The Solovay-Kitaev Theorem guarantees that for any single-qubit gate U and precision ε > 0, it is possible to approximate U to within ε using Θ(logc(1/ε)) gates from a fixed finite universal set of quantum gates. The proof involves first using a "shrinking lemma" to show that any gate in an ε-net can be approximated to within Cε using a constant number of applications of gates from the universal set. This is then iterated to construct an approximation of the desired gate U to arbitrary precision using a number of gates that scales as the logarithm of 1/ε.

1. The Solovay-Kitaev Theorem
James Ma
November 2018
1 Introduction
The Solovay-Kitaev Theorem is one of the most fundamental concepts in the
theory of quantum computation. Consider someone wants to design an algo-
rithm for a single-qubit, but the types of gate which he can use are restricted.
In the case of classical circuits, only a small set of gates(e.g. AND, OR, NOT)
can construct any arbitrary classical function. But for quantum computation,
sometimes we can only construct an approximate circuits for an unitary opera-
tion for arbitrary accuracy with a finite set of quantum gates. Such set is said
to be universal for quantum computation.
The SK Theorem guarantees for any single-qubit gate U, and >0. It is pos-
sible to approximate U to a precision using Θ(logc 1
) gates from a fixed finite
set.
The goal of this note is to show the SK theorem and it’s proof. But as I am
going to formally introduce the SK theorem, here comes some definitions and
notations.
2 Prior knowledge
Definition 1. (U(n) and SU(n))[3] The unitary group U(n) is the subgroup
of GL(n, C) consisting of unitary matrices. The special unitary group SU(n)
is the subgroup of an unitary group consisting of matrices of determinant 1.
We need a notation of distance to quantify what we mean by an approx-
imation to a unitary matrix, although the exact measure used is not all that
important. It is convenient to use the trace distance D(U, V ) ≡ tr(|U − V |),
where |X| ≡
√
X†X. Also, define · ≡ tr(| · |) as the trace distance.
Definition 2. [1] A subset S of SU(2) is said to be dense in SU(2) if for any
element U of SU(2) and >0, there is an element s ∈ S such that D(s, U)< .
Definition 3. [1] Suppose S and W are subsets of SU(2). Then S is said to
form an -net for W, where >0, if every point in W is within a distance of
some point in S.
1

2. Definition 4. [2] An instruction set G for a d-dimensional qudit is a finite
set of quantum gates satisfying:
1. All gates g ∈ G are in SU(d)
2. For each g ∈ G, the inverse g†
is also in G
3. G is an universal set for SU(d), i.e., the group generated by G is dense in
SU(d).
For convenience, define
Gl ≡ {g1...gl ∈ SU(2)|gi ∈ G} (1)
S ≡ {U ∈ SU(2)|D(U, I) ≤ } (2)
3 The SK Theorem
Theorem 1. (Solovay-Kitaev Theorem)
Let G be an instruction set for a 2-dimensional qudit. Let >0 be given. Then
Gl is an -net in SU(2) for l = O(logc
(1/ )), where c ≈ 4
Claim 1. Let A be an arbitrary matrix, U be unitary, then
tr(|AU|) = tr(|A|) (3)
Proof. Claim that for an arbitrary matrix X,
tr(|X|) = tr(
√
X†X) = tr(
√
XX†). (4)
Since X†
X is Hermitian, it is normal. Recall from the spectral theorem, any
normal operator M on a vector space V is diagonal with respect to some orthog-
onal basis for V . Hence X†
X = S†
DS where each column of S is an eigenvector
of X†
X, D is an diagonal matrix composed of eigenvalues of X†
X. Note that
all of these eigenvalues are real since X†
X is Hermitian. Hence
tr(
√
X†X) = tr(
√
S†DS) = tr(S†
D
1
2 S) = tr(D
1
2 ) =
i
σi (5)
where i σi sums up all singular values of X.
Since the set of singular values of T†
is equal to T, the claim is proofed. Then
we have
tr(|AU|) = tr(| (AU)†(AU)|) = tr(| (AU)(AU)†|) = tr(|A|) (6)
2

3. Lemma 1 (Shrinking Lemma). Let G be an instruction set for a 2-dimensional
qubit. There exists an universal constant 0 independent of G such that for any
≤ 0, if Gl is an 2
-net for S , then G5l is a C 3
-net for S√
C
3
2
, for some
constant C.
The proof of the lemma will be shown later, but first let’s see how it implies
the SK theorem.
Proof of The SK Theorem. Since G is an instruction set, it is dense in SU(2).
There exists l0 such that Gl0
is an 2
0-net for SU(2), also for S . Let = 0 and
l = l0. Then by Lemma 1, G5l0
is a C 3
0-net for S√
C
3
2
0
. Iterating the procedure
k times, we can conclude that G5kl0
is an (k)2
-net for S (k), where
(k) =
(C 0)( 3
2 )k
C
(7)
We can choose 0 such that C 0 < 1. Also for 0 is chosen small enough, then
(k)2
< (k + 1).
The second step is to show that given U ∈ SU(2), we are going to construct
an approximation using products of elements of G. Let U0 ∈ Gl0
be an (0)2
-
approximation to U. Define V1 ≡ UU†
0 , then
D(V1, I) = tr|V1 − I| = tr|(U − U0)U†
0 | = tr|U − U0| < (0)2
< (1) (8)
Also from the iterated application of Lemma 1, there exists U1 ∈ G5l0 such that
D(U1, V1) < (1)2
(9)
Thus,
D(U1U0, U) = tr(|(U1 − V1)U0|) = tr(|(U1 − V1)|) < (1)2
(10)
Continue the similar procedure. Define V2 ≡ UU†
0 U†
1 , then
D(V2, I) < (1)2
< (2) (11)
Again, there exists U2 ∈ G52l0
such that
D(U2, V2) < (2)2
(12)
Thus,
D(U2U1U0, U) < (2)2
(13)
In general, we can construct Uk ∈ G5kl0
such that
D(Uk...U0, U) < (k)2
(14)
3

4. In order to approximate an arbitrary unitary gate U to an accuracy (k)2
,
totally we need
l0 + 5l0 + ... + 5k
l0 =
5k+1
− 1
4
(15)
gates to implement. To approximate with accuracy , we must therefore choose
k such that (k)2
< . Substituting (7) this can be restated as
(
3
2
)k
<
log(1/C2
)
2log(1/C 0)
(16)
It follows that the number of gates n required to approximate to within
satisfies(c = log5/log(3/2) ≈ 4)
n <
5
4
5k
l0 =
5
4
(
3
2
)kc
l0 <
5
4
(
log(1/C2
)
2log(1/C 0)
)c
l0 = O(logc
(
1
)) (17)
Now, we are going to proof Lemma 1. But there are still a few facts to be
noted before we start.
Let su(n) denote the set of all n × n traceless Hermitian matrices. Here are
some important facts about su(n)
• If H ∈ su(n), then e−iH
∈ SU(n), since dete−iH
= e−i∗tr(H)
= 1 and
e−iH
(e−iH
)†
= e−iH
eiH
= I.
• If A, B ∈ su(n),then e−iH
∈ SU(n) since (i[A, B])†
= −i(BA − AB) =
i[A, B] and tr(i[A, B]) = i ∗ tr(AB − BA) = 0
Suppose U, V ∈ SU(2), U, V ≡ UV U†
V †
is the group commutator.
Claim 2. Let A, B ∈ su(n) such that A , B ≤ for some sufficiently small
. Then e−[A,B]
− e−iA
, e−iB
= O( 3
)
Proof.
e−[A,B]
= I − [A, B] +
1
2
[A, B]2
− .... (18)
e−iA
, e−iB
= (I − iA −
A2
2
+ ...)(I − iB −
B2
2
+ ...) (19)
(I + iA −
A2
2
+ ...)(I + iB −
B2
2
+ ...) (20)
= I − [A, B] − .... (21)
The expansions for both terms agree up to the second order. Hence,
e−[a,B]
− e−iA
, e−iB
= c3[A, B]3
+ c4[A, B]4
+ .... (22)
The claim follows by using triangle inequality and submultiplicativity of the
norm.
4

5. Notation. Let u : R3
SU(2) be the map u(r) := exp(− i
2 r·σ). It provides
a correspondence between the Lie algebra su(2) and the Lie group SU(2).
Claim 3. Let y, z ∈ R3
. Then exp(−[1
2 y · σ, 1
2 z · σ]) = u(y × z)
Proof. Since,
[y · σ, z · σ] = 2i(y × z) · σ (23)
We have
u(y × z) = e− i
2 y×z·σ
= e−[ 1
2 y·σ, 1
2 z·σ]
(24)
Claim 4. Let r ∈ R3
. Then D(u(r), I) = 4sin|r|
4
Proof. The eigenvalues of r · σ are ±|r|, because the characteristic polynomial
of r · σ is
det(rσ − λI) =
rz − λ rx − iry
rx + iry −rz − λ
(25)
= λ − (r2
x + r2
y + r2
z) (26)
Thus, the eigenvalues of u(r) − I are e± i
2 |r|
− 1.
|e± i
2 |r|
− 1| = cos
|r|
2
− 1
2
+ sin
|r|
2
2
(27)
= 2 − 2cos
|r|
2
= 2
1
2
1 − cos
|r|
2
= 2sin
|r|
4
(28)
hence u(r) − I = 4sin|r|
4
Claim 5. Let r ∈ R3
. If U(r) ∈ S , then |r| < + O( 3
)
Proof. We have u(r) − I = 4sin|r|
4 < .Thus,|r| < 4arcsin 4 . Follows from
Taylor expansion arcsinz = z + 1
2 · z3
3 + 1·3
2·4 · z5
5 + ....
Claim 6. If y, z ∈ R3
and |y|, |z| < , then u(y) − u(z) = |y − z| + O( 3
)
Proof.
u(y) − u(z) = (u(y) − u(z))u(z)†
= u(y)u(z)†
− I (29)
= e− i
2 y·σ
e
i
2 z·σ
− I = u(y − z) − I = 4sin
|y − z|
4
(30)
and the result follows from the Taylor expansion sinα = α − α3
3! + α5
5! − ....
5

6. Proof The Shrinking Lemma. We are going to proof in this way:
(l, 2
, ) =⇒ (4l, C 3
, 2
) =⇒ (5l, C 3
,
√
C 3
) (31)
Step 1. (l, 2
, ) =⇒ (4l, C 3
, 2
)
Let U ∈ S 2 , choose x ∈ R such that U = u(x). By Claim 5, |x| < 2
+ O( 6
).
We can find y, z ∈ R such that x = y × z, |y|, |z| < + O( 5
)
(Notation. |x| = |y × z| ≤ |y||z| = 2
+ O( 6
))
And, u(y), u(z) ∈ S because
u(y) − I = 4sin
|y|
4
= 4
|y|
4
−
1
3
|y|
4
3
+ ... < + O( 3
) (32)
Then, we are going to approximate y, z by Gl.
Since G is an 2
-net for S , we can choose y0, z0 ∈ R such that u(y0), u(z0) ∈
Gl S .Then,
u(y0) − I < , u(y0) − u(y) < 2
(33)
u(z0) − I < , u(z0) − u(z) < 2
(34)
By Claim 5,
|y0| < + O( 3
) (35)
|z0| < + O( 3
) (36)
By (35),(36) and Claim 6,
u(y0) − u(y) = |y0 − y| + O( 3
) < 2
(37)
Hence,
|y0 − y| < 2
+ O( 3
) (38)
|z0 − z| < 2
+ O( 3
) (39)
The main goal of the proof in this part is to show that U can be approximated
well enough using a sequence of 4l gates from G. We want to show
u(x) − u(y0), u(z0) < C 3
(40)
where u(y0), u(z0) are those 4l gates in G.
u(x) − u(y0), u(z0) (41)
≤ u(x) − u(y0 × z0) + u(y0 × z0) − u(y0), u(z0) (42)
Where
u(x) − u(y0 × z0) = u(y × z) − u(y0 × z0) (43)
= |y × z − y0 × z0| + O( 6
) (44)
= |[(y − y0) + y0] × [(z − z0) + z0] − y0 × z0| + O( 6
) (45)
= |(y − y0) × (z − z0) + (y − y0) × z0 + y0 × (z − z0)| + O( 6
) (46)
≤ 4
+ 2 3
+ O( 6
) < c 3
+ O( 4
) (47)
6

7. where c ≥ 2.
u(y0 × z0) − u(y0), u(z0) (48)
= u(y0 × z0) − e− i
2 y0·σ
, e− i
2 z0·σ
(49)
= e−[ 1
2 y0·σ, 1
2 z0·σ]
− e− i
2 y0·σ
, e− i
2 z0·σ
≤ d 3
(50)
Thus, combine the two part into (42), we have
u(x) − u(y0), u(z0) ≤ C 3
(51)
For some constant C.
Step 2. (4l, C 3
, 2
) =⇒ (5l, C 3
,
√
C 3
) Recall that the initial parameters
were (l, 2
, ). Thus, for a given u ∈ S√
C 3 we can find V ∈ Gl
such that
U − V = UV †
− I < 2
meaning that UV †
∈ S 2 . Since G4l
is an s 3
-net
for S 2 , we can find y0, z0 ∈ R3
such that
U − u(y0), u(z0) V = UV †
− u(y0), u(z0) ≤ C 3
(52)
References
[1] Michael A. Nielsen & Isaac L. Chuang. Quantum Computation and Quantum
Information. Cambridge University Press, 2010.
[2] Christopher M. Dawson and Micheal A. Nielson. The Solovay-Kitaev Algo-
rithm. Rinton Press, 2008.
[3] Alistair Savage. Introduction to lie groups. Technical report, Department
of Mathematics and Statistics, University of Ottawa, 2015.
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