Epsilonnets, unitary designs and random quantum circuits
Abstract
Epsilonnets and approximate unitary designs are natural notions that capture properties of unitary operations relevant for numerous applications in quantum information and quantum computing. The former constitute subsets of unitary channels that are epsilonclose to any unitary channel. The latter are ensembles of unitaries that (approximately) recover Haar averages of polynomials in entries of unitary channels up to order .
In this work we establish quantitative connections between these two seemingly different notions. Specifically, we prove that, for a fixed dimension of the Hilbert space, unitaries constituting approximate expanders form nets in the set of unitary channels for and , where is a numerical constant. Conversely, we show that nets with respect to this metric can be used to construct approximate unitary expanders for .
We further apply our findings in conjunction with the recent results of Varju (2013) in the context of quantum computing. First, we show that that approximate tdesigns can be generated by shallow random circuits formed from a set of universal twoqudit gates in the parallel and sequential local architectures considered in Brandão et al. (2016). Importantly, our gate sets need not to be symmetric (i.e. contains gates together with their inverses) or consist of gates with algebraic entries. Second, we consider a problem of compilation of quantum gates and prove a nonconstructive version of the SolovayKitaev theorem for general universal gate sets. Our main technical contribution is a new construction of efficient polynomial approximations to the Dirac delta in the space of quantum channels, which can be of independent interest.
I Introduction
Approximate designs and nets are natural proxies of the set of all unitary transformations of a finitedimensional Hilbert space. They capture complementary aspects of unitary channels. We start by reviewing here relevance and contexts in which they appear in quantum information theory.
Unitary approximate designs Dankert et al. (2009) are tailored to reproduce statistical moments of degree at most of the Haar measure on the unitary group. As such, they find numerous applications throughout quantum information, including randomized benchmarking Epstein et al. (2014), efficient estimation of properties of quantum states Huang et al. (2020), decoupling Szehr et al. (2013), information transmission Abeyesinghe et al. (2009) and quantum state discrimination Sen (2005). Pseudorandom unitaries are also used to model equilibration of quantum systems Brandão et al. (2016); Masanes et al. (2011), quantum metrology with random bosonic states Oszmaniec et al. (2016) and in order to model scrambling inside black holes Roberts and Yoshida (2017a); Nakata et al. (2017); AAAAA (2013) . Recently, approximate unitary designs got a lot of attention in the context of proposals for attaining the socalled quantum computational advantage Harrow and Montanaro (2017), especially random circuit sampling Boixo et al. (2018) that was recently realized experimentally by Google Arute et al. (2019). The reason for this is the anticoncentration property Hangleiter et al. (2018); Yoganathan et al. (2019), which seems essential in the proofs of quantum speedup.
Recently, there was a lot of interest in efficient implementations of pseudorandom quantum unitaries. First, it is known that the multiqubit Clifford group forms an exact unitary design but fails to be a unitary design Zhu et al. (2016). Second, in Brandão et al. (2016) it was shown that random circuits built form Haarrandom 2qubit gates acting (according to the specified layout) on qubit systems of the depth polynomial in form approximate designs. This result holds also if the random twoqubit gate set is replaced by a universal gate set that is symmetric (i.e. contains gates together with their inverses) and consists of gates with algebraic entries. Importantly, both of these requirements are crucial as the arguments of Brandão et al. (2016) heavily rely on the work by Bourgain and Gamburd Bourgain and Gamburd (2011). These results were later improved in 2018 in Harrow and Mehraban (2018), where even faster convergence in was proved using specially design layouts in which random twoqubit gates were placed. Additionally, recent work Mezher et al. (2020) (partially) lifted these stringent requirements. Moreover, the authors of Haferkamp et al. (2020) showed that random circuits constructed form Clifford gates and a small number of nonClifford can be used to efficiently generate approximate designs. Finally, there exist proposals for efficient generation of approximate designs using diagonal gates Nakata et al. (2017) and via Hamiltonian Nakata et al. (2017) and stochastic Banchi et al. (2017) dynamics.
Epsilon nets form (often discrete) subsets of the set of unitary channels that approximate every unitary operation up to some accuracy. They appear naturally in the context of compilation of quantum gates, i.e. the task is to approximate a target unitary gate via the sequence of elementary gates belonging to some ”simple” gateset . Traditionally, compilation of quantum gates is carried out using the celebrated SolovayKitaev algorithm Kitaev et al. (2002); Dawson and Nielsen (2005) which states that for any universal and symmetric gateset and any target quantum gate , there exist a sequence of gates from that approximates and has length for . Moreover, the aforementioned sequence can be found efficiently. Importantly, the SolovayKitaev algorithm requires the gateset to be symmetric as in the course of the compilation it is necessary to perform group commutators. There have recently appeared works which partially lifted this restriction by assuming that the gateset in question contains an irreducible representation of a group Sardharwalla et al. (2016); Bouland and Ozols (2017). We also note that the relation between efficient gate approximations and spectral gaps (here we study spectral gaps on restricted spaces rather then on the full space of functions on the unitary group) have been previously used in Harrow et al. (2002).
The notions of approximate designs and epsilonnets seem to be intuitively related but, according to our best knowledge, the quantitative connection between them has not been systematically studied before. We would like to remark however that analysis of the proof of Theorem 5 in Hastings and Harrow (2009) allows to infer that approximate designs define nets for and . Moreover, a related problem was recently studied in the context of harmonic analysis on Lie groups. Specifically, recent work in Varju (2013) established quantitative relation between spectral gaps on groups and epsilon nets on these manifolds and . We will comment on the relation of these findings to our Result 1 in Section VII.
Overview of the results and their significance— In our work we aim to provide quantitative relation between nets and approximate unitary designs. We follow closely the approach that was put forward in Varju (2013), where for a semisimple compact connected Lie group it was shown that nets follow from spectral gaps of certain ”transition operators” (defined via the gateset of interest, and acting on the function spaces built from the irreps of ). We translate these to the quantum information language and observe that when a Lie group is a group of quantum channels (isomorphic to the projective unitary group), spectral gaps of the aforementioned transition operators are in one to one correspondence with the parameter in the definition of approximate expanders Low (2010) (see Eq. (3)). Making use of this correspondence, we show that approximate designs can form nets. We modify the construction proposed in Varju (2013) and attain better dependence of on and , the dimension of the Hilbert space (see Section VII for a detailed discussion). Moreover, our arguments do not depend on the detailed knowledge of the representation theory and are instead based solely on the geometry of quantum channels (Result 1). We also prove a converse result i.e. that nets can be used to define approximate expanders.
These general results are then applied to different problems in quantum computation. First, we give a necessary and sufficient criterion for universality of any collection of quantum gates. Second, Result 3 shows a (nonconstructive) variant of SolovayKitaev theorem for gatesets that, in contrasts do the existing results, does not require inverses (see the discussion on SolovayKitaev theorem above). Finally, we prove in Result 4 that short random quantum circuits generated from twoqubit universal gatesets placed in the parallel and sequential layouts considered in Brandão et al. (2016) form approximate designs. Crucially, compared to previous approaches (see the discussion above) we do not require to be symmetric or to have algebraic entries.
Structure of the paper— In Section II we introduce basic concepts and notation. In particular, we describe designs and approximate designs, and introduce a notion of distance with respect to which we define epsilon nets. This allows us to formulate our main results in Section III. Then, in Section IV we discuss open problems and possible further applications of our results. In Section V we introduce notion of mixing operator defined on functions acting on unitary channels, and its gap. We relate the operator to the moment operators introduced in Section II.
After these preliminary sections, we are in position to give formal statements and proofs of our findings. The first group of results concerns arbitrary measures on unitary channels. And so, in Section VI we prove that for large enough, an exact design forms an net, in Section VII we show that approximate design also does. In that part we also prove the result in converse direction, namely that from a net with small enough one can construct a design. In Then we move to measures obtained from uniform distribution on sequences of gates from some gate set. In section VIII we apply the above results (employing some additional results from Varju (2013)) to prove a nonconstructive version of SolovayKitaev theorem which does not require assumption that gate set contains inverses.
Finally, we consider much more structured measures  namely random circuits on qudits. In Section IX we prove that random circuits (local and parallel) form approximate designs without assuming that the gate set contains inverses, and the gates have algebraic entries. We conclude the main part of the article with Section X where we outline the construction of the polynomial approximation of the Dirac delta on the group of unitary channels. This polynomial function plays a crucial role in the proofs of the results from Sections VI and VII.
The Appendix is largely devoted to technical results needed in the construction of the aforementioned polynomial approximation of the Dirac delta. Some of the results presented there can be of independent interest because of the intriguing connection with the random matrix theory (specifically, TracyWidom distribution Aubrun and Szarek (2017) and distribution of operator norm of GUE matrices).
Ii Main concepts and notations
Throughout this work we will be concerned with unitary channels acting on a unitary dimensional Hilbert space . A unitary channel is a CPTP map defined by , where is a quantum state and is a unitary operator on . In what follows we will denote by the set of all unitary quantum channels on . Note that every unitary operator uniquely defines a quantum channel but the converse is not true: a quantum channel specifies a unitary up to a global phase. For this reason we can identify with the projective unitary group . Therefore, is a compact connected semisimple Lie group Hall (2000) (we will use this observation in what follows).
In order to define the notion of net we need to first specify the distance in the set of unitary channels. We will consider the metric induced by the diamond norm . This notion of distance has strong operational interpretation of in terms of maximal statistical distinguishability of quantum channels Nielsen and Chuang (2011). We will use the following equivalent expression for the diamond norm (see Theorem 26 in Johnston et al. (2009))
(1) 
where denotes the operator norm and are unitaries representing channels and respectively. We say that a subset is an net (with respect to the metric ), if for every there exist such that . A set of gates is called universal if sequences of gates from form nets in for arbitrary small .
The set of unitary channels inherits the unique invariant normalized measure from the unitary group according to the following prescription. For we set , where are Haar measures on and respectively, and is the set of all unitary operators that define quantum channels belonging to . Haar measure on can be also defined via the action on functions of unitaries that are invariant under the global phase (i.e. , for arbitrary and ), . In what follows we will not differentiate between unitary channels and unitary operators, as well as Haar measures defined on these sets, unless it leads to ambiguity. In particular, will denote by the Haar measure of a subset of unitary channels or unitary group , depending on the context. We will also use the notation and for ”densities” of Haar measures on and respectivelly.
An ensemble of unitaries characterized by the probability measure is called a design Dankert et al. (2009) iff
(2) 
where is arbitrary balanced polynomial in , i.e. a function of the form , where is an operator on . Note that balanced polynomials on are well defined functions on . In this work we will be predominantly interested in discrete ensembles, i.e. the ones that take the form , for which . Approximate unitary designs (see for example Brandão et al. (2016, 2016)) are ensembles of unitaries that satisfy (2) up to some desired accuracy. In this work we will focus on a version of approximate designs called approximate expanders defined as ensembles satisfying
(3) 
where for any measure (in particular for the Haar measure ) we define a moment operator
(4) 
The quantity is sometimes called expander norm of . There exist other related definitions of approximate designs that use different quantifiers to gauge how well approximates the properties of Haar measure (see for example Low (2010)).
Iii Summary of main results
Here we present our main findings regarding the relation between approximate designs (expanders) and epsilonnets.
Result 1 (Approximate expanders define nets).
Consider an ensemble of unitaries described by the discrete measure on . Let and assume that ensemble is a approximate expander with (up to logarithmic factors in and ) and , where is a numerical constant. Then, the channels defined via the elements of form an net in with respect to the distance induced by the dimond norm.
We note that setting gives the connection between exact designs and nets. We give the technical formulation of the above result in Theorems 2 and 3. There we state the explicit dependence of and on the dimension of the Hilbert space and generalise the above statements to arbitrary probability measures (ensembles) on . Our proofs follow the method presented in Varju (2013). Our technical contributions are twofold. First, we simplify the original arguments making them largely independent of the machinery of group theory and thus more accessible for the broader audience. Second, in Theorem 1 we construct an efficient polynomial approximation of the Dirac delta on which allows us to attain better dependence of on the dimension and . Our construction can be of independent interests and its details are provided in Section X.
Result 1 can be used to find out how many times one needs to iterate gates comprising the approximate design so that they form an net. Specifically in Proposition 2 we prove that that it is enough to iterate them
We also prove the connection in the opposite direction. Namelly, we show that epsilon nets can be used to construct approximate designs (see Theorem 4 for the formal statement).
Result 2 (nets define approximate designs).
Consider a gateset forming an net in . Then, there exists an ensemble of quantum gates from which forms an approximate expander.
We apply the results established above in the context of quantum computing. To this end we use additional ingredient which follows from Theorem 6 of Varju (2013) (see Section V and Theorem 5 for the translation of representationtheoretic concepts to the formalism of tensor expanders). Specifically, the spectral gap of the moment operator associated to a measure supported on a universal gateset closes not faster than . Note that the above relies solely on universality of so that the assumptions made e.g. in Bourgain and Gamburd (2011) on algebraic entries of gates and the property that is symmetric (i.e. implies ) are not relevant. Leveraging this and the recent results of Sawicki and Karnas (2017b, a), it is possible to prove that universality of of any gateset is equivalent to being approximate expander, where and depends solely on . This finding complements recent results Zimborás et al. (2015); Oszmaniec and Zimborás (2017) that classified semisimple compact Lie subgroups of in terms of their second order commutants).
Finally, we use the above strong spectral gap results of Varju to show the following two results which are relevant to theoretical underpinnings of quantum computing.
Result 3 (Nonconstructive inversefree SolovayKitaev).
Let be a universal gateset in (not necessarily symmetric i.e. does not imply ). Then, every unitary channel can be approximated by sequences of gates from of length .
The formal proof can be found in Section VIII. We note that Result 3 does not give a constructive algorithm to find the approximating sequence of gates. Our last result shows that approximate designs can be generated efficiently by local random circuits without assuming inverses and algebraic entries. The formal proof is given Section IX.
Result 4.
Let be a set of universal twoqudit gates. Consider two types of random circuits on line of qudits Brandão et al. (2016).

Local random circuits: we pick uniformly at random two neighboring qudits, and apply gate chosen from according to uniform measure . We denote the resulting distribution by .

Parallel random circuits: we apply with probability either or , where each is picked independently from according to . We denote the resulting distribution by .
Let () be lengths of random local (parallel) circuits which are approximate texpanders, where instead of we take Haar measure over twoqudit gates. There exist a constant such that if
(5) 
then, the corresponding random circuits ( and ) are approximate expanders.
Note that in Brandão et al. (2016) it was shown that local (parallel) random quantum circuits with Haar distributed gates of lengths satisfying
(6) 
are approximate texpanders. It then follows that circuits constructed from scale efficiently with , too.
Remark 1.
It is straightforward to derive analogous bounds for other notions of approximate designs (based, for example, on the diamond norm). The conclusions are analogous. Let us stress, however, that our proof technique does not immediately apply to the scenarios considered in Harrow and Mehraban (2018) and hence we cannot use it to get convergence faster than for  qubits square lattice. We however believe that this technical problem can be overcome with some effort.
Acknowledgements We are grateful to Stanisław Szarek for explaining to us the intricacies of computing volumes of balls in the unitary group and related manifolds. AS acknowledges financial support from National Science Centre, Poland under the grant SONATA BIS: 2015/18/E/ST1/00200. MH acknowledges support from the Foundation for Polish Science through IRAP project cofinanced by EU within the Smart Growth Operational Programme (contract no.2018/MAB/5). MO acknowledges the financial support by TEAMNET project (contract no. POIR.04.04.000017C1/1800).
Iv Open problems
We conclude the introductory part of our wrok with a list of interesting problems which we left for further research.

Optimal scaling of and : Can one improve scaling in the results connecting nets with designs? We conjecture that the depenance is essentailly optimal. The same concerns scalling of with . We conjecture that with some work it should be possible to obtain (for fixed ).

Explicit constant in SK theorem: Unlike in all other results, our version of SolovayKitaev theorem contains an unknown constant depending on the dimension and the gate set. To what extent we can determine it (at least to leading order in the dimension)?

Improve constants: dependence on dimension in our estimates is not necessarily optimal, as we have applied quite crude estimate of volume of projective ball. There is also much room for improvement in other places  e.g. estimates such as in Lemma 2 can be sharpened at least for small . It is not however excluded, that dependence on dimension is, at least asymptotically, optimal.

Termination of the universality checking algorithm: The explicit value of constant in our version of SolovayKitaev theorem and the connection between approximate designs and nets can shed a new light on complexity of universality checking algorithms proposed in Sawicki and Karnas (2017b).

Connection with black hole dynamics and complexity growth: Recently, there were some interesting works connecting complexity of random circuits with black hole dynamics (cf. Roberts and Yoshida (2017b); Susskind (2018); AAAAA (2013)). It is conceivable that our findings may provide some useful tools, especially in the high complexity regime. In this context it is also natural to explore the possible generalizations of our results to approximate projective designs and nets in the set of pure quantum states.
V Mixing operators on unitary group, their gap and approximate designs
In this section we establish the connection between spectral gaps of mixing operators on unitary channels and approximate unitary designs (expanders). Let be the Hilbert space space of squareintegrable functions on , i.e. functions satisfying , where denotes the Haar measure on . For every we introduce a shift operator defined via . For every measure on we can consider an operator which is defined as a convex combination of operators according to measure , . Its action on functions on can be exlicitelly written as
(7) 
The operator can be understood it as a transition operator of a random walk on in which at every step a unitary is applied at random according to the measure .
We shall also consider restriction of to the subspace spanned by balanced polynomials of degree up to in as well as in i.e. subspace of functions on of the form . In particular, if we choose to be the Haar measure on then the operators and are projectors  they project onto the space of constant functions on . Let us denote the space orthogonal to the constant functions on by . We define the gap of as:
(8) 
We note that the sodefined function is a gap, when the support of the measure includes a set of universal gates. We are only interested in such situation, so we will keep name gap for .
We define a gap for analogously as for and denote it by . By straightforward calculations we get
(9) 
The following proposition establishes a very useful connection between and moment operator introduced in Eq.(4).
Proposition 1.
For any measure on we have
(10) 
and consequently we have , where is the expander norm of .
Proof.
The action of and on is determined by the left regular representation. Under this action decomposes into irreducible components and we have
(11) 
where is the matrix corresponding to via the irreducible representation with the highest weight and the symbol denotes unitary equivalence. On the other hand the representation is reducible and decomposes into
(12) 
Thus the operator can be written as
(13) 
We notice, however, that the space is spanned by the matrix elements of the representation and hence, by the decomposition (12), by matrix elements of irreducible representations . Let be the linear span of functions . It can be verified by direct computation that for every we have
(14) 
where is the dimension of the multiplicity space equal to , the dimension of carrier space of representation . Thus it follows that that collection of weights and agree, up to multiplicities. The theorem now follows from comparing decompositions (11) and (13).
∎
As an immediate consequence we get that for a design, the gap is equal to . We conclude this part by noting that composition of operator is compatible with taking convolutions in the sense that for all we have . This implies the following wellknown result.
Fact 1.
If is a approximate expander, then is a approximate expander.
Vi Exact tdesigns and epsilonnets
In this part we will show that elements of exact designs form nets with respect to the diamond norm distance provided (up to logarithmic factors in and ). We follow the ideas from Varju (2013) with two important differences. First, we significantly reduce the usage of representation theory. Second, we construct a new polynomial approximation of the Dirac delta on the group of quantum channels (see Theorem 1 and Section X where we provide details of the construction). This allows us to obtain improved dependence of on and in Theorem 2.
We start with giving the intuition beyond the proof of our result. We consider a family of realvalued balanced polynomials (i.e. polynomials of degree at most ) that has the following properties:

Normalisation: , for all .

Vanishing integrals on balls sufficiently far from identity : for every and for every such that we have
(15) where .
Functions can be regarded as polynomial approximation of the Dirac delta localized at , the identity channel (see Fig. 1).
We then consider the following integral,
(16) 
where , and for any measure and a function on we define (analogously as before for functions on ) . Next, under the assumption that is an exact design we show in Lemma 1 that equals the Haar measure of . On the other hand if channels from the support of do not form an net in we can use Eq.(15) to prove that vanishes as (see Lemma 2 and Theorem 1). We finally look for such that is smaller than , value of which is controlled by Fact 2. This number gives a degree of exact design that is ensured to form net. The graphical presentation of this general reasoning is given in Fig. 2 while technical details are given below. The main result connecting exact designs with nets is Theorem 2.
Lemma 1.
Let be a measure on which is an exact unitary design. Then for arbitrary function (i.e. a balanced polynomial of degree at most in and in ) satisfying
(17) 
and for any , we have
(18) 
where .
Proof.
As explained above, our goal is to upper bound the integral defined in Eq.(16) in terms of . To this aim we will use the following technical Lemma.
Lemma 2.
Let be a measure on . Suppose that the support of is not an net in . Then, there exists such that for any function on , and any satisfying we have
(19) 
Proof.
For simplicity we assume that the measure is discrete i.e. . The proof is analogous in the general case. Let be a unitary channel that cannot be approximated by elements form the supprort of : . From the definition of the moment operator (see Eq. (7)) we have
(20) 
By changing the variables in each summand and denoting we get
(21) 
Finally, using the defining property of and employing the unitary invariance of the diamond norm we obtain
(22) 
We conclude the proof by using the above ineqiality in each summand of Eq.(21). ∎
The following statement about the volume of the Ball in the space o unitary channels is known as folclore in quantum information community. Here we adapt a rigorous result of Szarek (1998).
Fact 2 (Estimates for the volume of Ball in the manifold of quantum channels Szarek (1998)).
Let be a ball centered around , where is the diamond norm distance from Eq.(1). There exist absolute constants such that for all
(23) 
where and .
Remark 2.
The last necessary element in our proof strategy is the existance of efficient polynomial approximation of the Dirac in the space of unitary channels. Here we present only the final result, while details of the construction and the necessary technical details are presented in Section X and the appendix.
Theorem 1 (Efficient polynomial approximation of the Dirac on unitary channels).
Consider a set of Unitary channels on dimensional quantum system equipped with a metric induced from the diamond norm (see Eq.(1)). Let be positive numbers satisfying , , . There exist a function with the following properties

Normalisation: .

Vanishing integrals on balls sufficiently far from identity channel : for every such that we have
(24) 
Low degree polynomial: can be represented as a balanced polynomial in and of degree
(25) 
Bounded norm:
(26)
We are now ready to prove the main result of this section. In the course of the proof we will make use of properties 1, 2 and 3 listed above. Property 4, which bounds the second norm of will be usesed in the subsequent section while discussing connection between approximate designs and nets.
Theorem 2 (Exact expanders define nets for sufficiently large ).
Let and let be a measure on which is an exact design with
(27) 
for
Then, the set of unitary channels from the support of , forms an net in with respect to the distance defined in Eq.(1).
Recall that for a discrete measure we have simply iff .
Proof.
Assume that is an exact design and that the set of unitary channels from the support of , is not an net. Let be a unitary channel that cannot be approximated by elements form the support of : . Let be the function satisfying conditions described in Theorem 1. By Fact 2 and Lemma 1 we have
(28) 
where . On the other hand by Lemma 2 we have
(29) 
Using Theorem 1 for
(30) 
we get
(31) 
As decreases (and the degree increases according to (30)) eventually the righthand side of (31) becomes smaller than the lower bound from (28). In particular, by inserting which satisfies
(32) 
to Eq.(30) we get (by contradiction) the degree such that that unitaries from the support of an exact design form an net in . It is easy to see that taking
(33) 
suffices to satisfy (32). Inserting to (30) gives equal to the righthand side of inequality (27). Finally, we remark that dropping the factor of in inequality (32) yields essentially identical scaling.
∎
Vii Approximate tdesigns and epsilonnets
In this section, we will establish even a closer connection between approximate  designs and nets. Specifically, we prove that under suitable conditions approximate expanders define nets and vice versa.
We first extend the reasoning established in the preceding section. Namely, for approximate designs the integral from Eq. (16) is not anymore equal . Therefore, we need to argue that the integral is not too small with respect to the volume. This is expected, as is almost a design. The following Lemma expresses this intuition quantitatively.
Lemma 3.
Let be an arbitrary measure on which is a approximate unitary expander. Then for arbitrary , and a function (i.e. a balanced polynomial of degree at most in and in ) satisfying
(34) 
we have the following inequality
(35) 
Proof.
We start with the following identity
(36) 
where is the inner product in , and is the indicator function of a set . Using CauchySchwartz inequality we obtain
(37) 
Furthermore, condition (34), the assumption and definitions of and the infinity norm allows us to write an estimate
(38) 
where in the last equality we used Proposition 1. By combining bounds (38) and (37) with (36) we obtain the desired result, i.e. we get (35). ∎
With the help of the above Lemma and due to properties of a carefully chosen polynomial approximation to the Dirac delta given in Theorem 1 we are in the position to prove the main result of this section.
Theorem 3 (approximate expanders define nets).
Suppose that a measure on is a approximate unitary expander with
(39) 
and
Remark 3.
A similar result follows from arguments given in the proof of Theorem 5 in Hastings and Harrow (2009). From careful analysis of the arguments presented there it can be shown that approximate expanders with and define nets with respect to the distance between unitary channels induced from the HilbertSchmidt norm
(40) 
Our result gives a more favorable scaling of and in both and . This is because the inequality implies that net with respect to distance is automatically net with respect to distance . In order to attain the scaling claimed above we used tight bounds on volumes of HilbertSchmidt balls in (cf. Aubrun and Szarek (2017) Theorem 5.11).
Proof.
We proceed analogously as in the proof of Theorem 2. Assume that is a approximate design and that the set of unitary channels from the support of , , is not an net. We choose to be a unitary channel that cannot be approximated by elements form the supprort of : . Moreover, we take to be the polynomial function described in Theorem 1 for (c.f Eq.(33)) and .
From Lemma 3 and Eq.(26) it follows that
(41) 
for any . On the other hand, by repeating the same arguments as in the proof of Theorem 2 we have
(42) 
It is now clear that if is such that
(43) 
then we obtain inequality (32). However, already form the proof of Theorem 2 we know that this inequality cannot be satisfied for