PHYSICAL REVIEW A 86, 052329 (2012)
Magic-state distillation with low overhead
Sergey Bravyi
1
and Jeongwan Haah
2
1
IBM Watson Research Center, Yorktown Heights, New York 10598, USA
2
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
(Received 3 October 2012; published 27 November 2012)
We propose a family of error-detecting stabilizer codes with an encoding rate of 1/3 that permit a transversal
implementation of the gate T = exp (−iπZ/8) on all logical qubits. These codes are used to construct protocols
for distilling high-quality “magic” states T
|
+
by Clifford group gates and Pauli measurements. The distillation
overhead scales as O( log
γ
(1/)),where is the output accuracy and γ = log
2
(3) ≈ 1.6. To construct the desired
family of codes, we introduce the notion of a triorthogonal matrix, a binary matrix in which any pair and any
triple of rows have even overlap. Any triorthogonal matrix gives rise to a stabilizer code with a transversal T
gate on all logical qubits, possibly augmented by Clifford gates. A powerful numerical method for generating
triorthogonal matrices is proposed. Our techniques lead to a twofold overhead reduction for distilling magic
states with accuracy ∼ 10
−12
compared with previously known protocols.
DOI: 10.1103/PhysRevA.86.052329 PACS number(s): 03.67.Pp
I. INTRODUCTION
Quantum error-correcting codes provide a means of trading
quantity for quality when unreliable components must be used
to build a reliable quantum device. By combining together
sufficiently many unprotected noisy qubits and exploiting
their collective degrees of freedom insensitive to local errors,
quantum coding allows one to simulate noiseless logical qubits
and quantum gates up to any desired precision provided that the
noise level is below a constant threshold value [1–4]. Protocols
for fault-tolerant quantum computation with an error threshold
close to 1% have been proposed recently [5–7].
An important figure of merit of fault-tolerant protocols is the
cost of implementing a given logical operation such as a unitary
gate or a measurement with a desired accuracy . Assuming
that elementary operations on unprotected qubits have a unit
cost, all fault-tolerant protocols proposed so far, including
the ones based on concatenated codes [4] and topological
codes [6–8], enable implementation of a universal set of logical
operations with the cost O( log
β
(1/)), where the scaling
exponent β depends on a particular protocol.
For protocols based on stabilizer codes [9] the cost of a
logical operation may also depend on whether the operation is
a Clifford or a non-Clifford one. The set of Clifford operations
(CO) consists of unitary Clifford group gates, such as the
Hadamard gate H ,theπ/4 rotation S = exp (−iπZ/4), and
the controlled-
NOT (CNOT) gate, preparation of ancillary
|
0
states, and measurements in the |0,|1 basis. Logical CO
usually have a relatively low cost as they can be implemented
either transversally [9] or, in the case of topological stabilizer
codes, by the code deformation method [7,8,10]. On the other
hand, logical non-Clifford gates, such as the π/8 rotation
T = exp (−iπZ/8), usually lack a transversal implementation
[11,12] and have a relatively high cost that may exceed the
one of CO by orders of magnitude [8]. Reducing the cost of
non-Clifford gates is an important problem since the latter con-
stitute a significant fraction of any interesting quantum circuit.
The present paper addresses this problem by constructing
low overhead protocols for the magic-state distillation, a
particular method of implementing logical non-Clifford gates
proposed in [13]. A magic state is an ancillary resource state
ψ that combines two properties.
Universality. Some non-Clifford unitary gate can be imple-
mented using one copy of ψ and CO. The ancilla ψ can be
destroyed in the process.
Distillability. An arbitrarily good approximation to ψ can
be prepared by CO, given a supply of raw ancillae ρ with the
initial fidelity
ψ
|
ρ
|
ψ
above some constant threshold value.
Since the Clifford group augmented by any non-Clifford
gate is computationally universal [14], magic-state distillation
can be used to achieve universality at the logical level provided
that logical CO and logical raw ancillae ρ are readily available.
Below we shall focus on the magic state
|
A
= T
|
+
∼
|
0
+ e
iπ/4
|
1
.
A single copy of
|
A
combined with a few CO can be
used to implement the T gate [15], thereby providing a
computationally universal set of gates [13,16]. It was shown
by Reichardt [17] that state
|
A
is distillable if and only
if the initial fidelity
A
|
ρ
|
A
is above the threshold value
(1 + 1/
√
2)/2 ≈ 0.854.
Our main objective will be to minimize the number of raw
ancillae ρ required to distill magic states
|
A
with a desired
accuracy . To be more precise, let σ be a state of k qubits
which is supposed to approximate k copies of
|
A
. We will say
that σ has an error rate iff the marginal state of any qubit
has an overlap of at least 1 − with
|
A
. Suppose such a state
σ can be prepared by a distillation protocol that takes as input
n copies of the raw ancilla ρ and uses only CO. We will say
that the protocol has a distillation cost C = C()iffn Ck.
For example, the original distillation protocol of Ref. [13]
based on the 15-qubit Reed-Muller code has a distillation cost
O( log
γ
(1/)), where γ = log
3
(15) ≈ 2.47.
II. SUMMARY OF RESULTS
Our main result is a family of distillation protocols for
state |A with a distillation cost O( log
γ
(1/)), where γ =
log
2
(
3k+8
k
) and k is an arbitrary even integer. By choosing
large enough k the scaling exponent γ can be made arbitrarily
close to log
2
(3) ≈ 1.6. The protocol works by concatenating
an elementary subroutine that takes as input 3k +8 magic
states with an error rate p and outputs k magic states with an
error rate O(p
2
). For comparison, the protocol found by Meier
052329-1
1050-2947/2012/86(5)/052329(10) ©2012 American Physical Society
SERGEY BRAVYI AND JEONGWAN HAAH PHYSICAL REVIEW A 86, 052329 (2012)
et al. [18] has a distillation cost as above with the scaling
exponent γ = log
2
(5) ≈ 2.32. Distillation protocols with the
scaling exponent γ = 2 were recently discovered by Campbell
et al. [19], who studied extensions of stabilizer codes, CO, and
magic states to qudits. We conjecture that the scaling exponent
γ cannot be smaller than 1 for any distillation protocol and
give some arguments in support of this conjecture in Sec. VI.
Our distillation scheme borrows two essential ideas from
Refs. [13,18]. First, as proposed in [13], we employ stabilizer
codes that admit a special symmetry in favor of transversal T
gates and measure the syndrome of such codes to detect errors
in the input magic states. Secondly, as proposed by Meier
et al. [18], we reduce the distillation cost significantly by
using distance-2 codes with multiple logical qubits. The new
ingredient is a systematic method of constructing stabilizer
codes with the desired properties. To this end we introduce the
notion of a triorthogonal matrix, a binary matrix in which any
pair and any triple of rows have even overlap. We show that any
triorthogonal matrix G with k odd-weight rows can be mapped
to a stabilizer code with k logical qubits that admit a transversal
T gate on all logical qubits, possibly augmented by Clifford
gates. Each even-weight row of G gives rise to a stabilizer
which is used in the distillation protocol to detect errors in the
input magic states. Finally, we propose a powerful numerical
method for generating triorthogonal matrices. To illustrate its
usefulness, we construct the first example of a distance-5 code
with a transversal T gate that encodes 1 qubit into 49 qubits.
While the asymptotic scaling of the distillation cost is of
great theoretical interest, its precise value in the nonasymptotic
regime may offer valuable insights on the practicality of a given
protocol. Using raw ancillae with the initial error rate 10
−2
and
the target error rate between 10
−3
and 10
−30
, we computed
the distillation cost C() numerically for the optimal sequence
composed of the 15-to-1 protocol of Ref. [13], and the 10-to-2
protocol of Ref. [18]. Combining these protocols with the
ones discovered in the present paper, we observed a twofold
reduction of the distillation cost for = 10
−12
and a noticeable
cost reduction for the entire range of (see Table I in Sec. VIII).
Since a magic-state distillation is meant to be performed
at the logical level of some stabilizer code, throughout this
paper we assume that CO themselves are perfect. Whether
or not this simplification is justified depends on the chosen
code. More precisely, let the cost of implementing logical
CO and the distillation cost be log
β
(1/) and log
γ
(1/),
respectively, where is the desired precision. In the case
β<γ, high-quality CO are cheap, and one can safely assume
that CO are perfect. The opposite case, when high-quality
CO are expensive (i.e., β>γ), is realized, for example,
in the topological one-way quantum computer based on the
three-dimensional cluster state introduced by Raussendorf
et al. [8], where β = 3. As was pointed out in [8], in this
case it is advantageous to use expensive high-quality CO only
at the final rounds of distillation and to use relatively cheap
noisy CO for the initial rounds. Using the 15-to-1 distillation
protocol of Ref. [13] with γ = log
3
15 ≈ 2.47, the authors of
Ref. [8] showed how to implement a universal set of logical
gates with the cost O( log
3
(1/)). A detailed analysis of errors
in logical CO was performed by Jochym-O’Connor et al. [20].
The rest of the paper is organized as follows. We begin with
the definition of triorthogonal matrices and state their basic
properties in Sec. III. The correspondence between triorthog-
onal matrices and stabilizer codes with a transversal T gate is
described in Sec. IV. We introduce our distillation protocols for
magic state
|
A
in Secs. V and VI and Appendix A. A family
of distance-2 codes with an encoding rate of 1/3 that admit a
transversal T gate is presented in Sec. VII. We compute the
distillation cost of the new protocols and make a comparison
with the previously known protocols in Sec. VIII. A numerical
method of generating triorthogonal matrices is presented in
Sec. IX. Finally, Appendix B presents the [[49,1,5]] code with
a transversal T gate.
Notation. Below we adopt standard notation and termi-
nology pertaining to quantum stabilizer codes [21]. Given
a pair of binary vectors f,g ∈ F
n
2
,let(f,g) =
n
j=1
f
j
g
j
(mod 2) be their inner product and |f | be the weight of
f , that is, the number of nonzero entries in f .Givena
linear space G ⊆ F
n
2
, its dual space G
⊥
consists of all vectors
f ∈ F
n
2
such that (f,g) = 0 for any g ∈ G. We shall use the
notation X,Y,Z for the single-qubit Pauli operators. Given any
single-qubit operator O and a binary vector f ∈ F
n
2
, the tensor
product O
f
1
⊗···⊗O
f
n
will be denoted O(f ). In particular,
X(f )Z(g) = (−1)
(f,g)
Z(g)X(f ). The Pauli group P
n
consists
of n-qubit Pauli operators i
ω
P
1
⊗···⊗P
n
, where P
j
∈
{I,X,Y,Z}, and ω ∈ Z
4
. The Clifford group C
n
consists of all
unitary operators U such that U P
n
U
†
= P
n
. It is well known
that C
n
is generated by one-qubit gates H = (X + Z)/
√
2
(the Hadamard gate), S = exp (iπZ/4) (the S gate), and
the controlled-Z gate (Z) = exp (iπ
|
11
11
|
). All quantum
codes discussed in this paper are of the Calderbank-Shor-
Steane (CSS) type [22,23]. Given a pair of linear spaces
F,G ⊂ F
n
2
such that F ⊆ G
⊥
, the corresponding CSS code
has stabilizer group {X(f )Z(g),f∈ F ,g∈ G} and will be
denoted as CSS
(
X,F; Z,G
)
.
III. TRIORTHOGONAL MATRICES
To describe our distillation protocols let us define a new
class of binary matrices.
Definition 1. A binary matrix G of size m × n is called
triorthogonal iff the supports of any pair and any triple of its
rows have even overlap, that is,
n
j=1
G
a,j
G
b,j
= 0(mod2) (1)
for all pairs of rows 1 a<b m and
n
j=1
G
a,j
G
b,j
G
c,j
= 0(mod2) (2)
for all triples of rows 1 a<b<c m.
An example of a triorthogonal matrix of size 5 × 14 is
G =
⎡
⎢
⎢
⎢
⎣
1111111
1111111
11111111
11 11 11 11
1111 1111
⎤
⎥
⎥
⎥
⎦
,
(3)
052329-2
MAGIC-STATE DISTILLATION WITH LOW OVERHEAD PHYSICAL REVIEW A 86, 052329 (2012)
where only nonzero matrix elements are shown. The two
submatrices of G formed by even-weight and odd-weight rows
will be denoted G
0
and G
1
respectively. The submatrix G
0
is
highlighted in bold in Eq. (3). We shall always assume that G
1
consists of the first k rows of G for some k 0. Define linear
subspaces G
0
,G
1
,G ⊆ F
n
2
spanned by the rows of G
0
, G
1
, and
G, respectively. Using Eq. (1) alone, one can easily prove the
following.
Lemma 1. Suppose G is triorthogonal. Then (i) all rows
of G
1
are linearly independent over F
2
, (ii) G
0
∩ G
1
= 0,
(iii) G
0
= G ∩ G
⊥
, and (iv) G
⊥
0
= G
1
⊕ G
⊥
.
Proof. Let f
1
,...,f
m
be the rows of G such that the first k
rows form G
1
. By definition, any vector f ∈ G
1
can be written
as f =
k
a=1
x
a
f
a
for some x
a
∈ F
2
.FromEq.(1) we infer
that (f
a
,f
b
) = δ
a,b
for all 1 a,b k and (f
a
,g) = 0for
any g ∈ G
0
. Hence x
a
= (f,f
a
). If f = 0orf ∈ G
0
, then
x
a
= 0 for all a. This proves (i) and (ii). Since any row of G
0
is orthogonal to itself and any other row of G, we get (f,g) = 0
for all f ∈ G
0
and g ∈ G. This implies G
0
⊆ G ∩ G
⊥
.Iff =
m
a=1
x
a
f
a
∈ G ∩ G
⊥
, then x
a
= (f,f
a
) = 0 for all 1 a
k, that is, f ∈ G
0
. This proves (iii). Finally, (iv) follows from
G
1
⊕ G
⊥
⊆ G
⊥
0
, G
1
∩ G
⊥
= 0, and dimension counting.
As we show in Sec. IV, any binary matrix G with n
columns and k odd-weight rows satisfying Eq. (1) gives rise
to a stabilizer code encoding k qubits into n qubits. Condi-
tion (2) ensures that this code has the desirable transversality
properties, namely, the encoded |A
⊗k
state can be prepared
by applying the transversal T gate T
⊗n
to the encoded |+
⊗k
,
possibly augmented by some Clifford operator. To state this
more formally, define n-qubit unnormalized states
|
G
0
=
g∈G
0
|
g
,
|
G
=
g∈G
|
g
. (4)
Define also a state
|
A
⊗k
=
k
a=1
[I + e
iπ/4
X(f
a
)]
|
G
0
, (5)
where f
1
,...,f
k
are the rows of G
1
.
Lemma 2. Suppose a matrix G is triorthogonal. Then there
exists a Clifford group operator U composed of (Z) and S
gates only such that
|
A
⊗k
=UT
⊗n
|
G
. (6)
Proof. Below we promote the elements of binary field F
2
to the normal integers of Z; we associate F
2
0 → 0 ∈ Z
and F
2
1 → 1 ∈ Z. Unless otherwise denoted by (mod2)
or (mod4), every sum is the usual sum for integers, and no
modulo reduction is performed.
When y = (y
1
,...,y
m
) is a string of 0 or 1, let (y) ≡
|y| (mod 2) be the parity of y. Let us derive a formula for a
phase factor e
iπ(y)/4
as a function of components y
a
. Observe
that
(y) =
1
2
[1 − (1 − 2)
|y|
] =
|y|
p=1
|y|
p
(−2)
p−1
. (7)
Since the binomial coefficient (
|y|
p
) is the number of ways to
choose p nonzero components of y, we may write
e
iπ(y)/4
=exp
iπ
4
m
a=1
y
a
−
iπ
2
a<b
y
a
y
b
+iπ
a<b<c
y
a
y
b
y
c
.
(8)
By definition of state
|
G
, one has
T
⊗n
|
G
=
f ∈G
e
iπ|f |/4
|
f
.
Since
|
G
depends on the linear space G rather than the matrix
presentation G, we may assume that all rows of G are linearly
independent over F
2
.Letg
1
,...,g
m
be the rows of G, and
decompose f =
m
a=1
x
a
g
a
(mod 2), where x
a
∈{0,1} are
uniquely determined by f .
Each component f
j
of f is the parity of the bit string
(x
1
g
1
j
,x
2
g
2
j
,...,x
m
g
m
j
), and |f |is the sum of f
j
. Hence, Eq. (8)
implies
e
iπ|f |/4
= exp
iπ
4
m
a=1
x
a
|g
a
|−
iπ
2
a<b
x
a
x
b
|g
a
· g
b
|
+ iπ
a<b<c
x
a
x
b
x
c
|g
a
· g
b
· g
c
|
, (9)
where g
a
· g
b
denotes the bitwise AND operation. Triorthogo-
nality condition (2) implies that the triple overlap |g
a
· g
b
· g
c
|
is even, so we may drop the last term in Eq. (9).Thisis,in
fact, one of the main reasons why we consider triorthogonal
matrices.
Let the first k rows of G have odd weight and all others
have even weight, and put
|g
a
|=
2
a
+ 1if 1 a k,
2
a
otherwise.
In addition, Eq. (1) implies for distinct a,b that
|g
a
· g
b
|=2
ab
.
Here all
a
and
ab
are integers. Thus
e
iπ|f |/4
= exp
iπ
4
k
a=1
x
a
exp
iπ
2
Q(x
1
,...,x
m
)
,
where
Q(x) =
m
a=1
a
x
a
− 2
a<b
ab
x
a
x
b
.
Let us show that the unwanted phase factor e
iπQ/2
can be
canceled by a unitary Clifford operator that uses only (Z)
and S gates. To this end, we rewrite Q(x)asafunctionoff .
As noted earlier, x
a
are uniquely determined by f . Indeed,
052329-3
SERGEY BRAVYI AND JEONGWAN HAAH PHYSICAL REVIEW A 86, 052329 (2012)
there is a matrix B over F
2
such that x
a
=
p
B
ap
f
p
(mod 2)
since {g
a
} is a basis of the linear space G. (There could be
many such B.) We again use Eq. (7) with the observation that
x
a
is the parity of the bit string (B
a1
f
1
,...,B
an
f
n
) to infer
x
a
=
p
B
ap
f
p
− 2
p<q
B
ap
B
aq
f
p
f
q
(mod 4),
2x
a
x
b
= 2
p,q
B
ap
B
bq
f
p
f
q
(mod 4)
for all a,b = 1,...,m. Therefore, we can express Q(x)as
Q(x(f )) =
n
p=1
p
f
p
− 2
p<q
pq
f
p
f
q
(mod 4),
where
p
,
pq
are some integers determined by B,
a
, and
ab
, all of which depend only on our choice of the matrix G.
Explicitly,
p
=
a
a
B
ap
− 2
a<b
ab
B
ap
B
bp
and
pq
=
a
a
B
ap
B
aq
−
a<b
ab
(B
ap
B
bq
+ B
bp
B
aq
).
The extra phase factor e
iπQ/2
is canceled by applying the
(Z)
pq
gate for each pair of qubits p<qand the gate (S
†
)
p
to every qubit p. This defines the desired Clifford operator U
composed of (Z) and S gates such that
UT
⊗n
|
f
= exp
iπ
4
k
a=1
x
a
|
f
(10)
for all f =
m
a=1
x
a
g
a
(mod 2) ∈ G. Therefore,
UT
⊗n
|
G
=
k
a=1
[I + e
iπ/4
X(g
a
)]
|
G
0
=|
A
⊗k
.
For the later use let us state the following simple fact.
Lemma 3. Let G be a triorthogonal matrix without zero
columns. If G
1
is not empty and G
0
has fewer than three rows,
then G
0
must have at least one zero column.
Proof. Suppose, on the contrary, all columns of G
0
are
nonzero. If G
0
has only one row, it must be the all-ones vector
1
n
. Then, the inner product between 1
n
and any row f of G
1
is
the weight of f modulo 2, which is odd. But, the orthogonality
equation (1) requires it to be even. This is a contradiction.
Suppose now that G
0
has two rows g
1
,g
2
. By permuting
the columns we may assume that G
0
=
ABC
, where
A =
1···1
0···0
,B=
0 ···0
1 ···1
,C=
1 ···1
1 ···1
.
Choose an odd-weight row f of G
1
, and let w
A
,w
B
,w
C
be the
weight of f restricted to the columns of A,B,C, respectively.
The (tri)orthogonality equations (1) and (2) imply
|g
1
· f |=w
A
+ w
C
= 0(mod2),
|g
2
· f |=w
B
+ w
C
= 0(mod2),
|g
1
· g
2
· f |=w
C
= 0(mod2).
This is a contradiction since |f |=w
A
+ w
B
+ w
C
= 1
(mod 2).
IV. STABILIZER CODES BASED ON
TRIORTHOGONAL MATRICES
Given a triorthogonal matrix G with k odd-weight rows,
define a stabilizer code CSS
X,G
0
; Z,G
⊥
with X-type stabi-
lizers X(f ), f ∈ G
0
, and Z-type stabilizers Z(g), g ∈ G
⊥
.The
inclusion G
0
⊆ G implies that all stabilizers pairwise commute.
Lemma 4. The code CSS
X,G
0
; Z,G
⊥
has k logical qubits.
Its logical Pauli operators can be chosen as
X
a
= X(f
a
), Z
a
= Z(f
a
),a= 1,...,k, (11)
where f
1
,...,f
k
are the rows of G
1
. The states |G
0
, |G,
and |
A
⊗k
defined in Eqs. (4) and (5) coincide with encoded
states |0
⊗k
, |+
⊗k
, and |A
⊗k
, respectively.
Proof. Indeed, the assumption that f
a
have odd weight
and Eq. (1) ensure that the operators defined in Eq. (11)
obey the correct commutation rules, that is,
X
a
Z
b
=
(−1)
δ
a,b
Z
b
X
a
. It remains to be checked that X
a
and Z
a
commute with all stabilizers. Given any Z-type stabilizer
Z(g), g ∈ G
⊥
, one has X(f
a
)Z(g) = (−1)
(f
a
,g)
Z(g)X(f
a
) =
Z(g)X(f
a
) since f
a
∈ G and g ∈ G
⊥
. Given any X-
type stabilizer X(f ), f ∈ G
0
, one has Z(f
a
)X(f ) =
(−1)
(f
a
,f )
X(f )Z(f
a
) = X(f )Z(f
a
) since f
a
∈ G and G
0
⊆
G
⊥
; see Lemma 1. This shows that X
a
and Z
a
are indeed
logical Pauli operators on k encoded qubits.
Property (iii) of Lemma 1 implies that Z(g) |f =|f for
any f ∈ G
0
and any g ∈ G + G
⊥
. Thus the state |G
0
defined
in Eq. (4) coincides with the encoded |0
⊗k
state. It follows
that |G=
k
a=1
(I + X
a
)|G
0
is the encoded |+
⊗k
state,
while |
A
⊗k
=
k
a=1
(I + e
iπ/4
X
a
)|G
0
is the encoded |A
⊗k
(ignoring the normalization).
Using Lemma 4 one can show that the operator UT
⊗n
defined in Lemma 2 implements an encoded T gate on each
logical qubit of the code CSS(X,G
0
; Z,G
⊥
). Indeed, for any
x ∈ F
k
2
, the encoded state |x≡|x
1
,...,x
k
is
|
x=X
x
1
1
···X
x
k
k
|G
0
=
f ∈G
0
+x
1
f
1
+···+x
k
f
k
|f .
Using Eq. (10) from the proof of Lemma 2, one arrives at
UT
⊗n
|x=e
i
π
4
k
a=1
x
a
|x.
This provides a generalization of a transversal T gate to
multiple logical qubits.
V. DISTILLATION SUBROUTINE
We are now ready to describe the elementary distillation
subroutine. It takes as input n copies of a (mixed) one-qubit
ancilla ρ such that
A
|
ρ
|
A
= 1 − p. We shall refer to p as
the input error rate. Define single-qubit basis states
|
A
0
≡
|
A
and
|
A
1
≡ Z
|
A
. We shall assume that ρ is diagonal in the A
basis; that is,
ρ = (1 − p)
|
A
0
A
0
|
+ p
|
A
1
A
1
|
. (12)
This can always be achieved by applying operators I and
A ≡ e
−iπ/4
SX with a probability of 1/2 each to every copy
052329-4
MAGIC-STATE DISTILLATION WITH LOW OVERHEAD PHYSICAL REVIEW A 86, 052329 (2012)
of ρ. Note that A
|
A
α
= (−1)
α
|
A
α
; that is, the random
application of A is equivalent to the dephasing in the A basis,
which destroys the off-diagonal matrix elements
A
0
|
ρ
|
A
1
without changing the fidelity
A
0
|
ρ
|
A
0
.
Define linear maps
T (η) = TηT
†
, E(η) = (1 − p)η + pZηZ (13)
describing the ideal T gate and the Z error, respectively. Using
Clifford operations and one copy of ρ as in Eq. (12), one
can implement a noisy version of the T gate, namely, E ◦ T .
A circuit implementing E ◦ T is shown in Fig. 1, where the
Z error E isshownbytheZ-gate box with a subscript p
indicating the error probability. One can easily show that this
circuit indeed implements E ◦ T by commuting E through the
CNOT gate and the classically controlled SX gate.
The entire subroutine is illustrated in Fig. 2. The first step is
to prepare k copies of the state
|
+
and encode them using the
code CSS
X,G
0
; Z,G
⊥
. This results in the state
|
G
defined
in Eq. (4) and requires only CO.
State
|
G
is then acted upon by the map (E ◦ T )
⊗n
.The
latter can be implemented using CO and n copies of ρ,as
shownonFig.1. This results in a state
η
1
≡ (E ◦ T )
⊗n
(|GG|) = E
⊗n
(
ˆ
T |GG|
ˆ
T
†
),
where
ˆ
T ≡ T
⊗n
. Next, we apply the Clifford unitary operator
U constructed in Lemma 2. Since U involves only (Z) and
S gates, it commutes with any Z-type error. Hence the state
prepared at this point is
η
2
≡ UηU
†
= E
⊗n
(U
ˆ
T |GG|
ˆ
T
†
U
†
) = E
⊗n
(|A
⊗k
A
⊗k
|),
where we have used Eq. (6). The next step is a non-
destructive eigenvalue measurement for X-type stabilizers
of the code CSS(X,G
0
; Z,G
⊥
), that is, the Pauli operators
X(f
k+1
),...,X(f
m
), where f
k+1
,...,f
m
are the rows of
G
0
. If at least one of the measurement returns the outcome
−1, the subroutine returns FAILED, and the final state is
discarded. If all measured eigenvalues are +1, state η
2
has been
projected onto the code space of the code CSS(X,G
0
; Z,G
⊥
),
and the subroutine is deemed successful (since we do not have
any X-type errors, the syndrome of all Z-type stabilizers is
automatically trivial). This results in a state
η
3
=
0
η
2
0
/P
s
,
where
0
is the projector onto the code space of
CSS(X,G
0
; Z,G
⊥
) and P
s
= Tr (η
2
0
) is the success proba-
FIG. 1. (Color online) Implementation of the T gate using CO
and one copy of the ancillary state
|
A
. If the ancilla is a mixture of
|
A
and Z
|
A
with probabilities 1 − p and p, respectively, the circuit
enacts a noisy version of the T gate, namely, ρ
out
= (1 − p)Tρ
in
T
†
+
pZTρ
in
T
†
Z = E ◦ T (ρ
in
). The above circuit is used n times in the
subroutine of Fig. 2.
FIG. 2. (Color online) The distillation subroutine for the magic
state
|
A
based on a triorthogonal matrix G. The encoder pre-
pares k copies of the state
|
+
encoded by the stabilizer code
CSS(X,G
0
; Z,G
⊥
). Implementation of each T gate consumes one
ancillary
|
A
state, as shown in Fig. 1. If the ancillae
|
A
have error
rate p, each ideal T gate is followed by a Z error with probability
p. The Clifford operator U is constructed in Lemma 2. Note that U
is diagonal in the Z basis and thus commutes with any Z error. The
syndrome s is measured only for X-type stabilizers X(f
a
), where
f
a
are the rows of G
0
. In the case when all stabilizers X(f
a
)have
eigenvalue +1 (trivial syndrome) the decoder is applied. It returns
k copies of state
|
A
with the overall error probability O(p
d
). The
trivial syndrome is observed with probability 1 − O(p).
bility. State η
3
only has a contribution from errors Z(f ) with
f ∈ G
⊥
0
= G
1
⊕ G
⊥
; see Lemma 1 since these are the only
Z-type errors commuting with all X-type stabilizers. Hence
the success probability is
P
s
=
f ∈G
⊥
0
(1 − p)
n−|f |
p
|f |
=
1
|G
0
|
f ∈G
0
(1 − 2p)
|f |
, (14)
where the second equality uses the MacWilliams identity [24].
Any vector f ∈ G
1
⊕ G
⊥
can be written as f = g + x
1
f
1
+
...+ x
k
f
k
, where g ∈ G
⊥
and f
1
,...,f
k
are the rows of G
1
.
Since Z(g) is a stabilizer, we conclude that
Z(f )|
A
⊗k
=Z(x
1
f
1
+···+x
k
f
k
)|A
⊗k
=
Z
x
1
1
···Z
x
k
k
|A
⊗k
.
Here we used the definition of the logical Z-type operators;
see Eq. (11). Hence state η
3
coincides with an encoded k-qubit
mixed state
ρ
out
=
1
P
s
x∈F
k
2
p
out
(x)
|
A
x
A
x
|
, (15)
where |A
x
=|A
x
1
⊗···⊗|A
x
k
and
p
out
(x) =
f ∈G
⊥
+x
1
f
1
+···+x
k
f
k
(1 − p)
n−|f |
p
|f |
. (16)
The last step of the subroutine is to decode CSS(X,G
0
; Z,G
⊥
),
thereby mapping η
3
to ρ
out
.Thek-qubit state ρ
out
is the output
state of the distillation subroutine. The reduced density matrix
describing the ath output qubit can be written as
ρ
out,a
= (1 − q
a
)
|
A
0
A
0
|
+ q
a
|
A
1
A
1
|
,
where q
a
is the output error rate on the ath qubit:
q
a
= 1 −
1
P
s
x : x
a
=0
p
out
(x).
052329-5
![FIG. 3. (Color online) Distillation cost C as a function of the target error rate = 10−δ for a fixed input error rate p = 0.01. The top curve is obtained from [18], and the bottom curve is obtained from the optimization using the triorthogonal matrices G(k). The lines are a guide to the eye.](/figures/fig-3-color-online-distillation-cost-c-as-a-function-of-the-29i4hg5y.webp)