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Algorithms for a Scalable Quantum Computer
Optimal Pulse Sequences, Montgomery Factoring, and Bayesian Inference
Guang Hao Low, Rich Rines, Theodore Yoder, Isaac Chuang
Montgomery Factoring
Introduction
Bayesian Inference
Quantum computers promise speedups to a whole host of
classical algorithms, from exponential advantage in factoring to
a plethora of polynomial advantages in machine learning tasks.
However, as the size of the problem grows, the quantum circuit
required to solve it also grows, and small errors in single gates
add to have a substantial effect on the output. Scaling up a
quantum computer therefore requires either simplifying the
circuits or reducing errors. To that end, we present open-loop
error correction for single-qubit gates in the form of pulse
sequences of optimal length. We also show how to reduce the
circuit for Shor’s algorithm using the Montgomery product from
number theory, bringing the factoring of 35 within experimental
grasp. Finally, we demonstrate how a Bayesian inference task
can be performed with polynomial advantage on a scalable
quantum computer.
Say that we have a probability distribution P (~
x) on n binary
variables. After observing some set of evidence variables E = e how
can we sample from P (Q|e) for a set of query variables? Such
sampling allows us to perform approximate inference, inferring the
most likely values of Q given e by evaluating argmaxQ P (Q|e).
X1
X2
X3
P (x2 |x1 )
P (x3 |x1 )
Say that we wish to perform a single-qubit rotation R [✓] by
= X̂ cos + Ŷ sin but we only
angle ✓ around the axis
have access to a faulty operation M [✓] = R [✓(1 + ✏)] which
over- or under-rotates by a factor ✏ . In the presence of these
amplitude errors, how can we still perform R [✓] accurately?
The answer comes from the non-commutativity of single-qubit
operations which allows us to chain faulty pulses in sequence
such that
L
Y
n+1
U0 [✓] =
M'j [✓0 ] ·M0 [✓] = R0 [✓] + O(✏
). (1)
j=1
|
{z
}
⌘S(~
')
?
E =e
N ⇥ O(nm)
q-sample state [3]: |
O(N n)
Pi
q-sample
|
Pi
) N = O(1/P (e))
X p
=
x1 ,...,xn
Figure 3: (Right) This circuit prepares the qsample | P i based on the structure of the
x1
Bayes net in Fig. 2. Once compiled, the circuit
complexity is O(n2m ). (Below) The quantum
x2
rejection sampling algorithm produces one
sample of P (Q|e) per q-sample prepared. It
does so by enhancing the component of | P i x3
that has the correct evidence |ei using
amplitude amplification. This effectively
x4
speeds up the classical post-selection by a
square-root.
Quantum Rejection Sampling
1⇥
(5)
represents the computational bottleneck of Shor’s algorithm.
Montgomery reduction is a commonly used classical computational
technique for efficient modular exponentiation under a large
modulus, reducing the algorithmic complexity of modular reduction
by division to that of multiplication:
c Binary
modular addition of a constant for shift-and-add (S+A) multiplication is assumed to have the circuit cost fo
constant additions.
Remarkably,
we multiplication
can reclaimcircuit.
the algorithmic advantage by adapting
d The Zalka/Kutin method
is an approximate
to a uniquely quantum mechanical arithmetic model acting in
various multiplication
procedures.multiplier
Gate counts
and depths
are measured in
quantumofFourier
space. The resulting
provides
a unique
P (x1 , . . . , xn )|x1 , . . . , xn i TABLE
(4) I: Circuit characteristics
of multi-qubit gates.
scalable building block for constructing Shor’s circuits:
|0i
U1
U2
|0i
U4
|0i
Q|e i|ei
p
m
O(n2 / P (e))
|
U3
|0i
Amp. Amp.
|
3-Qub
3n
3n
O n2
O n2
0
0
ability
to
implicitly
calculate
U
with
no
computational
overhead.
a For the QFMP circuit costs, some deterministic optimizations were made, such as combining sequential controlled rot
Unfortunately,
this
implicit
calculation
is
not
immediately
reversible,
same control and target qubits
b For n-bit binary arithmetic
mitigating
advantage
of size
Montgomery
reductionaddition
when and
naively
circuits, the
we assume
a circuit
of 6n for non-modular
3n for non-modula
q
⇠
P
(Q|e)
O(N P (e))⇥
adapted to quantum modular exponentiation.
constant
post-select
N ⇥ x ⇠ P (x)
The dependence of L on n that is implicit in this equation is
crucial. We prove that L > n and argue that L = 2n is
achievable. This is in contrast to the next best sequences for
arbitrary ⌘ ✓/2⇡, which scale as L = O(n3.09 ) [1].
|xi ! |a x mod N i,
Figure 2: (Left) A Bayes net is a directed acyclic graph
with n nodes and maximum indegree m showing the
Given: T = xy < N R
conditional dependences of random variables of a
0
1
x) (Eq.(3)). With each node Multiplication
N
:=
N
mod R Depth Qubits
probability distribution P (~
is
Method
Gate Count
Parallelized
stored a table of probabilities. Sampling nodes top to
0 (n)
QFMP, Simplea
10n2 9n
+
2
O
2n
U
:=
T
N
mod
R
x) in time O(nm).
bottom, gives a sample of P (~
2
QFMP,
General
12n
+ 3n + 6
O1(n)
2n + 2
(Below) Classical rejection sampling takes N samples
(T +18n
UN
(mod N )
2 )/R ⌘ T R
from P (~
x) and rejects those with incorrect evidence
Quantum Montgomery, Generalb
+ O (n)
O n2
3n + O (1)
E` 6= e . Since the correct evidence only appearsS+A,
with
2 )/R < 2N
UN
X4
Ripple-Carry Addersc [18] (T
[12]+24n
+ O (n)
O n2
2n + O (1)
probability P (e) , if we desire one sample from the
3
2
2
S+A,
QFT
Adders
[4]
[2]
2n
+
O
n
O
n
2n + 2
P
(Q|e)
posterior
we
must
set
and
N
=
O(1/P
(e))
P (x4 |x1 , x3 )
The
classical
advantage
of
Montgomery
reduction
results
from
the
d
2
take O(nm/P (e)) time.
Zalka/Kutin [18] [6]
2n + O (n log2 n) O (n)
3n + O (log2 n)
Classical Rejection Sampling
Optimal Pulse Sequences
2k
(3)
P (~x) = P (x1 )P (x2 |x1 )P (x3 |x1 )P (x4 |x1 , x3 )
P (x1 )
Repeated application of the controlled modular product operation,
1⇥ q ⇠ P (Q|e)
Pi
Multiplication Method
Fourier Montgomery
Shift-and-Add [4]
Fourier Shift-and-Add, Draper [5]
Fourier Shift-and-Add, Beauregard [6]
Depth
O (n)
O n2
O n2
O n3
Gates
12n2 + 3n + 6
24n2 + O (n)
2n3 + O n2
2n3 + O n2
Qubits
2n + 2
2n + O (1)
3n
2n + 2
Small circuit width and
small CONCLUSIONS
lower-order terms in the circuit depth
VIII.
and gate count make the Fourier Montgomery method immediately
applicableperforms
to foreseeable
factoring
experiments,
such
as linear
N = 35.
The overall QFMP operation
Montgomery
modular
multiplication
with
circuit depth
width of 2n + 2 qubits. For the uncontrolled operator, no three-qubit gates are required; however 3n wi
for the controllable version
O(n2m )
|p3 ! in the ci
|0! required for modular exponentiation. Further, small constant terms
depth, a low quadratic gate count, and the ability to perform Montgomeryization within classical pre
|p2 ! N . This
|0! advantage of the operator to exist both asymptotically and for small
allows the computational
+ N2
QF T †
N
introduces opportunities
|p1 !
|0! for quantum factoring+ circuits.
2
N
We notice that, in practice, the QFMP+operation
will likely be more efficient than the version presente
2
N
+
|0!
0 ! approxim
as all of our operations are wrapped2 up in Fourier rotations, we note that for larger N we |pcan
×y
angles to greatly reduce
output [1]. We also
notice tha
N
|a
!
|x3 !circuit size with little change •in the− experimental
3
2
N
gates required to make the QFMP controllable are acting on a subset
− 2 of the possible Hilbert space of thre
j
N
~='
~R ('
~= '
~ R ) result in Im L (~
The symmetries '
') = 0 for
j
•
•
−
|x
!
2 ! state. Th
instance, one of the three
in the|a|0i
2 qubits input to the initial Fredkin gates is deterministically
2
÷y
even (odd) and the resulting sequences are called initialed PD the
(AP).
opportunity for hardware-specific
optimization,
including potentially
breaking the operation
into tw
•
•
|a1 !
|x1 !
All such sequences have L = 2n.
These optimizations are very specific to the particular implementation of the QFMP operation, and the
•
•
|a0 !
|x0 !
thoroughly explored here.
SolvingMethods
for Sequences
We introduce three techniques for solving (2):
•  Analytical: The substitution tk = tan( k /2) converts (2) into a
system of polynomial equations, which can be solved by Groebner
[1] Adriano Barenco, Artur Ekert, Kalle-Antti Suominen, and Päivi Törmä. Approximate quantum fourier
bases to yield AP1, 2, 3 and PD2, 4.
@ j
Phys. Rev. A, 54:139–146, Jul 1996.
•  Perturbative: Given a solution at 0 and nonzero Jacobian @'k Ldecoherence.
0
[2] Stephane Beauregard. Circuit for shor’s algorithm using 2n+3 qubits. arXiv, 2003. http://arxiv.o
solutions '( ) for ⇡ 0 can be obtained. For a certain AP
2
ph/0205095v3.
1.  Ken Brown, Aram Harrow, Isaac Chuang, Phys. Rev. A 70,
Figure 1: We compare the transition probability p = |h0|U0 [⇡]|1i| effected by
ToP
sequence ToP, 'k (0) = ⇡ and we prove a nonzero Jacobian[3]
at C.
0. H. Bennett. Logical 052318
reversibility
of computation. IBM J. Res. Dev., 17:525 – 532, Nov 1973.
pulse sequences with amplitude error ✏ in the cases of length L = 9 (blue) and
(2004).
[4] Thomas
G. Draper. Addition
on
a
quantum
computer.
arXiv,
2000.
http://arxiv.org/abs/quant-ph/00080
also L = 25 (red). Included are BB4,12 of this work, Vitanov’s V8,24 ,
•  Numerical: We use Mathematica [2] to find roots to equation (2)
up
2.  Wolfram Research Inc., Mathematica, Version 9.0,
2
Shaka’s 2, Cho’s C9 , Husain’s F2 , and Tycko’s S2 .
[5] Lieven M. K. Vandersypen et. al. Experimental realization of shors quantum factoring algorithm using nu
to n = 12.
IL (2012).
resonance. arXiv, 2001.Champaign,
http://arxiv.org/abs/quant-ph/0112176.
3.  Dorit
Aharonov,
Amnon Ta-Shma,
STOC
ACM,
pp. 20-29
Name
Length Notes
Our sequences are obtained algebraicallya pulse
by expanding
We of L faulty pulses M'j [✓j ].
[6] Samuel A. Kutin. Shor’s
algorithm
on a nearest-neighbor
machine.
arXiv,’03,
2006.
http://arxiv.org/abs/quantsequence S (1).
consisting
(2003).
[7] Igor L. Markov and Mehdi
Saeedi. Constant optimized quantum circuits for modular multiplication and e
Denote simplify
by '
~ the the
vector
of phase angles ('1 , '2 , . . . , 'L ),
SCROFULOUS
3
n = 1, non-uniform ✓j [9]
make the assumption ✓0 = 2⇡ to dramatically
result.
2
n
which
are
our
free
parameters.
Leaving
each
amplitude
✓
arXiv, 2012. http://arxiv.org/abs/1202.6614.
4.  Christof Zalka, arXiv:quant-ph/9806084 (1998)
j
Pn, Bn
O(e ) Closed-form [5]
~ are
The resulting n complex equations to be solved for '
as a free parameter (e.g. SCROFULOUS [9]) may help
[8] Alfred J. Menezes, Scott
A. Vanstone,
and arXiv:quant-ph/0008033
Paul C. Van Oorschot. Handbook
of Applied Cryptography. CR
5.  Thomas
Draper,
(2000)
3 n  30, numerical [5]
SKn
O(n )
j
j
reduce
sequence
length,
but
we
find
that
a
fixed
value
Boca Raton, FL, USA,
1st
edition, 1996.
n
>
30,
conjectured
(2)
(~
'
)
=
f
(
),
j
=
1,
.
.
.
,
n
6. 
Stephane
Beauregard, arXiv:quant-ph/0205095 (2003)
L
L
✓j = ✓0 leads to the most compelling results.
[9] P. L. Montgomery. Modular Multiplication without Trial Division. Mathematics of Computation, 44(170):5
n  3(4), closed-form
where
◆✓ a target
◆ rotation R'0 [✓T ], or
The
!
j goal is to✓implement
APn (PDn)
2n n  12, analytic continuation
Bibliography
j
')
L (~
=
X
{hk }
0
exp
i
j
X
k=1
k
( 1) 'hk
X
L T
j
equivalently Rk0 [✓T
T ] with the replacement 'j ! 'j
(
1)
.
, fL ( ) =
including the correct
global
phase, with a small
k
j
k
n+1
'0 ,
n > 12, conjectured
error
ToPn
2n arbitrary n, perturbative
O(✏k=0). Specifically, we seek to eliminate the error of
I: Summary
of existing
pulse sequences
and our
optimalon
the faulty pulse M0 [✓T ] with a pulse sequence satisfying Table
TABLE
I: Comparison
of known
pulse sequences
operating
The primed sum enforces 1  h1 < · · · < hj  L and T = ( + L)/2 .
S = R0 [ ✏✓T ] + O(✏n+1 )
so that UT = S · M0 [✓T ] = R0 [✓T ] + O(✏
(1)
n+1
) imple-
sequences
PD,states
and ToP.
arbitrary AP,
initial
that suppress systematic amplitude errors to order n for arbitrary target rotation angles. Arbitrary
accuracy generalizations are known or conjectured for all with
the exception of SCROFULOUS. The sequences APn, PDn,
The authors acknowledge the support of the NSF CUA, NSF
iQuISE IGERT, IARPA QCS ORAQL, and NSF CCF-RQCC
projects.
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