Dual unitaries as maximizers of the distance to local product gates Shrigyan Brahmachari Department of Mechanical Engineering Indian Institute of Technology Madras Chennai India 600036

2025-05-03 0 0 626.18KB 11 页 10玖币

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Dual unitaries as maximizers of the distance to local product gates

Shrigyan Brahmachari∗

Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, India 600036

Rohan Narayan Rajmohan,†Suhail Ahmad Rather,‡and Arul Lakshminarayan§

Department of Physics, Indian Institute of Technology Madras, Chennai, India 600036

The problem of ﬁnding the resource free, closest local unitary, to any bipartite unitary gate U

is addressed. Previously discussed as a measure of nonlocality, the distance KD(U)to the near-

est product unitary has implications for circuit complexity and related quantities. Dual unitaries,

currently of great interest in models of complex quantum many-body systems, are shown to have

a preferred role as these are maximally and equally away from the set of local unitaries. This is

proved here for the case of qubits and we present strong numerical and analytical evidence that it

is true in general. An analytical evaluation of KD(U)is presented for general two-qubit gates. For

arbitrary local dimensions, that KD(U)is largest for dual unitaries, is substantiated by its analytical

evaluations for an important family of dual-unitary and for certain non-dual gates. A closely allied

result concerns, for any bipartite unitary, the existence of a pair of maximally entangled states that

it connects. We give efﬁcient numerical algorithms to ﬁnd such states and to ﬁnd KD(U)in general.

Dual-unitary quantum circuits are of intense current

interest to many research communities [1–7,10–12,15]

as they provide non-trivial models of both integrable

and chaotic many-body quantum systems and allow

for universal computation. For example, these have

been used for the evaluation of dynamical correlation

functions [3,5], spectral statistics [4], construction of a

quantum ergodic hierarchy [11], entanglement genera-

tion [7], exact emergence of random matrix universality

[13], and measurement induced phase-transitions [14].

It has been shown to be classically simulatable for short

times or circuit depths for certain initial states and for

local expectation values [9]. However, for late times the

problem has been shown to be BQP-complete and the

dual-unitary circuits are capable of universal quantum

computation, while classical simulation of the problem

of sampling has been shown to be hard [9].

The building blocks of dual-unitary circuits are arbi-

trary single-particle gates and two-particle dual unitary

operators. The dual-unitary operators remain unitary

on reshufﬂing the indices, a property interpreted as a

space-time duality [3]. From a quantum information

theoretic viewpoint these are maximally entangled uni-

taries [22,24]. Nevertheless, their place in the space of

general bipartite unitary operators, U(d2)in local di-

mension d, is not understood and their construction for

∗sb818@duke.edu

Current address: Department of Electrical & Computer Engi-

neering, Duke University Pratt School of Engineering, Box 90291

Durham, NC 27708 USA.

†Current address: Department of Physics and Astronomy, North-

western University, Evanston, IL 60208, USA

‡Current address: Max Planck Institute for the Physics of Complex

Systems, N¨

othnitzer Str. 38,01187 Dresden, Germany.

§arul@physics.iitm.ac.in

d>2 is incomplete [3]. However, there are numerical

algorithms [22] to generate ensembles of dual-unitary

matrices in any dimension dand several analytic con-

structions [5,8,10,11].

As local unitary operators are considered to be a free

resource, a basic geometric question is the distance of

dual unitaries and in general any bipartite unitary op-

erator to the closest local unitary. This has been previ-

ously studied as a “strength measure” of bipartite uni-

tary operators and denoted as KD(U)[16], and satisﬁes

the conditions required of a quantum complexity mea-

sure. Formally, for a general metric D,KD(U)for some

bipartite unitary U∈U(d2)is deﬁned as,

KD(U):=min

uA,uB∈U(d)D(U,uA⊗uB). (1)

all unitary operators

local unitary

products

KD(U)

u1⊗u2

K∗

FIG. 1. A caricature of the geometry of KD(U). For a given

bipartite unitary operator, the projection to the subset of local

unitary products is found, and the distance is calculated.

arXiv:2210.13307v2 [quant-ph] 4 Dec 2023

This work uses the Hilbert-Schmidt metric ∥A∥=

ptr(AA†), as it seems both most appropriate and ac-

cessible to analytical considerations. As a strength mea-

sure, one can choose the metric to satisfy desirable

properties based on the application [16]. It should lead

to a function on unitaries that satisfy: (i) f(uA⊗uB) =

0, (ii) f(UV)≤f(U) + f(V)and (iii) f(U⊗I) = f(U).

The ﬁrst two properties are satisﬁed by all strength

measures, and we show that if we pick the Hilbert-

Schmidt metric, the third property, known as stability, is

satisﬁed by KD(U), see Appendix A. A particular appli-

cation of interest is in circuit complexity; one can show

that the strength measure on the operator norm of a

bipartite gate can be used to understand its capacity

to create entanglement when acting upon pure states,

based on recent arguments in Ref. [23].

Thus far there are partial results concerning KD(U)

for d=2 and no known results for d>2. Apart from

deriving several new results concerning general bounds

and exact evaluations of this measure, we show that

the set of dual-unitaries are maximally and equally away

from product unitaries for the case of qubits, d=2, and

present strong numerical evidence that this is the case

even for d>2. This motivated the caricature in Fig. (1)

which indicates the special place of dual-unitaries in the

space of bipartite operators.

These considerations are intimately related to an in-

teresting property that is easy to see for qubits, but also

appears to hold in general: for any bipartite unitary U

in Hd⊗ Hd, there exists a pair of maximally entangled

states |Φ1⟩and |Φ2⟩such that U|Φ1⟩=|Φ2⟩. A dy-

namical map is devised that converges to such a pair

of maximally entangled states for any U, which can be

used to ﬁnd the closest local unitaries to dual-unitary

gates. A closely related procedure works surprisingly

well also for general (non-dual) unitaries, which allows

for detailed numerical explorations of KD(U). We ﬁnd

KD(U)analytically for several special families of uni-

taries which also veriﬁes the numerical procedure.

I. GENERAL CONSIDERATIONS AND BOUNDS FOR

KD(U)

Let the operator Schmidt decomposition of an arbi-

trary two-qudit unitary gate Ube:

∑

i=1pλimA

i⊗mB

i. (2)

Here the set {mA

i,i=1, ··· ,d2}forms an orthonormal

operator basis in subspace Athat is

trmA

imA†

j=dδij,

and mBis a similar set in subspace B. The operator

Schmidt coefﬁcients λkare chosen to be in decreasing

order, 1 ≥λ1≥ ··· ≥ λd2≥0, and from the unitarity

of Uit follows that ∑d2

k=1λk=1.

The Schmidt decomposition provides measures of

operator entanglement, for example one such is the lin-

ear entropy

E(U) = 1−

∑

k=1

λ2

k. (3)

This range is 0 ≤E(U)≤E(S) = (d2−1)/d2, and is

0 iff Uis a product unitary (in which case KD(U) = 0).

The maximum value is attained when for all k,λk=

1/d2, and are the operator equivalents of Bell states.

The swap operator S(S|kl⟩=|lk⟩) achieves this value

and is an important example of a maximally entangled

unitary operator. However, Sis by far not the only uni-

tary operator to achieve this value. If an unitary Uis

such that E(U) = E(S), this may also be taken as the

deﬁnition of dual-unitary operators. Restriction of an

unitary Uto this special set will be generically denoted

as Udual, that is E(Udual) = 1. One of our goal is to

calculate KD(Udual), but we ﬁrst turn to general state-

ments and bounds.

For the distance measure KD(U)in Eq. (1), for the rest

of the paper we use the Hilbert-Schmidt metric. Deﬁne

K∗

D(U):=∥U−mA

1⊗mB

1∥=q2d2−2d2pλ1. (4)

As the Schmidt decomposition already provides the

nearest product operator (for a proof in the case of

states, see [17]), this quantity is the distance to the near-

est product operator, removing the unitarity constraint

from the local operators in Eq. (1). As KD(U)is the

distance to a more constrained set, the following lower-

bound follows:

KD(U)≥K∗

D(U). (5)

Alternatively, the deﬁnition in Eq. (1) implies

D(U) = min

uA,uB∥U−uA⊗uB∥2

=min

uA,uB2d2−2Re htr(U†(uA⊗uB))i

=2d2−max

uA,uB

2tr(U†(uA⊗uB)).

(6)

The second equality follows from the fact that the

phase can be absorbed by the local unitaries. Ex-

pand local unitaries uAand uBin the orthonormal

bases from the Schmidt decomposition of Uin Eq. (2),

uA=∑d2

i=1αimA

i,uB=∑d2

i=1βimB

i, where ∑i|αi|2=

∑i|βi|2=1. This leads to tr(U†(uA⊗uB))=



d2d2

∑

i=1pλiαiβi≤d2pλ1

∑

i=1|αi||βi| ≤ d2pλ1. (7)

Here the ﬁrst inequality holds as λ1is the largest

Schmidt coefﬁcient, and the second follows from the

Cauchy-Schwarz inequality. Using this in Eq. (6) we

indeed obtain the lower-bound as K∗

D(U)in Eq. (5)

It is possible to obtain an upper-bound as

KD(U)≤K∗

D(U) + q2d2−2∥mA

1∥1∥mB

1∥1, (8)

where ∥A∥1=tr √AA†is the trace norm. If mA

uAqmA†

1mA

1is its polar decomposition and similarly

for B,uA,Bare the nearest unitaries to mA,B

1[25]. The

upper bound is obtained as by deﬁnition the distance

of Uto the product uA⊗uBcannot be less than KD(U):

KD(U)≤ ∥U−uA⊗uB∥

=∥U−mA

1⊗mB

1+mA

1⊗mB

1−uA⊗uB∥

≤ ∥U−mA

1⊗mB

1∥+∥mA

1⊗mB

1−uA⊗uB∥.

(9)

The ﬁnal step follows from the triangle inequality and

the upper-bound is immediately obtained.

These lower and upper-bounds on KD(U)are de-

pendent only on the principal Schmidt eigenvalue λ1

and its corresponding Schmidt operators mA,B

1. These

bounds are tight, and it will be seen that the lower-

bound is quite good, and is, in any case, far from the

average value of the distance (squared) from local uni-

taries, which is 2d2. For the most part, we will concen-

trate on the lower bound that is maximized when λ1is

the minimum possible value =1/d2. This happens only

when all the λiare equal and =1/d2, which implies

that Uis necessarily maximally entangled, which is the

same as dual-unitary. Therefore K∗

D(U)≤K∗

D(Udual) =

√2d2−2d. For convenience we deﬁne

K∗

D:=max

U∈U(d2)K∗

D(U) = p2d2−2d. (10)

Providing a proof for d=2 and evidence for d>2, we

conjecture that

max

U∈U(d2)KD(U) = K∗

D. (11)

II. KD(U)FOR THE TWO-QUBIT CASE

As KD(U)is a local-unitary invariant, it is sufﬁcient

to consider the nonlocal part of the canonical decompo-

sition [18,19]:

U=exp[i(c1σ1⊗σ1+c2σ2⊗σ2+c3σ3⊗σ3)]. (12)

As the products of the Pauli matrices in this expression

commute we have:

∏

k=1

exp(ickσk⊗σk)=

∑

k=0

√µk(eiθkσk)⊗σk. (13)

Here the σ0is the 2 identity matrix and µkare func-

tions of the ci, for explicit expressions see [30]. This

is itself a Schmidt decomposition and the λkare found

by arranging µkin decreasing order, in particular λ1=

max{µ0,µ1,µ2,µ3}. This implies that for any 2-qubit

unitary operator, there exists a Schmidt decomposition

where both pairs of orthonormal bases are unitary as

well. Hence,

KD(U) = K∗

D(U) = q8(1−pλ1), (14)

. In this case both the lower and upper bounds coincide

and are exact. It immediately follows that KD(U)≤

K∗

D=2. Therefore maxU∈U(4)KD(U) = K∗

D=2. This is

achieved only for the case of dual-unitary gates. These

are parameterized in the Cartan form by the one pa-

rameter family: c1=c2=π/4, and c3is arbitrary. We

will ﬁnd this to be true for higher dimensions through

numerical investigations, as discussed later. In Fig. (2)

FIG. 2. The distance to local gates, KD(U), plotted as a color

map in the space of entangling properties of the gate. We ob-

serve that the highest values are attained on the dual-unitary

boundary line.

the KD(U)is shown for all possible two-qubit opera-

tors as a function of their entangling power ep(U)and

gate-typicality gt(U)which are local unitary invariants

[26–28]. For completeness, we recall their deﬁnitions.

The entangling power is the average linear entropy of

U|ϕA⟩|ϕB⟩when ϕA,Bare uniformly (Haar) sampled

from the subspaces and is related to the operator en-

tanglement, Eq. (3. Under appropriate scaling, we take

ep(U) = (E(U) + E(US)−E(S))/E(S), where Sis the

swap operator and 0 ≤ep(U)≤1. If Uis dual-

unitary, E(U) = E(S)and hence ep(U) = E(US)/E(S).

This is maximum and is equal to 1 iff US is also dual-

unitary. In general if Uis dual-unitary, US is deﬁned

to be Γ-dual (or T-dual), as the partial transpose of

the unitary is also unitary [11]. As the swap does

not create any entanglement ep(S) = ep(I) = 0, how-

ever E(S)is the maximum possible, that is the swap

is a very nonlocal gate. A complementary quantity,

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DualunitariesasmaximizersofthedistancetolocalproductgatesShrigyanBrahmachari∗DepartmentofMechanicalEngineering,IndianInstituteofTechnologyMadras,Chennai,India600036RohanNarayanRajmohan,†SuhailAhmadRather,‡andArulLakshminarayan§DepartmentofPhysics,IndianInstituteofTechnologyMadras,Chennai,India600036...

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Dual unitaries as maximizers of the distance to local product gates Shrigyan Brahmachari Department of Mechanical Engineering Indian Institute of Technology Madras Chennai India 600036

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