How smooth is quantum complexity?

Vir B. Bulchandani1 and S. L. Sondhi2

1Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA2Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

(Dated: August 12, 2021)

The “quantum complexity” of a unitary operator measures the difficulty of its construction froma set of elementary quantum gates. While the notion of quantum complexity was first introduced asa quantum generalization of the classical computational complexity, it has since been argued to holda fundamental significance in its own right, as a physical quantity analogous to the thermodynamicentropy. In this paper, we present a unified perspective on various notions of quantum complexity,viewed as functions on the space of unitary operators. One striking feature of these functions isthat they can exhibit non-smooth and even fractal behaviour. We use ideas from Diophantineapproximation theory and sub-Riemannian geometry to rigorously quantify this lack of smoothness.Implications for the physical meaning of quantum complexity are discussed.

I. INTRODUCTION

The computational complexity of a classical algorithmmeasures the physical resources that the algorithm re-quires to run. Although its precise definition depends onthe model of computation in question, the classical com-putational complexity is robust in the sense that equiva-lent models of computation give rise to equivalent com-plexity measures, up to an overhead in computationalresources that is polynomial in the input size. For exam-ple, the computational complexity of an algorithm on Ninput bits can be defined as its halting time C1(N) on aTuring machine or as the number of elementary Booleanoperations C2(N) in its Boolean circuit representation.Equivalence of the Turing machine and Boolean circuitmodels of classical computation then guarantees thatC2(N) = O(poly(C1(N)). This idea of equivalence upto polynomial functions yields a notion of “asymptoticcomplexity” that is independent of one’s preferred modelof classical computation.

The extension of these classical definitions to quan-tum computers is not unique, allowing for various no-tions of quantum complexity with very different proper-ties. A quantum algorithm on N qubits can be thoughtof as a unitary operator on the N -qubit Hilbert space; inthis sense, each definition of quantum complexity yieldsa distinct function on the space of unitary operatorsG = SU(2N ). A standard choice, that mimics the clas-sical circuit complexity, is to fix some set of elementarygates A ⊂ G and define the complexity C(U) of an op-erator U ∈ G as the number of elementary gates thatmust be multiplied to obtain U . However, this formula-tion of quantum complexity is rife with ambiguities thatdo not arise in the classical setting. For example, the setof elementary gates A can be chosen to be discrete andfinite or continuous and infinite, leading to very differentfunctions C(U) in each case. Nevertheless, this notionof quantum computational complexity turns out to berobust enough for practical purposes, allowing one to de-fine quantum complexity classes by analogy with theirclassical counterparts1.

Thus, at least from the viewpoint of quantum com-puter science, such ambiguities in the definition of quan-tum complexity C(U) are not very serious. However, re-cent work by Susskind and collaborators on the late-timedynamics of black holes2–5 has argued that the quantumcomplexity should be viewed as a fundamental physicalquantity, analogous to the usual thermodynamic entropyin some respects but with important differences in oth-ers3. In particular, it has been proposed6 that withinthe AdS/CFT correspondence, the dynamics of quantumcomplexity on the boundary of AdS spacetime uniquelycaptures certain non-trivial late-time features of blackhole dynamics in the bulk, long after the timescale atwhich thermalization sets in.

If correct, this intriguing conjecture has consequencesfar beyond the specific context of black hole physics.First, it implies that there may be a universal definitionof quantum complexity that is determined by physicalprinciples alone. This stands in contrast to the usualdefinition of classical computational complexity, whichis rather arbitrary from the physicist’s viewpoint. Sec-ond, it suggests that the quantum complexity might be auseful tool in the broader setting of many-body physics,for example, in distinguishing highly entangled states ofcondensed matter systems. Both these possibilities callfor a study of the quantum complexity as a physicallyinteresting quantity in its own right.

With this motivation, our paper provides a unified per-spective on the various notions of quantum complexity.We first show how three standard measures of the com-plexity of a quantum circuit – the discrete gate com-plexity, the circuit depth and the continuous complex-ity distance – define related generalizations of classicalcomputational complexity to the quantum setting. Wenext explore the analytical properties of these complex-ity measures, viewed as functions on the space of uni-tary operators, which have previously been argued toexhibit non-smooth and possibly fractal behaviour4,5,7.The main technical result of our paper is a rigorous ac-count of this lack of smoothness.

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II. THREE NOTIONS OF QUANTUMCOMPLEXITY

The classical notion of computational complexity, forexample based on Boolean circuits, operates in discretetime with a discrete set of elementary operations. In con-trast, the unitary group has the structure of a smoothmanifold. Thus in attempting to define the computa-tional complexity of a quantum algorithm on N qubits,viewed as a unitary operator U ∈ G = SU(2N ), there isno a priori reason to restrict oneself to either a discretenotion of computational “time” or a discrete set of ele-mentary operations, beyond a desire for harmony withestablished classical definitions.

Successively relaxing these requirements of discrete-ness in time and discreteness of gates leads naturally tothree distinct notions of quantum complexity. First letus recall that there are two fundamentally distinct typesof gate set A ⊂ G:

1. exactly universal : A is infinite and uncountable,and every U ∈ G can be expressed as a finite prod-uct of elements in A.

2. computationally universal : A is finite, and thegroup generated by elements ofA and their inversesis dense in G.

One further distinction, of potential importance forapplications of quantum complexity to condensed mat-ter systems or local quantum fields, is between spatiallylocal and k-local gate sets. Previous discussions of thequantum complexity tend to focus on elementary gatesets A that are k-local and all-to-all4,8. Demanding in-stead that the gate set A be local in space leads to somesmall differences compared to the non-local case, whichwe address in later sections.

The distinction between computationally universal andexactly universal gates sets leads to correspondingly dis-tinct notions of quantum complexity:

1. (discrete time, discrete gates) Let the gate setA = g1, g2, . . . , gr ⊂ G be computationally uni-versal. Then for a given tolerance ε > 0, the gatecomplexity of a unitary operator U ∈ G is definedas

Cε(U) = mink∑j=1

|nj | : ‖gn1i1gn2i2. . . gnk

ik− U‖ < ε. (1)

Here the minimization is over all k ∈ N, ij ∈1, 2, . . . , r and n1, n2, . . . , nk ∈ Z. The matrixnorm can be chosen freely; in this paper we willuse the Hilbert-Schmidt norm ‖A‖ = [Tr(A†A)]1/2.Among the three definitions of quantum complexityconsidered here, the gate complexity is most closelyanalogous to the classical computational complex-ity.

2. (discrete time, continuous gates) Now suppose thatthe gate set A is exactly universal, but compu-tational “time” is discrete, and at most W =O(poly(N)) gates may be applied in a given timestep. For example, A might be the set of all op-erators on two qubits, of which at most W = N/2can be applied simultaneously. In this setting, anatural measure of the quantum complexity is thecircuit depth σ(U) of the shallowest circuit requiredto simulate U . Up to a factor polynomial in N , thisis equivalent to the least number of distinct gatesin A required to simulate U .

3. (continuous time, continuous gates) Finally, onecan consider both an exactly universal gate set Aand allow the unitary U of interest to be simulatedin continuous time. The “cost” of simulating Uis then most naturally formulated as a problem inHamiltonian control8,9. Concretely, one introducesa cost function f(H) that defines a norm on thespace of 2N ×2N Hermitian matrices, and for givenU, V ∈ G, defines their complexity distance C(U, V )in terms of the least costly path from the identityU to V in G, to wit

C(U, V ) = infγ a.c.

∫ 1

0

dt f(H), (2)

where the infimum is over all absolutely continu-ous10 paths γ : [0, 1]→ G such that

d

dtγ = −iH(t)γ, γ(0) = U, γ(1) = V. (3)

The complexity of a specific operator U ∈ G isthen given by C(U) := C(1, U), where 1 denotesthe identity in G. This formulation of quantumcomplexity is usually referred to as “complexity ge-ometry”. For suitable choices of f , C(U, V ) is thegeodesic distance between U and V with respectto a Riemannian metric on G. More generally, fendows G with the structure of a sub-Finsler man-ifold, with respect which the complexity distanceC(U, V ) defines a so-called Carnot-Caratheodorydistance10,11.

These three broad notions of quantum complexity andtheir basic properties are summarized in Fig. 1 and TableI. Among these definitions, viewed as functions on G, thecircuit depth is simplest to capture analytically12. A par-ticularly tractable example is the case of a single qubitG = SU(2), for which the gate set A = eiφZ , eiθY :φ, θ ∈ [0, 2π) is exactly universal by Euler angles. Thenunitaries U ∈ G that are generic in measure are realizedby circuits of minimum depth σ(U) = 3, corresponding totheir Euler angle parameterization U = eiχ1Zeiχ2Y eiχ3Z .More generally, consider N qubits with G = SU(2N ) andsuppose that the exactly universal gate set A in ques-tion is parameterized by K = O(poly(N)) continuousparameters. Since the real dimension of the manifold G

3

FIG. 1. Left to right: the gate complexity, the circuit depth and the complexity distance define three increasingly smoothnotions of quantum complexity on the space of unitary operators.

is 4N − 1, the circuit depth of a unitary U that is genericin measure must satisfy the inequality

σ(U) ≥ dimG

KW∼ 4N

poly(N), N →∞. (4)

Thus the circuit depth σ(U) is bounded below almosteverywhere by a function exponentially large in N .

In fact, both the discrete gate complexity Cε(U) andthe continuous complexity distance C(U, V ) exhibit sim-ilar worst-case exponential scaling4 in N to the circuitdepth σ(U). It is therefore usually assumed13 that thesedefinitions are more-or-less equivalent from the viewpointof quantum computing1,14, and differ only in their easeof calculation. However, if the complexity is viewed as afundamental physical quantity in its own right6, then thedistinction between these different complexity measuresbecomes important.

The most striking such distinction arises in the localanalyticity properties of these complexity measures. Inparticular, both the gate complexity and the complexitydistance have a more intricate mathematical structurethan the circuit depth and can exhibit highly non-smoothbehaviour. The purpose of the remaining sections is toquantify this lack of smoothness rigorously. Our resultsare organized as follows. We first prove the existence ofefficiently universal gate sets for which Cε(U) exhibitsworst-case scaling in ε, Cε(U) = Ω(log 1/ε), densely inG. This clarifies the manner in which Cε(U) tends to anowhere continuous function as ε → 0. We next provethat the complexity distance C(U, V ) is at worst continu-ous but not differentiable as V → U in generic directions.These results give precise meaning to various qualitativediscussions of the “fractal” geometry of quantum com-plexity in the literature.

III. GATE COMPLEXITY

A. Background to results

For the gate complexity Eq. (1), let 〈A〉 denote thegroup generated by elements of A and their inverses,which by assumption is dense in G. To gain some intu-ition, first consider the limit C0(U) := limε→0 Cε(U). Be-cause the group 〈A〉 has measure zero in G, C0(U) =∞almost everywhere. Thus as ε → 0, Cε(U) tends to anowhere continuous function on the group G, analogousto the nowhere continuous Dirichlet function of real anal-ysis. (Notice that for a finite group with 〈A〉 = G, C0

recovers the “word metric” studied in geometric grouptheory; we refer to Ref. [15] for a discussion of analogiesbetween geometric group theory and quantum complex-ity.)

The lack of continuity of C0, viewed as a function fromG to the extended real line, is usually assumed to beregulated by the introduction of ε > 0; in this sense thequantum complexity defines a regularized notion of wordmetric for Lie groups. Indeed, for any computationallyuniversal gate set, the Solovay-Kitaev theorem provides aconstructive algorithm16 to approximate any given U ∈G within ε > 0 with Cε(U) = O(logc 1/ε) where c ≈4. It was shown by Harrow, Recht and Chuang17 thatthe Solovay-Kitaev estimate could be improved for theclass of so-called “efficiently universal” gate sets, whichallow all unitaries U to be ε-approximated using at mostCε(U) = O(log 1/ε) elementary gates. This result wasalso shown to be optimal, in the sense that the worst-casecomplexity of U ∈ G is bounded below by Ω(log 1/ε).

The log 1/ε scaling of the worst-case complexity fol-lows by an elementary counting argument, which works

4

Time Gate set Complexity measure Notation Mathematical interpretation

Discrete Discrete Gate complexity Cε(U) Regularized word metric

Discrete Continuous Circuit depth σ(U) Dimension of group

Continuous Continuous Complexity distance C(U, V ) Carnot-Caratheodory distance

TABLE I. A summary of the various notions of quantum complexity that we consider in this paper.

by thickening points of 〈A〉 in G by ε-balls until the entiregroup is covered17. However, it is not clear from this ar-gument how such “complicated unitaries” are distributedwithin the Lie group G. The main result of this sectionis that there exist efficiently universal gate sets for whichcomplicated unitaries are dense in G. In other words,every unitary in G is arbitrarily close to another unitarywhose complexity scales as Ω(log 1/ε) as ε → 0. Since〈A〉 is also dense in G, it follows that every neighbour-hood of G contains both a unitary with finite complexityindependent of ε and a unitary with complexity divergingas log 1/ε as ε → 0. This illustrates how the discontinu-ous behaviour of C0 is inherited by the regularized gatecomplexity, Cε with ε > 0.

In fact, the mathematical idea necessary to capturethis behaviour did not appear until relatively recently.The foundation of our analysis is the “non-commutativeDiophantine property” introduced by Gamburd, Jakob-son and Sarnak18 and subsequently refined by Bourgainand Gamburd19,20, that is essentially a non-Abelian ana-logue of the Diophantine properties satisfied by the realnumbers21. As a rule, Diophantine properties tend tobe the preserve of number theorists, rather than physi-cists; a notable exception is the KAM theory of perturbedintegrable systems, which demonstrates that quasiperi-odic tori whose frequencies satisfy Diophantine boundsare stable to the onset of chaos22. In the complexity lan-guage, KAM tori have “complicated resonances”, thatcannot be achieved at any finite order in perturbationtheory. Other examples include the sensitivity of theAubry-Andre model to the commensuration of its poten-tial23 and the sensitivity of the quasiparticle content ofthe gapless spin-1/2 XXZ chain to its anisotropy24. Ouranalysis of the gate complexity Cε(U) exploits its simi-larity to these phenomena, insofar as the gate complexityis also sensitive to the irrationality properties of the realnumbers17–20,25.

For the special case G = U(1), lower bounds on thegate complexity follow by standard results in the Dio-phantine approximation of real numbers21. The exten-sion to gate complexities in non-Abelian unitary groups,G = SU(d) with d ≥ 2, is only made possible by somerelatively recent mathematical advances in Lie grouptheory18–20,26, that were alluded to above. The remain-der of this section is structured as follows. We first famil-iarize the reader with Diophantine bounds in the simpli-fied setting of U(1), which contains most of the ideas nec-essary for understanding the non-Abelian case. We thenprovide a worked example of the singular behaviour ofCε(U) for the basic physical case of G = SU(2). Finally,

we extend the result to a class of efficiently universal gatesets in G = SU(d) with d > 2.

B. Gate complexity in U(1)

The basic ideas of our proof in the non-Abelian casecan be understood by applying standard results in Dio-phantine approximation21 to lower bound the gate com-plexity on a dense subset of the group G = U(1). Due tothe projective nature of quantum mechanics, this exam-ple is somewhat unphysical, but is nevertheless instruc-tive.

When G = U(1), a single generator g1 = eiα with α/πirrational is computationally universal, in the sense that〈g1〉 is dense in U(1). The complexity of U = eiφ ∈ Gassociated with the gate set A = g1 is given by

Cε(eiφ) = min|n| : |eiφ − einα| < ε. (5)

Thus we consider n such that∣∣∣∣sin(φ− nα2

)∣∣∣∣ < ε

2. (6)

Let m denote the unique integer such that

φ− nα2π

− 1

2≤ m <

φ− nα2π

+1

2. (7)

Then ∣∣∣∣φ− nα2−mπ

∣∣∣∣ < π

2. (8)

Using the inequality (2/π)|x| < | sinx| for |x| < π/2, itfollows that ∣∣∣∣ φ2π − n α2π −m

∣∣∣∣ < ε

4. (9)

To proceed further, we make the additional assumptionthat α

2π is an algebraic irrational number. We then havethe following result:

Proposition 1. Let φ ∈ [0, 2π) and δ > 0. Then thereexists φ′ with |φ′ − φ| < δ such that

logCε(φ′) >

1

3log

(1

ε

)+O(ε0). (10)

Proof. Since algebraic numbers are dense in R andα/2π has finite degree by assumption, there exists φ′ al-gebraic such that |φ′ − φ| < δ and φ′/2π is rationally

5

independent from α/2π. Let n ∈ Z with |eiφ′−einα| < ε.Then φ′ satisfies Eq. (9) for some m ∈ Z. Also, by astandard result in Diophantine approximation21, for anyexponent τ > 2 there is a constant K(α, φ′, τ) such that∣∣∣∣ φ′2π

− n α2π−m

∣∣∣∣ > K(α, φ′, τ)

(|n|+ |m|+ 1)τ. (11)

Using the definition of m, it follows by Eq. (9) that

logCε(φ′) >

1

τlog

(1

ε

)+O(ε0). (12)

The result follows upon setting τ = 3.

C. Gate complexity in SU(2)

Let us now turn to the more difficult case of G =SU(2). For concreteness, consider the gate set A2 =eiπαZ , eiπαY where Y, Z denote the usual Pauli matri-ces and cosπα ∈ Q\0,±1,±1/2. It follows by Niven’stheorem that α is irrational; thus the gates in A gen-erate dense subsets of the circles eitZ , t ∈ [0, 2π) andeitY , t ∈ [0, 2π). By Euler angles, 〈A2〉 is dense in Gand A2 yields a computationally universal gate set onSU(2).

Moreover, by a theorem of Swierczkowski27, the group〈A2〉 is free; this means that its Cayley graph is a tree andtherefore defines a (rooted) Bethe lattice with coordina-tion number z = 4 in the compact manifold SU(2). This“fractal” geometry suggests discontinuous behaviour ofthe quantum complexity, as has been suggested in theliterature4; we now make this intuition precise.

The main technical apparatus that we will require isas follows:

Definition 1. (Gamburd, Jakobson, Sarnak18) Let A =g1, g2, . . . , gr with gi ∈ SU(2). A satisfies the non-Abelian Diophantine condition if there exists a constantD = D(A) such that for all words Wl = gi1gi2 . . . gil 6=±12 of length l in 〈A〉, the inequality

‖Wl ± 12‖ ≥ D−l (13)

holds.

Proposition 2. (Gamburd, Jakobson, Sarnak18) LetA = g1, g2, . . . , gr with gi ∈ SU(2) ∩ M2(Q), whereM2(Q) denotes the set of 2 × 2 matrices with algebraicelements. Then A satisfies the non-Abelian Diophantinecondition.

It is clear that the definition Eq. (13) is analogous toa real Diophantine condition, although the lower boundin the non-Abelian case is exponential rather than poly-nomial in the word length. It is also clear that our two-element gate set A2 = eiπαZ , eiπαY satisfies the non-Abelian Diophantine condition. In fact, this implies thatA2 is efficiently universal17,19, in the sense that any el-ement of SU(2) can be ε-approximated with worst-case

complexity O(log 1/ε). We now demonstrate that uni-taries with complexity Ω(log 1/ε) are dense in SU(2).

Proposition 3. Let U ∈ SU(2) and Cε(U) denote itsgate complexity defined with respect to A2. Then for allδ > 0, there exists U ′ ∈ SU(2) with ‖U ′ − U‖ < δ and anon-universal constant D(α,U ′) such that

Cε(U′) >

1

logDlog

(1

ε

)+O(ε0). (14)

Proof. Pick U ′ ∈ SU(2) ∩M2(Q) with ‖U − U ′‖ < δ

and U ′ /∈ 〈A2〉 and let W(2)n denote a word of length n

in 〈A2〉 that ε-approximates U ′. Then by Proposition 2applied to the gate set A2 ∪U ′, there exists a constantD(α,U ′) such that

‖W (2)n U ′

−1 − 12‖ ≥ D−(n+1). (15)

By unitarity of U ′, it follows that

‖W (2)n − U ′‖ ≥ D−(n+1). (16)

Then by assumption

ε > D−(n+1), (17)

which implies

n+ 1 >1

logDlog

(1

ε

)(18)

(note that it is possible18 to set D > 1). Thus we haveachieved a lower bound on the length of words that ε-approximate U ′, and in particular

Cε(U′) >

1

logDlog

(1

ε

)+O(ε0), (19)

which was to be shown. It is immediate that the conclusions of Proposition 3

extend to any computationally universal gate set A =(g1, g2, . . . , gr) on SU(2), provided that the gi have al-gebraic elements. As noted previously19, all such gatesets are efficiently universal. It follows that our resultshold for the entire class of known (at the time of writing)efficiently universal gate sets on SU(2), which includesfamiliar examples such as the “Clifford+T” gate set act-ing on a single qubit25.

D. Gate complexity in SU(d)

Finally, we discuss the general (and most physical) caseof quantum complexity of unitaries in general special uni-tary groups, SU(d), with d > 2.

Unfortunately, this case is also the most difficult andthere are not very many explicit results available. We willtherefore proceed somewhat indirectly. Let us first notethat we can use A2 defined above for SU(2) to construct

6

an efficiently universal gate set on SU(d), given by theembedding17

A = βj(M) : j = 2, 3, . . . , d,M ∈ A2, (20)

where

βj(M) =

1j−2 0 0

0 M 0

0 0 1d−j

. (21)

Since this gate set is efficiently universal, it is cer-tainly computationally universal and therefore generatesa dense subgroup of G = SU(d). We then have the fol-lowing result, which follows by a direct generalization20,28

of Proposition 2 to SU(d):

Theorem 1. Let A = g1, g2, . . . , gr where gi ∈SU(d)∩Md(Q), with 〈A〉 dense in SU(d) and d ≥ 2. LetU ∈ SU(d) and Cε(U) denote its complexity defined withrespect to A. Then for all δ > 0, there exists U ′ ∈ SU(d)with ‖U−U ′‖ < δ and a non-universal constant D(A, U ′)such that

Cε(U′) >

1

logDlog

(1

ε

)+O(ε0). (22)

The proof proceeds similarly to that of Proposition3. Note that the lower bound of Eq. (22) is “fractal”,in the sense that the prefactor D(A, U ′) depends18 onthe irrationality properties of A and U ′. An analogousobservation holds for the simpler U(1) case, Eq. (12), forwhich the least permissible τ for given φ′ (not necessarilyalgebraic) is related to its irrationality measure. Such“fractal” lower bounds are quite unusual in physics; awell-studied example arises in the transport theory ofthe spin-1/2 XXZ chain29.

E. Locality versus non-locality

The discussion up to and including Theorem 1 provesthe existence of efficiently universal gate sets in SU(d)such that unitaries with Ω(log 1/ε) complexity are dense.One point that we have not considered so far is the is-sue of locality. For concreteness, consider a system of Nphysical qubits on a ring, with Hilbert space dimensiond = 2N , and suppose that the only experimentally ac-cessible gates are those acting on at most two qubits atonce. The space of all exact, all-to-all, two-qubit gates isdescribed by K = O(N2) continuous parameters. If oneadditionally imposes spatial locality, for example, allow-ing only two-qubit gates that act on nearest neighbourqubits, the number of continuous parameters shrinks toK = O(N).

A natural question is whether our results on the gatecomplexity can be formulated for sets of computation-ally universal gates A that are local in the above sense.This indeed turns out to be achievable, though the con-struction we propose is again somewhat indirect (and

certainly not unique). Let us take “local” here to meantwo-qubit gates acting on nearest neighbours, which al-lows for K = N distinct gate configurations (note thatour arguments extend straightforwardly to all-to-all gatesets). The first step is to choose a pair of rotations thatare algebraic and densely generate SU(2), for exampleeiπαZ , eiπαY with cosπα = 1/3. Then the embeddingEq. (20) can be used to construct a gate set A with 6Kelements that densely generates each nearest neighbourtwo-qubit copy of SU(4). Since the full set of nearest-neighbour two-qubit gates is exactly universal30 in thefull group G = SU(2N ), the group 〈A〉 must be densein G. Moreover, each matrix in 〈A〉 is algebraic by con-struction. It follows that the gate set A satisfies thehypotheses of Theorem 1.

The upshot is that it is possible to construct local,efficiently universal gate sets such that unitaries withΩ(log 1/ε) complexity are dense in G. This is truewhether one considers spatially local or all-to-all few-qubit gate sets. However, one shortcoming of our con-struction is that the geometry of the Cayley graph of 〈A〉,which has been argued to capture important features ofthe quantum complexity4,15, is not especially obvious.

If one instead relaxes the requirement of localityand asks only that the elementary gates be express-ible as finite products of local operations, then a moreexplicit characterization of the Cayley graph becomespossible. Specifically, applying the Breuillard-Gelandertheorem26,31 to the dense subgroup 〈A〉 < G yields apair of gates A′ = g1, g2 that freely generate a densesubgroup of SU(2N ), are expressible as finite productsof local gates, and satisfy the hypotheses of Theorem1. In particular, the Cayley graph of 〈A′〉 is a rootedBethe lattice with coordination number z = 4 (see Fig. 1for a depiction), belying the physical intuition4 that thecoordination number of the Cayley graph of a computa-tionally universal gate set on N qubits must scale withN .

IV. COMPLEXITY GEOMETRY

A. Riemannian complexity geometry

We now turn to Nielsen’s notion of complexitygeometry8 for continuous sets of elementary gates A,which applies ideas from the theory of Hamiltonian con-trol to the problem of constructing arbitrary unitary op-erators through the repeated action of few-qubit gates.While the control theoretical formulation of quantumcomplexity allows for fairly arbitrary cost functions inprinciple, it is simplest to focus on cost functions thatderive from an underlying Riemannian metric3,8.

For concreteness let us again consider quantum algo-rithms on N qubits. Then the Hilbert space dimension isd = 2N , and the goal is to describe the complexity geom-etry of the group G = SU(d). We shall refer to elementsof the Lie algebra g (viewed as the tangent space at the

7

identity) in terms of Hermitian matrices, i.e. by writingiH, with H Hermitian and traceless. Following Nielsen,let P denote a projector onto k-local Hermitian matriceswith k ≤ 2 and Q a projector onto those with k > 2.Further define the “penalty factor” q ≥ 1 and introducean inner product

〈H1, H2〉 =Tr(H1PH2) + qTr(H1QH2)

2N(23)

on g. By right translation, this can be extended to aRiemannian metric on all of G, and when q = 1, recovers(up to a constant factor) the bi-invariant metric definedby the Killing form.

Using this inner product, we define the complexity dis-tance of two unitary operators U, V to be

C(U, V ) = infγ a.c.

∫ 1

0

dt 〈H,H〉1/2, (24)

where the infimum is over all absolutely continuous pathsγ : [0, 1]→ G such that

d

dtγ = −iH(t)γ, γ(0) = U, γ(1) = V. (25)

For finite q, the complexity distance C(U, V ) is simply thegeodesic distance from U to V with respect to the Rie-mannian metric Eq. (23). However, if one takes q = ∞(the most physical choice from the viewpoint of locality),the inner product 〈. , .〉 ceases to be invertible.

In this limit, Nielsen’s metric is most naturally inter-preted as defining a sub-Riemannian geometry8,9,32, forwhich only tangent directions consisting of k-body oper-ators with k ≤ 2 are allowed. This observation allows oneto deduce several generic features of the q =∞ limit us-ing standard results on sub-Riemannian geometry, whichdo not seem to have been applied previously to the quan-tum complexity (although some of the resulting conceptswere discussed for the special case of a single qubit7).

B. Sub-Riemannian complexity geometry

We first introduce some key definitions10,11. Let M bea real n-manifold and ∆ ⊂ TM a smooth sub-bundleof the tangent bundle of M . The distribution ∆ iscalled bracket generating if any tangent vector v ∈ TpMcan be expressed as a linear combination of vectorsX1, [X2, X3], [X4, [X5, X6]], . . . with Xj ∈ ∆. A sub-Riemannian manifold consists of a triple (M,∆, g) where∆ is a bracket-generating distribution and g is a Rie-mannian metric on M . A curve γ : [0, 1] → M is calledhorizontal if γ(t) ∈ ∆γ(t) wherever γ is defined.

For Nielsen’s geometry with q = ∞, we take M = Gand define ∆ to be the space of k-body operators in gwith k ≤ 2, extended to all of G by right translation.The metric g on G can be taken to be the standard bi-invariant Riemannian metric

〈H1, H2〉 =Tr(H1H2)

2N, (26)

though more general choices are allowed. Then the triple(G,∆, g

∣∣∆

) is a sub-Riemannian manifold. Given thesedefinitions, the complexity distance of two unitary oper-ators U, V is defined as the length of the shortest hori-zontal curve connecting U to V , to wit

C(U, V ) = infγ a.c.γ⊂∆

∫ 1

0

〈H,H〉1/2, (27)

with γ satisfying Eq. (25). Letting d(U, V ) denote thegeodesic distance with respect to the standard (q = 1)metric, Eq. (26), it is clear that the complexity distanceis bounded below by the usual geodesic distance, i.e.

C(U, V ) ≥ d(U, V ). (28)

The metric defined by C(U, V ) is usually called a Carnot-Caratheodory distance11 (though this seems to be some-thing of a misnomer, given that the distributions appear-ing in Caratheodory’s formulation of the second law ofthermodynamics are integrable33).

When the distribution ∆ is bracket generating, the fol-lowing result holds:

Theorem 2. (Chow11) Let M be a connected manifoldand ∆ a bracket-generating distribution. Then any twopoints in M can be joined by a piecewise smooth curvethat is horizontal.

In the context of Nielsen’s complexity geometry, thismeans that for any U, V ∈ G, there exists a piecewisesmooth curve from U to V such that the instantaneousHamiltonian H(t) comprises purely k-body terms withk ≤ 2. In particular, the complexity distance C(U, V )is always finite. However, these curves are expected tolook much “rougher” than geodesics in the Riemannianmetric. This intuition can be made precise as follows.

Let us first define the flag of sub-bundles

∆ ⊂ ∆[2] ⊂ ∆[3] . . . ⊂ ∆[s] = TG (29)

inductively by allowing successive commutators. Namely,we set ∆[1] = ∆ and define

∆[k+1] = span∆[k], [X,Y ] : X ∈ ∆, Y ∈ ∆[k], k > 1.(30)

Thus, for example,

∆[2] = spanX1, [X2, X3] : X1, X2, X3 ∈ ∆. (31)

Now let X1, X2, . . . , Xn be a frame of sections of TGand define integers mk such that X1, X2, . . . , Xmk

is aframe of sections for ∆[k] (thus mk is the number of lin-early independent vectors in ∆[k] at a given point). Thedegree dj of the vector field Xj quantifies its commutatordepth, and is defined so that

Xj ∈ ∆[dj ]\∆[dj−1]. (32)

with ∆[0] := 0. In the complexity language, vectorfields with dj = 1 correspond to “easy” directions in the

8

space of Hermitian operators while vectors with dj > 1correspond to “hard” directions. One of the basic re-sults of sub-Riemannian geometry relates this structureof “easy” and “hard” directions to the local behaviour ofthe complexity metric.

To formulate this result, let us first define the flowexpU (X) that takes U to the point γ(1) on the integralcurve

d

dtγ(t) = Xγ(t), γ(0) = U. (33)

This defines an exponential map expU : Rn → G, of theform

(t1, t2, . . . , tn) 7→ expU (t1X1 + t2X2 + . . .+ tnXn). (34)

We define the “box” Box(r) in Rn as

Box(r) = (t1, t2, . . . , tn) ∈ R : |tj | ≤ rdj. (35)

These anisotropic boxes in Rn capture the geometry ofballs BU (r) = V ∈ G : C(U, V ) < r in the complexitymetric. In particular, the following result holds:

Theorem 3. (the “ball-box theorem”11) Balls in thecomplexity metric are uniformly equivalent to images ofboxes under the exponential map. Specifically, there arestrictly positive continuous functions A, r : G→ R+ withA > 1 such that

expUBox(A−1r) ⊂ BU (r) ⊂ expUBox(Ar). (36)

This result has two important consequences for sub-Riemannian complexity geometry. First, it implies thisgeometry is fractal, in the sense that the Hausdorff di-mension of the metric space (G,C) is equal to11 nH =∑nj=1 dj > n, and therefore exceeds its real dimension

n = 4N − 1. This fact was recently noted in the litera-ture without proof5. It follows intuitively from Eq. (36),which suggests that the volume of a complexity ball scalesanomalously with its radius r, as ∼ r

∑nj=1 dj .

Second, this result suggests that moving in a genericdirection from a unitary U to another unitary V at a Rie-mannian distance d(U, V ) = δ, the complexity distancebehaves as C(U, V ) ∼ δ1/s, where s = maxj dj > 1. Moreformally, the identity map relating the metric spacesid : (G, d)→ (G,C) is known10,11 to be α-Holder contin-uous with Holder exponent α = 1/s, which implies theinequality

C(U, V ) ≤MUδ1/s (37)

for some constant MU > 0 and δ sufficiently small. Thisindicates that the complexity distance is continuous butpossibly not differentiable as V → U . A precise state-ment is as follows:

Proposition 4. The q =∞ complexity distance C(U, V )is continuous as V → U but does not have one-sideddirectional derivatives as V → U in generic directions.

Proof. First introduce “privileged coordinates”34

y1, y2, . . . , yn about the point U , which satisfy

aU (|y1|1/d1 + |y2|1/d2 + . . .+ |yn|1/dn) ≤ C(U, V ) (38)

for some constant aU > 0. Here, y1, y2, . . . , yn denote theprivileged coordinates of V , while U has privileged coor-dinates y = 0 by construction. Choosing V to approachU in a generic direction v, i.e. defining V (t) by its priv-ileged coordinates yj = vjt with |vj | > 0 and sufficientlysmall t, we have

C(U, V (t))

t> aU |vn|t1/s−1 →∞, t→ 0+. (39)

We deduce that approaching from generic directions inthe tangent space at U , the one-sided directional deriva-

tive limt→0+C(U,V (t))

t does not exist. Meanwhile theHolder condition Eq. (37) implies that C(U, V ) → 0as V → U .

At this point, several comments are in order. First,the results described above can be adapted to sub-Finslermetrics10, and in this sense capture the most general caseof Nielsen’s complexity geometry for bracket-generatingdistributions ∆. Second, the lack of one-sided directionalderivatives of the sub-Riemannian complexity distanceC(U, V ) as V → U distinguishes it from the Riemanniangeodesic distance d(U, V ). While neither function is dif-ferentiable at V = U , the Riemannian distance admitsone-sided directional derivatives as V → U along smoothcurves.

We now briefly discuss the question of local versus non-local gate sets in the context of the sub-Riemannian com-plexity distance. As noted above, the local geometry ofcomplexity balls is entirely determined by the structureof “easy” and “hard” vector fields encoded by the se-quence of integers (d1, d2, . . . , dn). Suppose now that inNielsen’s definition we impose spatial locality rather thantwo-locality, i.e. demand that the “easy” directions ing comprise only spatially local two-qubit operators. Asdiscussed in Section III E, this modifies the dimension ofthe vector space K = dim(∆) of easy directions fromK = O(N2) for all-to-all, two-local gates to K = O(N)for spatially local two-qubit gates. This in turn leads toa redefinition of the integers (d1, d2, . . . , dn) and a cor-responding increase in the Hausdorff dimension nH , butsince both distributions are bracket generating30 thereare no other substantive changes.

C. Approaching the q = ∞ limit

For large but finite q, Nielsen’s metric Eq. (23) isRiemannian. This means that the complexity distancefunction C(U, V ) admits one-sided directional derivativesalong smooth curves through U (though it is still not dif-ferentiable at V = U). Moreover, the complexity geome-try is no longer truly fractal because the Riemannian ex-ponential map is locally well-behaved33, raising the ques-tion of how far the singular features of the q = ∞ limit

9

are inherited by the complexity geometry with q < ∞.This point was previously discussed for the special case ofa single qubit7. We now generalize these considerationsto the complexity geometry of N qubits; our discussionwill be less rigorous than in preceding sections.

As noted previously4,5,7, the relevant geometrical ideais the “cut locus” ΓU of a given U ∈ G, which is the set ofpoints V ∈ G for which there exists more than one length-minimizing geodesic connecting U and V . For large q, weexpect that the nearest points V ∗ ∈ ΓU to U have theproperty that the sub-Riemannian geodesic connectingU to V ∗ is equal in length to the Riemannian geodesicconnecting U to V ∗. Writing d(U, V ∗) = δ, we thereforeexpect

C(U, V ∗) ∼ q1/2δ ∼ δ1/s (40)

in a generic direction, where s was defined above.Thus the distance δ to the nearest cut point V 6= U

such that C(U, V ) is not smooth scales with q as

δ ∼ q−s/(2(s−1)). (41)

For a single qubit with s = 2, this recovers a previousresult7. Notice that as q →∞, the cut locus ΓU becomesarbitrarily close to U in almost all directions. This is oneway to understand the singular nature of the limit q =∞.

D. Complexity distance versus gate complexity

Finally, one can ask how far the continuous complexitydistance emerges naturally as a continuum limit of thediscrete gate complexity. The existence of such a limitwould be strong evidence in favour of a universal notionof quantum complexity.

Remarkably, an equivalence between discrete gatecomplexity (i.e. a word metric) and continuous complex-ity distance (i.e. a Carnot-Caratheodory metric) does ex-ist for nilpotent Lie groups, for example, the Heisenberggroup. Roughly speaking, this arises due to an equiva-lence between the volume of Cayley graphs at radius rand the volume of their limiting sub-Riemannian balls atradius r, both of which scale with the same anomalousHausdorff dimension, e.g. as ∼ r4 for the Heisenberggroup35,36.

However, for unitary groups any such local equivalencebetween discrete and continuous complexity geometriesbreaks down. This is easily seen by comparing the vol-ume growth of the Cayley graph on two free generatorsat radius r, which scales exponentially as ∼ 3r, with thevolume of a complexity ball, which scales polynomiallyas ∼ rnH . The existence of exponentially growing Cay-ley graphs in unitary Lie groups, compared to nilpotentones, reflects the basic distinction between semisimpleand solvable Lie algebras, and can be viewed as a conse-quence of the Tits alternative in group theory35,37. Thisappears to rule out a correspondence between discreteand continuous notions of quantum complexity down tothe smallest scales of the unitary group G in question.

V. DISCUSSION

We have proposed a unifying perspective on the vari-ous notions of quantum complexity and shown how theycan be distinguished by their smoothness properties, orlack thereof. Our results clarify how the gate complexitytends to a nowhere continuous function of its argumentas ε → 0, and make precise the sense in which Nielsen’scomplexity geometry becomes fractal as q →∞.

At the time of writing, it is still not clear how far thereexists a “unique”, or universal, definition of quantumcomplexity, nor indeed whether such a definition can beexpressed in terms of the complexity measures studiedin our paper. One possible route to tackling this ques-tion would be making sense of how the different notionsof complexity depicted in Fig. 1 emerge as limits of oneanother. The breadth of mathematical ideas involvedin their formulation, together with the group-theoreticalobstruction identified above, suggests that this task willbe far from trivial. Meanwhile, the idea that there mightexist such a universal notion of quantum complexity6 mo-tivates further study of complexity measures in realisticmany-body quantum systems.

Acknowledgments. We thank V. Khemani, S.Parameswaran, R. Nandkishore, R. Moessner and espe-cially A. Brown for helpful discussions. We are grateful toA. Harrow for comments on the manuscript. This workwas supported with funding from the Defense AdvancedResearch Projects Agency (DARPA) via the DRINQSprogram.

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