we have used in the last equality

R/σ lim = 0, (385) d→∞ d/σ

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 40 √ 2 which follows from Eq. (359). Finally, using A = 2/σ + εA = O(1/σ) from page 483 in [21] and the upper bound on in Eq. (324), we ﬁnd for all αl,m,p ∈ Dom(γl,m,p) and for all (m, p) ∈ Stot, l ∈ Id,

" 2 2! p 2 R 10 R R ||Eˆ||1 ≤ dLdP √ π Qmax + O Qmax (386) 3 σ 3 d d 2 ! # 2 4 d −π 4 R d − π σ2α2 + O e 3 ( σ ) + O e 4 0 , as d → ∞. (387) σ σ9/2

We have the error term ||Eˆ||1 in terms of R, and σ. These are free parameters that we can choose. The optimal scaling is given by σ = ln5/2 d R = ln4 d, (388) as the errors become

" 7 11 p 2 10π ln (d)Qmax ln (d) 2 ||Eˆ||1 ≤ dLdP √ + O Q (389) 3 3d d2 max ! !# 1 1 + O + O , as d → ∞, (390) d ln5/2(d) d ln45/4(d) for all αl,m,p ∈ Dom(γl,m,p) and for all (m, p) ∈ Stot, l ∈ Id. Asymptotically, the ﬁrst two error terms will be the dominant ones, and thus we conclude that for all αl,m,p ∈ Dom(γl,m,p) and for all (m, p) ∈ Stot, l ∈ Id,

" 7 11 # p 2 10π ln (d)Qmax ln (d) 2 ||Eˆ||1 ≤ dLdP √ + O Q , as d → ∞, (391) 3 3d d2 max

√ 7 dL dP ln (d)Qmax making the error eﬀectively scale as O d . Finally using Eq. (83), we achieve for all αl,m,p ∈ Dom(γl,m,p) and for all (m, p) ∈ Stot, l ∈ Id.

" 7 11 # 2 p 10π ln (d)Qmax ln (d) 2 f ≥ 1 − dLdP √ + O Q , as d → ∞, (392) worst 3d d2 max where Qmax = ∆hL + ∆hCo.

H.6 The Protocol Here we provide the protocol (based on the results from the previous subsection) which will achieve the statement of Theorem4 . Recall that the general protocol for all set-ups in this paper is described in AppendixB, thus here we only need to speciﬁcity how to choose the parameter kα given the clock measurement outcomes. 0 0 0 After obtaining the measurement outcomes k1, k2, k3 ∈ {0, 1, . . . , d − 1}, the second task will be to work out the quantities ˜l, ˜l +m ˜ , ˜l +p ˜ which are deﬁned via the relations  ˜l if q = 1, d  k0 = − + 1 + bk0c + Γ˜ , Γ˜ = ˜l +m ˜ if q = 2, (393) q 2 q q q (mod. d) ˜l +p ˜ if q = 3,

˜ ˜ ˜ ˜ where l, m,˜ p˜ ∈ {0, 1, . . . , d − 1} and Id = {0, 1, . . . , d − 1}. Importantly Γ1, Γ2, Γ3 are always calculable since d, 0 0 0 k1, k2, k3 are known. ˜ ˜ ˜ Next, out of the three angles Γ1, [Γ2](mod. d), [Γ3](mod. d) ∈ [0, d − 1], ﬁnd which one of these is the middle angle; denoted αmiddle. Speciﬁcally, this angle can be calculated as follows:

1. First, for q1 < q2 with q1, q2 ∈ {1, 2, 3}, calculate the quantities  [Γ˜ ] − [Γ˜ ] if [Γ˜ ] − [Γ˜ ] ≤ d − 1  q1 (mod. d) q2 (mod. d) q1 (mod. d) q2 (mod. d) 2 ∆ := (394) q1,q2 d − [Γ˜ ] − [Γ˜ ] if [Γ˜ ] − [Γ˜ ] ≥ d q1 (mod. d) q2 (mod. d) q1 (mod. d) q2 (mod. d) 2

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 41 2. Second, ﬁnd the set containing the smallest two out of the values ∆1,2, ∆1,3, ∆2,3:

Case 2.1) {∆1,3, ∆2,3} if ∆1,2 ≥ ∆1,3, ∆1,2 ≥ ∆2,3 (395)

Case 2.2) {∆1,3, ∆1,2} if ∆2,3 ≥ ∆1,2, ∆2,3 ≥ ∆1,3 (396)

Case 2.3) {∆1,2, ∆2,3} if ∆1,3 ≥ ∆1,2, ∆1,3 ≥ ∆2,3. (397)

3. Third, ﬁnd the common index w of the smallest two:  3 in case 2.1)  w := 1 in case 2.2) (398) 2 in case 2.3)

4. Fourth, the middle angle is ˜ αmiddle = Γw. (399)

We call w ∈ 1, 2, 3 in Eq. (399) corresponding to the middle angle, the location of the middle angle. Set α in Eq. (338) to the middle angle, α = αmiddle,

d d k = − + 1 + γ (α ) = − + 1 + α . (400) α 2 l,m,p middle 2 middle

Since Eq. (392) holds for all αl,m,p ∈ Dom(γl,m,p) and for all (m, p) ∈ Stot, l ∈ Id, the protocol would be complete, since calculation of the r.h.s. of Eq. (400) only requires the knowledge of known parameters 0 0 0 0 0 0 d, k1, k2, k3, k1, k2, k3. However, we are not quite done, since what is left to prove is that αmiddle ∈ Dom(γl,m,p) indeed holds true for all (m, p) ∈ Stot, l ∈ Id, since this has been assumed in Eq. (400) without justiﬁcation. To do so, we have to analyse two possible scenarios. First, however, recall Eq. (325); reproduced here for convenience:  l if q = 1, d  k0 := [k + l] = − + 1 + bk˜0c + Γ , Γ = l + m if q = 2, (401) q q (mod. d) 2 1 q q (mod. d) l + p if q = 3, where Id = {0, 1, . . . , d − 1} and ( 0 ˜0 kq + tphd/T0 if q = r, kq = 0 (402) kq otherwise, with r ∈ 1, 2, 3 the unknown location of the unknown phase tph ∈ R. The two scenarios are:

Case 1) The unknown location r ∈ 1, 2, 3 does not coincide with the location of the middle angle (w 6= r): In this case, by comparing Eqs. (393), (401), we see that the angle αmiddle corresponds to one of the angles Γ1, Γ2, Γ3. By deﬁnition, all three of these angles belong to the domain of γm,p for all (m, p) ∈ Stot and for all l ∈ Id. Case 2) The unknown location r ∈ 1, 2, 3 coincides with the location of the middle angle (w = r): In this case, it follows that the two angles out of the three angles Γ1, Γ2, Γ3 which do not correspond to the middle angle αmiddle, (namely angles Γh, Γk, where h 6= k 6= r) have not had an unknown phase applied to them (since ˜ ˜ by deﬁnition, there is only one unknown phase). Hence from Eqs. (393), (401) we have Γh = Γh, Γk = Γk. Hence since the middle angle is always between the other two angles Γh, Γk, and Dom(γl,m,p) includes all angles which ˜ are between all three angles Γ1, Γ2, Γ3 (See Fig.2), it follows that the middle angle αmiddle = Γr ∈ Dom(γl,m,p) for all (m, p) ∈ Stot, l ∈ Id.

Hence cases 1) and 2) together prove the assumption made in Eq. (400), namely that αmiddle ∈ Dom(γl,m,p) indeed holds true for all (m, p) ∈ Stot, l ∈ Id. This completes the proof.

I Quantum communication without a shared reference frame

0 Throughout the paper we have shown that the Quasi-Ideal clock states |ψnor(k1)i achieve the best bounds for the task of maximizing the entanglement ﬁdelity of a covariant code. Here, following [31], we also show how

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 42 they can improve on a pre-existing scheme for another related task: the transmission of quantum states between parties which do not share a common reference frame. The setting is as follows: Alice and Bob are connected by a noiseless quantum channel, but separated by an indeterminate amount of “time”. This means that the channel connecting then takes the form

Z T dt E(·) = G(·) := lim U(t)(·)U †(t), (403) T →∞ 0 T where U(t) is the time translation of the systems that are sent. This is a decoherence map in the basis of the generator of U(t). In order to damp the loss of information from the channel, what Alice can do is to send the message ρM together with a reference frame F, as G(ρM ⊗ |ψFi hψF|) which Bob then uses to “decode” the message. To decode, Bob measures the clock to learn the time that separates them, which can be used to retrieve the message up to some error (we label this decoding procedure with D). In [31] it was shown that for a qubit, the resultant channel is given by

Z T dt 2 † D ◦ E(ρ) = lim | hψ| UF(t) |ψi | UM (t)ρUM (t) = (1 − p)I + pG(ρ), (404) T →∞ 0 T where the probability of fully decohering p is given by

A − A p = 1 2 , (405) A1 where Z T dt 2 A1 = lim | hψ| UF(t) |ψi | , (406) T →∞ 0 T Z T dt 2 −i 2πt A2 = lim | hψ| UF(t) |ψi | e T . (407) T →∞ 0 T

1 In [31], it was found that for the Salecker-Wigner-Peres clock, pPSW = with d the dimension of the clock. dC+1 We now show how this changes when one uses the Quasi-Ideal clock |ψ(0)i as deﬁned in the main text. In that case, it can be shown that

2 2 2 π 2πdCt π 2πdCt 2 − 2 ω − 2 dC− ω | hψ| UF(t) |ψi | ∝ e 2σ + e 2σ + 1, (408) where ω is the frequency at which the qubit oscillates, and

d2 2 C −c1σ −c2 |1| ≤ O(e ) + O(e σ2 ), (409) where c1, c2 are positive constants independent of dC and σ. Let us now sketch how to derive Eq. (408). First one writes out the characteristic function | hψ| UF(t) |ψi |, as a ﬁnite sum of terms corresponding to the elements of the eigenbasis {|φki}. This is

π 2 2πdCt 2 π 2 2πdCt 2 2 X − [k +(k− ) ] 2 X − [k +(k+dC− ) ] | hψ| UF(t) |ψi | = A e σ2 ω + A e σ2 ω

dC dC 2πdCt dC 2πdCt dC k∈{− 2 +1,− 2 +b ω c} k∈{− 2 +b ω c+1, 2 } (410) 2 2πd t 2πd t d 2 X − π [k2+(k− C )2] − π [k2+(k+d − C )2] −c C = A e σ2 ω + e σ2 C ω + O(e 2 σ2 ) (411) k∈Z h 2 i − π 2πdCt +m2σ2−2id2 m 2πdCt )σ2 X σ 2 ω C ω = A2 √ e (412) 2dC m∈Z h 2 i − π d − 2πdCt +m2σ2−2id2 m(d − 2πdCt )σ2 d2 σ 2 C ω C C ω −c C + √ e + O(e 2 σ2 ) (413) 2dC 2 2 d2 π 2πdCt π 2πdCt 2 C 2 σ − − dC− −c σ −c = A √ e 2 ω + e 2 ω + O(e 1 ) + O(e 2 σ2 ), (414) 2dC

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 43 where in going from the second to the third line we have used the Poisson summation formula (see for instance Corollary C.0.2 from [21]), and in the third to the fourth we have kept the m = 0 term and bounded the size of the others. With that, we obtain ! 4 d2 σ σ 2 C −c1σ −c2 A1/C = + O + O(e ) + O(e σ2 ) (415) dC dC ! 3 4 d2 σ σ σ −c σ2 −c C 1 2 σ2 A2/C = − π 3 + O + O(e ) + O(e ), (416) dC dC dC with C > 0 independent of dC and σ. Thus, to leading order, it gives a probability of decoherence of ! 2 3 d2 σ σ −c σ2 −c C 1 2 σ2 pid = π 2 + O + O(e ) + O(e ). (417) dC dC

3 The best choice for the width of the clock is σ = (ln dC) 2 , which gives a scaling of

3 !2  3 !3 (ln dC) 2 (ln dC) 2 pid = π + O   , (418) dC dC which (up to the logarithmic factor) is quadratically smaller than pPSW . This is essentially the same advantage as the one obtained in covariant error correction as shown in the main text.

J Evolution without evolution: connection with the Page-Wootters mechanism

In this Section we brieﬂy re-cap the Page-Wooters mechanism (Section III. “Evolution in a Stationary Universe” in [46]) and show how our formalism and the results in this paper ﬁt into that picture. These authors consider a bipartite setup consisting of a clock and a system S of interest. The system and clock are considered as a closed system evolving under a Hamiltonian of the form

Hˆ = HˆS ⊗ 1clock + 1S ⊗ Hˆclock. (419)

Their idea is to show that there is a global state ρSclock which does not evolve in time, i.e. [ρSclock, Hˆ ] = 0 yet the dynamics of the system, after conditioning on the clock being in a state

−τiHˆclock |ψ(τ)iclock = e |ψ(0)iclock , (420) at time τ, is given by the free evolution of the system S according to its own Hamiltonian HˆS. Namely,

−iτHˆS iτHˆS trclock[Pˆclock(τ)ρSclockPˆclock(τ)] = e trclock[ρSclock]e , (421) where Pˆclock(τ) := 1S ⊗ |ψ(τ)ihψ(τ)|clock is the projector onto the clock state at time τ. So in the above sense, even though the global state ρSclock is stationary, the state of the system conditioned on the clock being at a particular time, is evolving according to the Schr¨odinger equation for its own Hamiltonian HˆS.

Before seeing how this is connected to our work, a few remarks are pertinent. Firstly, time is assumed to be continuous, i.e. if one wishes Eq. (421) to hold for all τ in some finite subinterval of the real line, then one needs to use an Idealised clock (these are discussed in Section1 in the main text). These, however, are unphysical in the sense of requiring infinite energy. To the best knowledge of the authors, while many aspects of the Page-Wooters mechanism have been explored (see [48–51] and references therein), it remains an open question to how well finite dimensional clocks (or infinite dimensional clocks with finite energy) can realize this. It is also noteworthy that in our setup, the clock and system are only classically correlated, and thus our results demonstrate that quantum entanglement between the clock and system S is not necessary to fulfil the Page- Wooters mechanism. We now find the Hamiltonian and time invariant state which realise the Page-Wooters mechanism up to a quantifiable error incurred due to using physical clocks. Firstly, we can identify the physical space with the system in the Page-Wooters model and the clocks in our setup with those in Page-Wooters model. The total Hamiltonian on the system and the clock will be the

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 44 Kronecker sum of the generators of the physical space Lie group and those of the clock’s Lie group, namely (c.f. Eq. (419)) M ˆ ˆ 1⊗M 1 M ˆ H = HP ⊗ C + P ⊗ HC. (422) i=1

Secondly, we use our covariant encoding channel Ecov(·) in Eq. (8) to realise the stationary state ρSclock of the Page-Wooters mechanism ρSclock = Ecov(ρL), (423) for any ρL ∈ S (HL) and for the choice of trivial representation on the Logical space, namely UL(t) = 1L for all t ∈ R. To see how this works, note that 1) the unitary group representation on the physical and clock space has the Hamiltonian in Eq. (422) as its generator, and 2) that the encoding map is covariant. Speciﬁcally

−itHˆ itHˆ ⊗M † ⊗M† † R e Ecov(ρL)e = UCo(t) ⊗ UC (t) Ecov(ρL) UCo(t) ⊗ UC (t) = Ecov(UL(t)ρLUL(t)) = Ecov(ρL) ∀ t ∈ , (424) where in the last line, we have used UL(t) = 1L for all t ∈ R. Hence diﬀerentiating w.r.t. t on both sides on Eq. (424), we achieve the Page-Wooters condition,

[H,ˆ Ecov(ρL)] = 0 ∀ t ∈ R. (425)

Thirdly, our clock state must, at least approximately, satisfy Eq. (421). We have already calculated the state dC−1 of Ecov(ρL) when the clocks were projected onto the time basis {|θkihθk|}k=0 in SectionB. Now, we want to perform a similar calculation but for when the clock is projected onto the state of the clock at time τ. In the case of one Quasi-Ideal clock, this corresponds to covariant positive-operator valued measure, which up ˆ 0 0 † to a vanishing error in the large dC/σ and σ limit, are {PQI(τ) := UC(τ)|ψnor(k1)ihψnor(k1)|UC(τ) }τ∈[0,T0]. Proceeding analogously to SectionB, but this time not applying the error map Ej, we ﬁnd

h i ˆ ˆ ⊗K† ˆ0 ⊗K ρP(τ) := trC PQI(τ)Ecov(ρL)PQI(τ) =UP (τ) Eτ (ρL) + E(ρL) UP (τ), (426) where dP−1 dL−1 FQ(τ) ˆ0 X X 2πiτ Q/T0 0 Eτ (·) := e − 1 Eq,q0,n,n0 (·)|qihq |, (427) F0(τ) q,q0=0 n,n0=0 with Eq,q0,n,n0 give by Eq. (36) and FQ(τ) given by

Z T0 1 −iωQt † FQ(τ) := dte hψnor(k0)| UC(τ)ρC,1(t)UC(τ) |ψnor(k0)i (428) T0 0 1 Z T0 −i2πQt/T0 2 = dte |hψnor(k0)| UC(t − τ) |ψnor(k0)i| , (429) T0 0 where |ψnor(k0)i is the Quasi-Ideal clock described previously in the manuscript. Note that Eq. (426) is † analogous to Eq. (72) but with a trivial error map Ej = I (no clock errors), ρ˜L = UL(tα)ρLUL(tα) = ρL (trivial T0 logical group representation), and a time τ rather than the discrete times tα := kα (since we are not projecting dC onto the time basis any-more). One can bound Eqs. (429) and (427), analogously to the calculations in Sections −2πiτQ/T0 DC. We will not perform such calculations here for brevity, but one ﬁnds that FQ(τ)/F0(τ) ≈ e where the approximation becomes exact in the large dC limit. So one can see that the quantity in square brackets in Eq. (427) vanishes in said limit. Hence for large dC, from Eq. (426)

h i ˆ ˆ ˆ ˆ ⊗K† ⊗K −iτHCo iτHCo R trC PQI(τ)Ecov(ρL)PQI(τ) ≈ UCo E(ρL)UCo (τ) = e E(ρL)e ∀ τ ∈ , (430) where the approximate equality becomes exact in the large dC limit. This is an approximate Page-Wooters condition (Eq. (420)).

Accepted in Quantum 2020-02-22, click title to verify. Published under CC-BY 4.0. 45