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Energetic cost of Hamiltonian quantum gates

Sebastian Deffner , Department of Physics, University of Maryland, Baltimore County, Baltimore, MD 21250, USA Instituto de F´ısica ‘Gleb Wataghin’, Universidade Estadual de Campinas, 13083-859, Campinas, S˜ao Paulo, Brazil

PACS – Thermodynamics

PACS – Quantum error correction and other methods for protection against decoherence

PACS – Control theory

Abstract –Landauer’s principle laid the main foundation for the development of modern ther-modynamics of information. However, in its original inception the principle relies on semiformalarguments and dissipative dynamics. Hence, if and how Landauer’s principle applies to unitaryquantum computing is less than obvious. Here, we prove an inequality bounding the change ofShannon information encoded in the logical quantum states by quantifying the energetic cost ofHamiltonian gate operations. The utility of this bound is demonstrated by outlining how it can beapplied to identify energetically optimal quantum gates in theory and experiment. The analysis isconcluded by discussing the energetic cost of quantum error correcting codes with non-interactingqubits, such as Shor’s code.

Introduction. –

Introductory textbooks oftenpresent thermodynamics as a phenomenological frame-work, whose processes are driven by the exchange of heatand work [1]. However, ever since the birth of Maxwell’sdemon [2], describing and understanding the properties of information has been an integral part of thermodynamics[3–9]. Nevertheless, the thermodynamics of information has only rather recently become a prominent area ofresearch, which has been attracting more and more at-tention [10–20]. This development arose out of stochasticthermodynamics [21–24], and its application to processeswith feedback [25, 26].A seminal result that laid the foundation for the theoryis Landauer’s principle [27, 28]. As a statement of thesecond law, it is typically written as, (cid:104) W (cid:105) ≥ k B T ln (2) , (1)which states that in order to erase one bit of classical in-formation at temperature T , at least k B T ln (2) of ther-modynamic work, (cid:104) W (cid:105) , are required. Although the orig-inal derivation [27] can be considered semiformal at best,the impact of Eq. (1) on the development of computinghardware can hardly be underestimated [29]. Neverthe-less, while recent experiments in classical [30–32] as wellas quantum systems [33, 34] veriﬁed the validity of Lan-dauer’s principle, they also demonstrated its shortcomings (a) E-mail: [email protected] and inadequacy when applied to reversible [29] or quan-tum computing [35].In particular, quantum computing [35] is designed fromunitary dynamics, which describes systems that are ther-mally isolated from the environment. Any external noiseinevitably thwarts so-called quantum advantage [36–38],as it harms the delicate nature of quantum states. Yet,Landauer’s principle (1) and its generalizations to opensystems [39–45] are speciﬁcally formulated for dissipativedynamics at ﬁnite temperature. Thus, some recent ef-forts have sought to remedy this issue, by either analyzingthe energetics of measurements and quantum operationsdirectly [46–48], or by generalizing notions of stochasticthermodynamics to zero temperature [49]. For a morecomprehensive review of recent developments, we refer tothe literature [50, 51].The present analysis is dedicated to an alternative treat-ment, and the main goal is to quantify the energetic cost ofsingle gate operations in unitary quantum computing. Tothis end, we derive an upper bound on the change of Shan-non information encoded in the marginal, logical states ofa quantum system, that evolves under a Hamiltonian gateoperation [52]. We ﬁnd that this upper bound is given bythe norm of the Hamiltonian, which has been proposedin the literature to quantify the energetic cost of quan-tum control protocols [53–55]. Thus, as a main result,we obtain an inequality relating the amount of processedinformation with the energetic cost of the operation – ap-1 a r X i v : . [ qu a n t - ph ] F e b ebastian Deﬀnergeneralized Landauer’s principle for quantum computing.The versatility of this bound is demonstrated for qubitreset, i.e., for a Hadamard gate. The discussion is con-cluded with a few remarks on the experimental relevanceof the study, and with remarks on its utility in assessingthe energetic overhead of quantum error correcting codes. Preliminaries: logical states and marginal infor-mation. –

In quantum statistical physics as well as inquantum information theory [35] the amount of informa-tion stored in quantum state is quantiﬁed by the von Neu-man entropy, S vN = − tr { ρ ln ( ρ ) } , (2)where ρ = (cid:80) (cid:82) Γ p (Γ) | ψ (Γ) (cid:105) (cid:104) ψ (Γ) | is the density operator.The issue now arises from the fact that S vN is invariantunder unitary transformations. Hence, the informationalcontent of a quantum state does not appear to changeunder a quantum computation realized with unitary gates.However, it also has been recognized that in assessingthe thermodynamics of a computation not only the bareinformation content, but also the algorithmic complexity,the self-entropy, i.e., the entropy carried in a single logicalstate, etc. need to be considered [7, 24, 56]. The conceptcan be easily illustrated for a simple operation on a qubit.Imagine a two-level system that is initially prepared inan even superposition of logical states | (cid:105) and | (cid:105) , | ψ in (cid:105) = 1 √ | (cid:105) + | (cid:105) ) . (3)Then a Hadamard gate is applied, which we write as H = 1 √ | (cid:105) (cid:104) | + | (cid:105) (cid:104) | + | (cid:105) (cid:104) | − | (cid:105) (cid:104) | ) . (4)In the present example, the Hadamard gate performsnothing but a qubit “reset”. The “output” state thenbecomes | ψ out (cid:105) = H | ψ i (cid:105) = | (cid:105) . (5)Since H is unitary, the von Neumann entropy, i.e., theinformational content of the pure state has not changedand remains to be zero.However, in the input state the probability to observethe logical state | (cid:105) is 1 /

2, and in the output state it is 1.In other words, the distribution over the logical states haschanged. This can be described by the Shannon informa-tion, S , of the marginal distribution, that is the entropy ofthe quantum state that has been measured in the logicalbasis, S = − (cid:88) n ∈{ , } p n ln ( p n ) , (6)where p n = | (cid:104) n | ψ (cid:105) | .For this simple example, the change of the Shannoninformation then becomes,∆ S = S out − S in = ln (2) . (7) The natural question arises, how to quantify the energeticcost for this change of information. In complete analogyto previous considerations [7, 13, 24, 57] it appears obviousthat such a change of information cannot be “for free”,and that some generalized version of Landauer’s principle(1) must apply. Cost of Hamiltonian gates. –

We now proceed toderive such a generalized Landauer’s principle for unitarydynamics. To this end, we consider pure quantum states, ρ ( t ) = | ψ ( t ) (cid:105) (cid:104) ψ ( t ) | , that evolve under the time-dependentSchr¨odinger equation i (cid:126) ∂ t | ψ ( t ) (cid:105) = H ( t ) | ψ ( t ) (cid:105) . (8)Only rather recently, it was shown how time-dependentHamiltonians can be constructed by inverse engineeringsuch that the evolution under Eq. (8) is equivalent to anydesired quantum gate [52]. Hence, for our present pur-poses it will be suﬃcient to work with arbitrary, time-dependent Hamiltonians.In the following, we will bound the change of the Shan-non information (6) under the driven dynamics (8). Inthe above example, we consider only a binary logical ba-sis. However, the following analysis is more general, sincewe formulate the derivation for any arbitrary marginal,which, e.g., includes the diagonal entropy [58]. We alsonote that the following arguments are somewhat reminis-cent of our previous work [59], yet the ﬁnal result andconclusions are conceptually markedly diﬀerent. Bound on marginal entropy.

We start by consideringthe magnitude of the change of Shannon information (6)under the time-dependent Schr¨odinger equation (8), whichcan be bounded from above with the triangle inequality, | ∆ S| ≤ (cid:90) τ d t | ˙ S ( t ) | . (9)Due to normalization, the rate of change of S simply reads,˙ S ( t ) = − (cid:80) n ˙ p n ln ( p n ). Hence, we continue by inspecting˙ p n ( t ). Noting that˙ p n ( t ) = ∂ t (cid:104) ψ ( t ) | n (cid:105) (cid:104) n | ψ ( t ) (cid:105) (10)we immediately have˙ p n ( t ) ≤ (cid:126) |(cid:104) n | H ( t ) | ψ ( t ) (cid:105) (cid:104) ψ ( t ) | n (cid:105)| . (11)It is then convenient to write Eq. (11) as˙ p n ( t ) ≤ (cid:126) | tr { H | n (cid:105) (cid:104) n | ψ (cid:105) (cid:104) ψ |}| (12)where we suppressed the explicit time-dependence to avoidclutter, and which can be further bounded by the H¨olderinequality [60]. This theorem states that (cid:12)(cid:12) tr (cid:8) A B † (cid:9)(cid:12)(cid:12) ≤ (tr {| A | p } ) /p (tr {| B | q } ) /q (13)for all non-negative p and q with 1 /p + 1 /q = 1. The rightside of Eq. (13) is given by the product of the Schatten- p p-2nergetic cost of Hamiltonian quantum gatesand Schatten- q norms of the operators A and B respec-tively, which can be expressed in terms of the singularvalues. For instance, the Schatten-1 norm, i.e., the tracenorm can be written as tr {| A |} = (cid:80) ν σ ν , where σ n arethe singular values of A .Now, choosing q = 1 and p = ∞ for A = H ( t ) and B = | n (cid:105) (cid:104) n | ψ (cid:105) (cid:104) ψ | , our case becomes particularly simple.Note that the operator | n (cid:105) (cid:104) n | ψ (cid:105) (cid:104) ψ | has only one singularvalue that is diﬀerent from zero, namely √ p n = | (cid:104) ψ | n (cid:105) | .Hence, we immediately obtain˙ p n ( t ) ≤ (cid:126) || H ( t ) || √ p n , (14)where || H ( t ) || is the operator norm, i.e., the largest singu-lar value of the time-dependent Hamiltonian.Using Eq. (14), the change of Shannon information (9)can now be bounded by | ∆ S| ≤ (cid:126) (cid:90) τ d t || H ( t ) || S ( t ) (15)where we introduce S ( t ) ≡ − (cid:80) n √ p n log p n ≥

0. Equa-tion (15) constitutes our ﬁrst main result. The change ofShannon information of the marginal (post-measurement)distribution is upper bounded by the time-convolution ofthe norm of the Hamiltonian and S ( t ). While Eq. (15) ismathematically simple and appealing, the physical inter-pretation is not quite as transparent as one would desire.Therefore, we continue by further bounding S ( t ) by itsmaximum that can be supported by the quantum system.For d -dimensional Hilbert spaces, with d < ∞ , we canwrite S ( t ) ≤ √ d ln ( d ) , (16)which follows from similar arguments as maximizing theShannon information. Further introducing the informa-tion in units of bits I ≡ S / ln (2), we ﬁnally obtain | ∆ I| ≤ (cid:126) √ d log ( d ) (cid:90) τ d t || H ( t ) || . (17)Quite remarkably, the time integrated norm of the Hamil-tonian has been discussed in the literature to quantify theenergetic cost of quantum control protocols [53–55,61–63].For instance, for spin systems this cost can be interpretedas the average power expended by magnetic control ﬁelds.Further comparing Eq. (17) with the original Landauer’sprinciple (1), we immediately observe that for single bitoperations, d = 2, we have indeed achieved a generalizedLandauer’s principle for Hamiltonian gates.Finally, Eq. (17) can also be generalized to inﬁnite-dimensional Hilbert spaces with bounded energy, E . Inthat case, the maximal Shannon information is given bythe Gibbs entropy with eﬀective, inverse temperature β ,such that the average energy is equal to E . See alsoRef. [59] for a related discussion. Reset of a logical qubit.

We conclude this sectionby returning to the aforementioned example of resettinga qubit. It has been shown by Santos [52] that the Hadamard gate H (4) can be implemented through thetime-dependent Hamiltonian H H ( t ) = ˙ ϕ ( t )2 √ (cid:126) ( σ x + σ z ) (18)where ϕ ( t ) is an arbitrary function fulﬁlling the bound-ary conditions ϕ (0) = 0 and ϕ ( τ ) = π . Note that physi-cally ˙ ϕ ( t ) is nothing but the strength of the magnetic ﬁeld, B ( t ) ∝ ˙ ϕ ( t ) (1 , , H = T > exp (cid:18)(cid:90) τ d t H H ( t ) (cid:19) , (19)where T > denotes time-ordering.Realizing now || σ x + σ z || = √

2, it is easy to see that thegeneralized Landauer’s principle (17) becomes | ∆ I| ≤ √ (cid:90) τ d t | ˙ ϕ ( t ) | . (20)Hence, we immediately observe that the energetic costfor resetting a qubit is simply given by the magnitude ofthe magnetic ﬁeld employed to realize the quantum gate.Moreover, it then becomes a problem of optimal controltheory to design energetically optimal Hadamard gates,which are characterized by minimizing the time-integratedmagnitude of the magnetic ﬁeld [64]. Interestingly, this isa problem that has already been studied extensively in theliterature on thermodynamic control [65–68].The full solution of the optimal control problem for ﬁnd-ing Hamiltonian quantum gates with minimal (average)intensity is beyond the scope of the present discussion.However, let us brieﬂy outline how such an analysis wouldwork. In Ref. [52] it was alluded to the fact that a Hamil-tonian Hadamard gate can, indeed, be realized in experi-ments with the linear protocol, ϕ ( t ) = πt/τ . (21)Optimal protocols can then be found by considering, e.g.,a Fourier ansatz ϕ ( t ) = ϕ ( t ) + (cid:88) k A k sin (2 π k t/τ ) . (22)Also see Ref. [55] for similar considerations in the contextof shortcuts to adiabaticity. Note, however, that is hasbeen shown in the literature [69] that more involved se-ries expansions, as for instance in terms of the Chebyshevpolynomials have better performance. In any case, theoptimization problem then reduces to ﬁnding the set ofcoeﬃcients { A k } k such that the right side of the inequal-ity (20) becomes minimal. However, it is worth emphasiz-ing that any polynomial ansatz must fulﬁll the boundaryconditions, ϕ (0) = 0 and ϕ ( τ ) = π , which does limit thechoice of possible driving protocols.In the simplest case, only one Fourier component in ad-dition to the linear protocol is available, and we have ϕ ( t ) = πt/τ + A sin (2 π t/τ ) . (23)p-3ebastian Deﬀner - - Fig. 1: Illustration of the generalized Landauer’s principle forqubit reset (20) for the linear protocol plus one Fourier mode(23). Left side of Eq. (20), | ∆ I| / √ / √

2, as red, dashedline, and

C ≡ (cid:82) τ d t | ˙ ϕ ( t ) | depicted as blue, solid line. It is then a simple exercise to show with

C ≡ (cid:82) τ d t | ˙ ϕ ( t ) | that C = (cid:2) − arccsc(2 A ) + √ A − (cid:3) , for A ≤ − / π, for A ∈ ( − / , , (cid:2) arccsc(2 A ) + √ A − (cid:3) , for A ≥ / . (24)Hence, we conclude that if only the linear protocol plus asingle Fourier mode is available, then the linear protocol(21) is energetically optimal. See also Fig. 1 for an illustra-tion of this ﬁnding. Finally we note that while Eq. (20) isof mathematically simple and appealing form, the boundis not particularly tight. For this simple example, theminimum of the right side turns out to be √ π , whereasthe left side is only 1. Signiﬁcantly tighter bounds can beobtained by directly working with Eq. (15). Applications and consequences. –

We concludethe analysis with a discussion of potential applications andconsequences of the generalized Landauer’s principle (17)for quantum computing. Broadly speaking, there are twoavenues of research [37] that appear plausible, namely theexperimental realization of logical qubits and the imple-mentation of quantum error correcting codes.

Energetics of experimental gate operations.

To dateseveral computational paradigms have been proposed, ofwhich quantum circuits or gate based quantum comput-ing [35], adiabatic quantum computing based on quantumannealing [70], and cluster state computing [71] have re-ceived some prominence. In addition, possible hardwarefor quantum computing has been developed in, e.g., quan-tum optics [72], ion traps [73], and solid state systems[74]. Yet, independent of the computational paradigmand in any experimental platform the actual processingof information is facilitated by applying time-dependent,external ﬁelds. Hence, Eq. (17) does apply to any version of a quantum computer.

Superconducting qubits.

As an illustrative example,consider transmon [75], charge [76], or ﬂux qubits [77],which are all essentially based on a Cooper pair box [78],and which are the basis of many currently available sys-tems, such as IBM’s Q Experience and the D-Wave ma-chine [37]. The Hamiltonian of such a Cooper pair boxreads [78] H = −

12 ( E J σ x + E σ z ) (25)where E = E C (1 − n g ) and n g = C g / (2 e ) U is the di-mensionless gate voltage at capacitance C g . Further, E J is the Josephson energy, which is proportional to the areaof the tunnel junction. Any quantum gate can then beimplemented by applying an external magnetic ﬁeld [52].Obviously, it is desirable to work with the magnetic ﬁeldsthat have – on average – the lowest intensity, to avoid ex-cessive dissipation and decoherence. Equation (17) thenprovides a simple tool, to determine the optimal ﬁelds withthe minimally required intensity to realize the requiredquantum gate. Cost of quantum error correction.

It has been rec-ognized that any reliable quantum computer will necessi-tate the implementation of quantum error correcting algo-rithms [35]. Loosely speaking, any such algorithm encodeslogical quantum states in several physical states that canbe controlled separately and in parallel. Hence, the logicalquantum states can be made resilient against the eﬀectsof noise, such as decoherence and dissipation.Reliable classical computing is typically implementedby repetition codes. [79]. In essence, this just means thatevery single bit operation is performed N times indepen-dently, and the outcome is determined from “majorityvotes”. Landauer’s principle (1) then quantiﬁes the ther-modynamic cost that arises from the “physical overhead”of the error correcting code. If the logical bit is encoded in N physical bits, we immediately have that the total ther-modynamic cost is simply given by N times the work toerase a single bit, i.e., N × k B T ln(2). The obvious ques-tion is if and how this argument carries over to quantumerror correcting codes. To date, a plethora of algorithmshas been proposed as e.g., Shor’s code [80], topologicalcodes [81], stabilizer codes [82], or entanglement-assistedschemes [83]. The generalized Landauer’s principle (17)can then be exploited to rank these various codes accord-ing to their energetic cost.The analysis becomes particularly simple for algorithmsthat rely on non-interacting qubits . Thus, the total Hamil-tonian can be written as a sum of identical, single qubitHamiltonians, H ( t ) = (cid:80) Nn =1 H ( t ). If a logical qubit isencoded in N physical qubits, Eq. (17) can be written as | ∆ I| ≤ (cid:126) √ d log ( d ) N (cid:90) τ d t || H ( t ) || . (26)Equation (26) follows from the linearity of the norm, sincethe H i ( t ) live only on the Hilbert space of the i th qubit.p-4nergetic cost of Hamiltonian quantum gatesNote that generally I (cid:54) = N I , since the N qubits are cor-related in the logical basis. Shor’s code.

Arguably the most prominent quantumerror correcting code was proposed by Shor [80]. Thisalgorithm is a generalization of classical repetition codes,and protects a single qubit against any arbitrary error,which has been demonstrated in several experiments, seefor instance Ref. [84–86].In this scheme, the logical qubit is encoded in 9 physicalqubits according to | (cid:105) → ( | (cid:105) + | (cid:105) ) √ | (cid:105) → ( | (cid:105) − | (cid:105) ) √ . (27)Correspondingly, Eq. (26) predicts that implementingShor’s code is 9 times as expensive as a single qubit oper-ation. Thus, the natural question arises whether there isa quantum error correcting code, that is less or even theleast expensive. Perfect quantum error correcting code.

It was shownin Ref. [87] that the minimal quantum error correctingcode requires only 5 physical qubits. Logical qubits areencoded (up to normalization) according to | (cid:105) → − | (cid:105) + | (cid:105) − | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) (28)and | (cid:105) → − | (cid:105) + | (cid:105) + | (cid:105) − | (cid:105) + | (cid:105) + | (cid:105) − | (cid:105) − | (cid:105) . (29)While the physical motivation and interpretation of thisalgorithm is somewhat obscure [87], it has been shownthat the perfect quantum error correcting code is, indeed,minimal and that it protects the logical qubits against anytype of environmental noise. Hence, the minimal energeticcost according to Eq. (26) of protecting single qubits is 5times the cost of a single qubit operation. Quantum error correction with interactions.

This sim-ple, linear scaling does not hold for more intricate quan-tum error correcting codes that involve “energy penalties”.Prominent examples include the toric code, which is atopological algorithm [81], and error correction in quan-tum annealers [88–92]. However, a thorough analysis ofsuch algorithms is beyond the scope of the present discus-sion, and is thus postponed to future work.

Concluding Remarks. –

In the present analysis,we have derived a generalized Landauer’s principle thatcan be used to quantify the energetic cost of Hamiltonianquantum gates. Remarkably, this novel bound holds forpurely Hamiltonian dynamics, and hence does not rely ondissipative dynamics. The utility of this bound has beendemonstrated by alluding to the optimal control problemof ﬁnding Hamiltonian quantum gates with minimal in-tensity, experimental considerations, and by assessing and ranking the energetic cost of (non-interacting) quantumerror correcting codes. This opens possibilities for a vari-ety of research avenues, and makes analyses of Landauer’sprinciple considerably more applicable to reversible andquantum computing. ∗ ∗ ∗

It is a pleasure to thank Steve Campbell for many in-sightful discussions on the thermodynamics of quantuminformation, and Marcus V. S. Bonan¸c and Nathan M.Myers for helpful comments on the manuscript. FurtherI am indebted to Tan Vu Van, who spotted a very unfor-tunate typo in the derivation. S.D. acknowledges supportfrom the U.S. National Science Foundation under GrantNo. DMR-2010127.

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