Fast Quantum Algorithms - Chalmers school... · 2019. 11. 15. · Figure 5.3: An overview circuit...

12
Chapter 5 Fast Quantum Algorithms For this part of the notes, we follow Ref. [Nielsen and Chuang, 2011]. 5.1 The Quantum Fourier Transform You should all have seen the discrete Fourier transform (DFT) where a set of N complex numbers {x 0 ,...,x N-1 } are Fourier transformed into N new complex numbers {y 0 ,...,y N-1 } according to y k = 1 N N-1 m=0 e i 2πmk N x m . The Fourier transformation is extremely useful, e.g. to detect periods in a signal where {x 0 ,...,x N-1 } could be the amplitude of some signal as a function of discretized time. The Fourier transformed signal {y 0 ,...,y N-1 } then describes the frequency content. Solving the Schrödinger equation on a lattice, DFT is what takes you between real space and momentum (k) space. The quantum Fourier transform (QFT) is a unitary n-qubit operation, transforming the initial N =2 n basis states {|0,..., |N - 1} into a new basis in a way which looks mathematically identical to the DFT, |j 1 N N-1 k=0 e i 2πjk N |k. (5.1) The action on an arbitrary state is N-1 j=0 x j |j N-1 k=0 y k |k, where the amplitudes y k are the DFT transforms of the amplitudes x m . One may easily verify that the new states are normalized and form an orthogonal set, and thus that the QFT is a unitary transform. The QFT can be used to find periods and also to extract eigenvalues of unitary operators to a high precision. But before discussing these issues in more detail, let’s see if we can find an effective implementation of the QFT. Remember that there are operators that need exponentially many single- and two-qubit gates for implementation, so what about the QFT? 5.1.1 Another definition We’ll now rewrite the definition of the QFT in a way that is more transparent for constructing a circuit. First we need a simple way to number the basis states. We simply number the n-qubit state |j using the 25

Transcript of Fast Quantum Algorithms - Chalmers school... · 2019. 11. 15. · Figure 5.3: An overview circuit...

Page 1: Fast Quantum Algorithms - Chalmers school... · 2019. 11. 15. · Figure 5.3: An overview circuit gure of the phase estimation circuit. (From Nielsen & Chuang), Fig. 5.3. probability

Chapter 5

Fast Quantum Algorithms

For this part of the notes, we follow Ref. [Nielsen and Chuang, 2011].

5.1 The Quantum Fourier Transform

You should all have seen the discrete Fourier transform (DFT) where a set ofN complex numbers {x0, . . . , xN−1}are Fourier transformed into N new complex numbers {y0, . . . , yN−1} according to

yk =1√N

N−1∑

m=0

ei2πmkN xm.

The Fourier transformation is extremely useful, e.g. to detect periods in a signal where {x0, . . . , xN−1}could be the amplitude of some signal as a function of discretized time. The Fourier transformed signal{y0, . . . , yN−1} then describes the frequency content. Solving the Schrödinger equation on a lattice, DFT iswhat takes you between real space and momentum (k) space.

The quantum Fourier transform (QFT) is a unitary n-qubit operation, transforming the initial N = 2n

basis states {|0〉, . . . , |N − 1〉} into a new basis in a way which looks mathematically identical to the DFT,

|j〉 → 1√N

N−1∑

k=0

ei2πjkN |k〉. (5.1)

The action on an arbitrary state isN−1∑

j=0

xj |j〉 →N−1∑

k=0

yk|k〉,

where the amplitudes yk are the DFT transforms of the amplitudes xm. One may easily verify that the newstates are normalized and form an orthogonal set, and thus that the QFT is a unitary transform.

The QFT can be used to find periods and also to extract eigenvalues of unitary operators to a highprecision. But before discussing these issues in more detail, let’s see if we can find an effective implementationof the QFT. Remember that there are operators that need exponentially many single- and two-qubit gatesfor implementation, so what about the QFT?

5.1.1 Another definition

We’ll now rewrite the definition of the QFT in a way that is more transparent for constructing a circuit.First we need a simple way to number the basis states. We simply number the n-qubit state |j〉 using the

25

Page 2: Fast Quantum Algorithms - Chalmers school... · 2019. 11. 15. · Figure 5.3: An overview circuit gure of the phase estimation circuit. (From Nielsen & Chuang), Fig. 5.3. probability

binary n-bit representation of j = j12n−1 + j22n−2 + · · ·+ jn−121 + jn20. E.g. in the 4 qubit case the state|5〉 = |0101〉 = |01〉|12〉|03〉|14〉. To proceed with the construction we also need to remember the definition ofbinary fractions

0.j1j2j3 . . . jn = j1/2 + j2/22 + j3/2

3 · · ·+ jn/2n,

e.g. 0.101 = 0.5 + 0.125 = 0.625, and more generally

0.jljl+1 . . . jm = jl/2 + jl+1/22 + · · ·+ jm/2

m−l+1.

Using this notation we can write the QFT in Eq. (5.1) as

|j〉 = |j1, j2, . . . , jn〉 →(|0〉+ ei2π0.jn |1〉

) (|0〉+ ei2π0.jn−1jn |1〉

). . .(|0〉+ ei2π0.j1j2...jn |1〉

)

2n/2. (5.2)

The algebraic manipulations connecting the two expressions are straightforward, but need some afterthought.Observing that

k

2n=k12n−1

2n+ · · ·+ kn20

2n= k12−1 + · · ·+ kn2−n =

n∑

l=0

kl2−l, (5.3)

from Eq.(5.1) we have:

|j〉 → 1√N

N−1∑

k=0

ei2πjkN |k〉 (5.4)

=1√N

1∑

k1=0

· · ·1∑

kn=0

ei2πj∑nl=1 kl2

−l |k1 . . . kn〉

=1√N

1∑

k1=0

· · ·1∑

kn=0

n⊗

l=1

ei2πjkl2−l |kl〉

=1√N

n⊗

l=1

1∑

kl=0

ei2πjkl2−l |kl〉

=1√N

n⊗

l=1

(|0〉+ ei2πj2−l |1〉)

=1√N

(|0〉+ ei2πj2−1 |1〉)(|0〉+ ei2πj2

−2 |1〉) . . . (|0〉+ ei2πj2−n |1〉)

=1√N

(|0〉+ ei2π0.jn |1〉)(|0〉+ ei2π0.jn−10.jn |1〉) . . . (|0〉+ ei2π0.j1...jn |1〉)

where in the last step we have used that

j2−n = j1/2 + j2/4 + . . . jn/2n = 0.j1 . . . jn (5.5)

...

j2−1 = j12n−2 + j22n−3 + . . . jn/2 = j12n−2 + j22n−3 + · · ·+ 0.jn

and that the integer part of j · 2l disappears in the exponent since it is multiplied by 2π.

5.1.2 An efficient implementation

Using the form of the QFT in Eq. (5.2) it is straightforward to implement the desired transformation witha quantum a circuit. We realize that we have to implement conditional phase shifts on each qubit, therefore

26

Page 3: Fast Quantum Algorithms - Chalmers school... · 2019. 11. 15. · Figure 5.3: An overview circuit gure of the phase estimation circuit. (From Nielsen & Chuang), Fig. 5.3. probability

we define the single qubit operator

Rk =

[1 0

0 ei2π/2k

].

Now let’s see what happens an input state |j1, j2, . . . , jn〉 when it passes through the circuit in Fig. 5.1. Thefirst Hadamard gate produces the state (|0〉+ |1〉)/

√2 if j1 = 0 and (|0〉 − |1〉)/

√2 if j1 = 1, i.e.

1√2

(|0〉+ ei2π0.j1 |1〉

)|j2, . . . , jn〉,

since ei2π0.j1 = ei2πj1/2 = −1 for j1 = 1 and +1 otherwise. The controlled-R2 gate rotates the component|1〉 of the first qubit by ei2π/2

2

if j2 = 1, i.e. it applies the phase ei2πj2/22

. Therefore, it produces the state

1√2

(|0〉+ ei2πj2/2

2

ei2πj1/2|1〉)|j2, . . . , jn〉 =

1√2

(|0〉+ ei2π0.j1j2 |1〉

)|j2, . . . , jn〉.

After all the controlled-Rk operations on the first qubit, the state is

1√2

(|0〉+ ei2π0.j1j2...jn |1〉

)|j2, . . . , jn〉.

The Hadamard on the second qubit produces

1√22

(|0〉+ ei2π0.j1j2...jn |1〉

) (|0〉+ ei2π0.j2 |1〉

)|j3, . . . , jn〉,

and the controlled R2 to Rn−1 gates yield the state

1√22

(|0〉+ ei2π0.j1j2...jn |1〉

) (|0〉+ ei2π0.j2...jn |1〉

)|j3, . . . , jn〉.

We continue in this fashion for all qubits, obtaining the final state

1√2n

(|0〉+ ei2π0.j1j2...jn |1〉

) (|0〉+ ei2π0.j2...jn |1〉

)...(|0〉+ ei2π0.jn |1〉

).

We now need to reverse the order of the qubits, which can be achieved using a series of SWAP gates.The number of gates needed are n on the first qubit plus n − 1 on the second and so on, adding up ton(n + 1)/2 = O(n2) gates. Then we need on the order of n SWAP gates, not changing the scaling. Thuswe can implement the QFT for n qubits using on the order of O(n2) gates. The best classical algorithm(FFT) needs O(n2n) gates, indicating why the QFT could be used for speedup. This does not translate in animmediate speed-up for computing classical FFT, because we cannot access the amplitudes when measuringthe Fourier-transformed quantum state, and we don’t even know how to efficiently prepare the input stateto be transformed. However, in the next section we’ll see one problem where the quantum Fourier transformis useful.

5.2 Phase estimation

The aim of this algorithm is to estimate the angle in the eigenvalue ei2πϕ corresponding to an eigenvector|u〉 of a unitary operator U . The vector |u〉 is given, as well as a circuit (black box, oracle) effectivelyimplementing controlled-Un operations. A circuit solving a first stage of this problem is shown in Fig. 5.2and an overview circuit showing the whole algorithm is given in Fig. 5.3. Two qubit registers are needed,the first has t qubits which are initialized to zero. The number of qubits t is determined by the required

27

Page 4: Fast Quantum Algorithms - Chalmers school... · 2019. 11. 15. · Figure 5.3: An overview circuit gure of the phase estimation circuit. (From Nielsen & Chuang), Fig. 5.3. probability

Figure 5.1: An efficient circuit to perform the quantum Fourier transform. (From Nielsen & Chuang), Fig.5.1.

Figure 5.2: A circuit performing the first step of the phase estimation algorithm. (From Nielsen & Chuang),Fig. 5.2.

accuracy in the estimate of ϕ. The second register is large enough to represent the eigenvector |u〉, and it isalso initialized to |u〉 and remains in this state throughout the computation. The initial set of Hadamard gatesputs all qubits of register 1 in an equal superposition of |0〉 and |1〉. If the k-th control qubit is in the state

|1〉 a unitary operation U2k will be performed on the second register, picking up a phase(ei2πϕ

)2k= ei2πϕ2k .

The first step (k = 0) gives for example:

CU1√2

(|0〉+ |1〉) |u〉 =1√2

(|0〉+ ei2πϕ|1〉

)|u〉. (5.6)

28

Page 5: Fast Quantum Algorithms - Chalmers school... · 2019. 11. 15. · Figure 5.3: An overview circuit gure of the phase estimation circuit. (From Nielsen & Chuang), Fig. 5.3. probability

The final state of the first register in this first step is

1√2t

(|0〉+ ei2π2t−1ϕ|1〉

)(|0〉+ ei2π2t−2ϕ|1〉

). . .(|0〉+ ei2π21ϕ|1〉

)(|0〉+ ei2π20ϕ|1〉

)=

1√2t

2t−1∑

k=0

ei2πkϕ|k〉.

(5.7)By comparison with (5.4) we see that this state is nothing else than the Fourier transform of the state|2tϕ〉 = |ϕ1ϕ2 . . . ϕt〉, where in the last step we have assumed that the phase ϕ has an exact representationin t bits as ϕ = 0.ϕ1ϕ2 . . . ϕt (with a slight abuse of notation). The final step is hence to make an Inversequantum Fourier transform of the first register. This allows recovering the latter state. Register 1 is thenread out and ϕ is recovered. If the phase is not an exact binary fraction in t qubits there will be some finite

Figure 5.3: An overview circuit figure of the phase estimation circuit. (From Nielsen & Chuang), Fig. 5.3.

probability of reading out some other state "close" to the best approximation. A careful analysis gives thatif we want n bits precision, with a failure probability less than ε, we need a register of size

t = n+

⌈log

(2 +

1

)⌉.

Phase estimation is interesting in its own right and may be used in quantum simulations. We will see howit enters Shor’s algorithm for factoring.

5.3 Factoring - Shor’s algorithm

We now know how to efficiently determine the phase of an eigenvalue to a unitary operator. In this sectionwe’ll see how this enables us to efficiently solve a number theoretical problem which is considered hard onclassical computers: order finding. Finally we show how factoring can be reduced to order finding.

5.3.1 Modular arithmetics

Order finding is defined in modular arithmetics. Modular arithmetics is based on the fact that, given anytwo positive integers x and n, x can uniquely be written as

x = k · n+ r,

where k is a non-negative integer and 0 ≤ r < n is the reminder

x = r mod n,

as an example2 = 5 = 8 = 11 mod 3.

29

Page 6: Fast Quantum Algorithms - Chalmers school... · 2019. 11. 15. · Figure 5.3: An overview circuit gure of the phase estimation circuit. (From Nielsen & Chuang), Fig. 5.3. probability

The greatest common divisor gcd(a, b) of two integers a and b is the largest integer dividing both a andb. If gcd(a, b) = 1 then a and b are called co-prime.

Multiplicative inverse

Now let’s look at modular multiplication by looking at the series

mk = k · a mod n, 0 < k < n.

As an example, take a = 6 and n = 15 with gcd(a, n) = 3 giving

mk = {6, 12, 3, 9, 0, 6, 12, 3, 9, 0, 6, 12, 3, 9},

showing that the equation x · 6 = y mod 15 has no solution for y ∈ {1, 2, 4, 5, 7, 8, 10, 11}. Note that inparticular it has no solution for y = 1. Then take a = 7 and n = 15 which are co-prime giving

mk = {7, 14, 6, 13, 5, 12, 4, 11, 3, 10, 2, 9, 1, 8},

showing that the equation x · 7 = y mod 15 may be solved for all y.The multiplicative inverse a−1 of an integer a modulus n is another integer which fulfils

a−1 · a = 1 mod n,

and it exists if and only if a and n are co-prime. If the inverse exists we can solve the equation

x · a = c mod n

for all integers a, b and c throughx = c · a−1 mod n.

Another way of formulating this is the following: all the integers between 1 and (n − 1) appear onceand only once in {mk} if and only if a and n are co-prime. If not we can write a = x · gcd(a, n) andn = y · gcd(a, n), where 0 < y < n. So for k = y we get

my = y · x · gcd(a, n) mod (y · gcd(a, n)) = 0,

and then the series {mk} will just repeat from the start.

5.3.2 Order finding

Consider the equationxr = 1 mod N,

which has solutions for the integers x and N being co-prime, and x < N . The smallest positive integerr solving the equation is called the order of x modulo N . One straightforward method to calculate r isto evaluate the series mk = xk mod N for 0 < k < N , then it’s clear that the series {mk} is periodicwith period r since xr+a = xr · xa = xa mod N . In other words, the order is the period of the modularexponentiation function mk = xk mod N . As an example let’s take x = 5 and N = 21 giving

mk = {5, 4, 20, 16, 17, 1, 5, 4, 20, 16, 17, 1, 5, 4, 20, 16, 17, 1, 5, 4},

and we have the order r = 6. There is no classical algorithm for finding r which scales polynomially in thenumber of bits L needed to represent the input, i.e. the integers x and N .

30

Page 7: Fast Quantum Algorithms - Chalmers school... · 2019. 11. 15. · Figure 5.3: An overview circuit gure of the phase estimation circuit. (From Nielsen & Chuang), Fig. 5.3. probability

5.3.3 Factoring as order finding

Factoring can be reduced to order finding as follows. Suppose we want to factor N = pq. Consider theperiod r of the function of k defined as xk mod N for some x. x is chosen such that r is even and thatxr/2 6= N −1 mod N . Then r allows to find p and q as follows. Define y = xr/2. Then y2 = xr = 1 mod N

by definition of period r. Therefore we have

y2 − 1 = (y − 1)(y + 1) = 0 mod N.

Therefore(y − 1)(y + 1) is multiple of N , i.e. it contains the two factors pq in his decomposition. Howeverneither (y − 1) nor (y + 1) are multiple N :

(y − 1) 6= 0 mod N, because otherwise the period would be smaller than r. (5.8)

(y + 1) 6= 0 mod N, by construction. (5.9)

Therefore (y − 1) and (y + 1) must slip the two factors q and p appearing in the decomposition of N . Sayfor instance:

(y − 1) = lp; (y + 1) = l′q.

We finally obtain therefore

p = gcd(N, xr/2 − 1); q = gcd(N, xr/2 + 1).

In other words, determining the order r of the modular exponentiation function xk mod N yields thedetermination of the two factors p and q such that N = pq. For n having L bits, this common factor can befound using Euclid’s algorithm in O(L3) steps. For uniformly chosen x one may calculate a lower bound forthe probability of r being even and that y = xr/2 is non-trivial,

p(r is even and xr/2 6= −1 mod N) ≥ 1− 1

2m,

where m is the number of different prime-factors in N , i.e. m ≥ 1.In the following, we are going to derived an efficient quantum algorithm for order finding.

5.3.4 A quantum algorithm for order finding

Given an integer x of which we want to find the order mod N , consider the L-qubit unitary operation

U |y〉 ≡{ |x · y mod N〉, 0 ≤ y ≤ N − 1

|y〉, N ≤ y ≤ 2L − 1.

The unitarity follows since it basically permutes the basis states and y has a multiplicative inverse modulusN since y and N are co-prime. The states

|us〉 =1√r

r−1∑

k=0

exp

{[−i2πskr

]}|xk mod N〉,

defined for integers 0 ≤ s ≤ r − 1 are eigenstates of U , since

U |us〉 =1√r

r−1∑

k=0

exp

{[−i2πskr

]}|xk+1 mod N〉 =

1√r

r∑

k=1

exp

{[−i2πs(k − 1)

r

]}|xk mod N〉 = exp

{[i2πs

r

]}|us〉,

31

Page 8: Fast Quantum Algorithms - Chalmers school... · 2019. 11. 15. · Figure 5.3: An overview circuit gure of the phase estimation circuit. (From Nielsen & Chuang), Fig. 5.3. probability

since xr = x0 mod N , and exp{[−i2πs(r−1)

r

]}= exp

{[i2πsr

]}. Using the phase estimation algorithm we can

now efficiently determine s/r with high accuracy. One requirement is that we can implement the operatorsU2k efficiently, which can be done using a procedure known as modular exponentiation (see Box 5.2. onpage 228 in N & C), needing O(L3) gates. Furthermore we need to produce one or more of the eigenstates|us〉 which is done by noting

1√r

r−1∑

s=0

|us〉 =1

r

r−1∑

s=0

r−1∑

k=0

exp

{[−i2πskr

]}|xk mod N〉 = |1〉.

To have enough accuracy in the phase estimation we should use t = 2L + 1 +⌈log(2 + 1

)⌉qubits in the

first register and prepare the second register in the |1〉 state. We’ll then get the phase ϕ = s/r, for a random0 ≤ s < r, with 2L+ 1 bits precision, with a probability of at least (1− ε). Knowing that the phase ϕ = s/r

is a rational number, where s and r are integers, not larger than L bits, we can classically determine s andr. The appropriate algorithm is called the continued fraction expansion and needs O(L3) gates.

5.3.5 Performance

The algorithm fails if s = 0, and also if s and r have common factors so that they cannot be extractedfrom s/r. The probability of failure can be shown to be small and one need only to repeat the procedure apolynomial (in L) number of times to obtain r with high probability. The number of gates needed are O(L)

for the initial Hadamards, inverse Fourier transform needs O(L2) gates, implementing U2k through modularexponentiation requires O(L3) gates, and the classical continued fraction algorithm needs O(L3) (classical)gates. If we need to repeat an O(L) number of times the overall scaling would be O(L4), but being moreclever there are ways to guarantee success in a constant number of attempts, giving the scaling O(L3) gates.

The algorithm

Find a factor of the composite L-bit integer N .

• 1. If N is even return the factor 2.

• 2. Determine whether N = ab, i.e. is if there is only one prime-factor. This can be done with O(L3)

operations. If so return the factor a.

• 3. Randomly choose 1 < x < (N − 1), and check whether x and N are co-prime (O(L3) operations).If not co-prime return the factor gcd(x,N).

• 4. Find the order r of x modulo N , which can be done using O(L3) quantum gates (quantum subrou-tine!!)

• 5. If r is even and xr/2 6= −1 mod N then compute gcd(xr/2 + 1, N) and gcd(xr/2 − 1, N) and checkif one is a non-trivial factor of N . Return this factor. If r is odd, or xr/2 = −1 mod N the algorithmfails.

The algorithm will succeed with a probability larger than 3/4.

32

Page 9: Fast Quantum Algorithms - Chalmers school... · 2019. 11. 15. · Figure 5.3: An overview circuit gure of the phase estimation circuit. (From Nielsen & Chuang), Fig. 5.3. probability

Bibliography

[Aaronson, a] Aaronson, S. The complexity zoo.

[Aaronson, b] Aaronson, S. Postbqp postscripts: A confession of mathematical errors.

[Aaronson, 2005] Aaronson, S. (2005). Quantum computing, postselection, and probabilistic polynomial-time. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences,461:3473.

[Aaronson, 2018] Aaronson, S. (2018). Lecture Notes for Intro to Quantum Information Science.

[Aaronson and Arkhipov, 2013] Aaronson, S. and Arkhipov, A. (2013). The computational complexity oflinear optics. Theory of Computing, 9:143.

[Albash and Lidar, 2018] Albash, T. and Lidar, D. A. (2018). Adiabatic quantum computation. Reviews ofModern Physics, 90(1):015002.

[Arute et al., 2019] Arute, F., Arya, K., Babbush, R., Bacon, D., Bardin, J. C., Barends, R., Biswas, R.,Boixo, S., Brandao, F. G. S. L., Buell, D. A., Burkett, B., Chen, Y., Chen, Z., Chiaro, B., Collins, R.,Courtney, W., Dunsworth, A., Farhi, E., Foxen, B., Fowler, A., Gidney, C., Giustina, M., Graff, R.,Guerin, K., Habegger, S., Harrigan, M. P., Hartmann, M. J., Ho, A., Hoffmann, M., Huang, T., Humble,T. S., Isakov, S. V., Jeffrey, E., Jiang, Z., Kafri, D., Kechedzhi, K., Kelly, J., Klimov, P. V., Knysh, S.,Korotkov, A., Kostritsa, F., Landhuis, D., Lindmark, M., Lucero, E., Lyakh, D., Mandrà, S., McClean,J. R., McEwen, M., Megrant, A., Mi, X., Michielsen, K., Mohseni, M., Mutus, J., Naaman, O., Neeley,M., Neill, C., Niu, M. Y., Ostby, E., Petukhov, A., Platt, J. C., Quintana, C., Rieffel, E. G., Roushan, P.,Rubin, N. C., Sank, D., Satzinger, K. J., Smelyanskiy, V., Sung, K. J., Trevithick, M. D., Vainsencher,A., Villalonga, B., White, T., Yao, Z. J., Yeh, P., Zalcman, A., Neven, H., and Martinis, J. M. (2019).Quantum supremacy using a programmable superconducting processor. Nature, 574(7779):505–510.

[Bartlett et al., 2002] Bartlett, S. D., Sanders, B. C., Braunstein, S. L., and Nemoto, K. (2002). Efficientclassical simulation of continuous variable quantum information processes. Phys. Rev. Lett., 88:9.

[Bartolo et al., 2016] Bartolo, N., Minganti, F., Casteels, W., and Ciuti, C. (2016). Exact steady state ofa kerr resonator with one-and two-photon driving and dissipation: Controllable wigner-function multi-modality and dissipative phase transitions. Physical Review A, 94(3):033841.

[Bouland et al., 2018] Bouland, A., Fefferman, B., Nirkhe, C., and Vazirani, U. (2018). Quantum supremacyand the complexity of random circuit sampling. arXiv preprint arXiv:1803.04402.

[Bremner et al., 2010] Bremner, M. J., Josza, R., and Shepherd, D. (2010). Classical simulation of commut-ing quantum computations implies collapse of the polynomial hierarchy. Proc. R. Soc. A, 459:459.

122

Page 10: Fast Quantum Algorithms - Chalmers school... · 2019. 11. 15. · Figure 5.3: An overview circuit gure of the phase estimation circuit. (From Nielsen & Chuang), Fig. 5.3. probability

[Bremner et al., 2015] Bremner, M. J., Montanaro, A., and Shepherd, D. (2015). Average-case complexityversus approximate simulation of commuting quantum computations. arXiv:1504.07999.

[Briegel et al., 2009] Briegel, H. J., Browne, D. E., Dür, W., Raussendorf, R., and Van den Nest, M. (2009).Measurement-based quantum computation. Nat. Phys., 5:19.

[Campagne-Ibarcq et al., 2019] Campagne-Ibarcq, P., Eickbusch, A., Touzard, S., Zalys-Geller, E., Frattini,N., Sivak, V., Reinhold, P., Puri, S., Shankar, S., Schoelkopf, R., et al. (2019). A stabilized logical quantumbit encoded in grid states of a superconducting cavity. arXiv preprint arXiv:1907.12487.

[Chabaud et al., 2017] Chabaud, U., Douce, T., Markham, D., van Loock, P., Kashefi, E., and Ferrini, G.(2017). Continuous-variable sampling from photon-added or photon-subtracted squeezed states. PhysicalReview A, 96:062307.

[Chakhmakhchyan and Cerf, 2017] Chakhmakhchyan, L. and Cerf, N. J. (2017). Boson sampling with gaus-sian measurements. Physical Review A, 96(3):032326.

[Choi, 2008] Choi, V. (2008). Minor-embedding in adiabatic quantum computation: I. the parameter settingproblem. Quantum Information Processing, 7(5):193–209.

[Choi, 2011] Choi, V. (2011). Minor-embedding in adiabatic quantum computation: Ii. minor-universalgraph design. Quantum Information Processing, 10(3):343–353.

[Douce et al., 2017] Douce, T., Markham, D., Kashefi, E., Diamanti, E., Coudreau, T., Milman, P., vanLoock, P., and Ferrini, G. (2017). Continuous-variable instantaneous quantum computing is hard tosample. Phys. Rev. Lett., 118:070503.

[Douce et al., 2019] Douce, T., Markham, D., Kashefi, E., van Loock, P., and Ferrini, G. (2019). Probabilisticfault-tolerant universal quantum computation and sampling problems in continuous variables. PhysicalReview A, 99:012344.

[Farhi et al., 2001] Farhi, E., Goldstone, J., Gutmann, S., Lapan, J., Lundgren, A., and Preda, D. (2001).A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem.Science, 292:472.

[Flühmann et al., 2018] Flühmann, C., Negnevitsky, V., Marinelli, M., and Home, J. P. (2018). Sequentialmodular position and momentum measurements of a trapped ion mechanical oscillator. Phys. Rev. X,8:021001.

[Gerry et al., 2005] Gerry, C., Knight, P., and Knight, P. L. (2005). Introductory quantum optics. Cambridgeuniversity press.

[Glancy and Knill, 2006] Glancy, S. and Knill, E. (2006). Error analysis for encoding a qubit in an oscillator.Phys. Rev. A, 73:012325.

[Glauber, 1963] Glauber, R. J. (1963). Coherent and incoherent states of the radiation field. Physical Review,131(6):2766.

[Gottesman et al., 2001] Gottesman, D., Kitaev, A., and Preskill, J. (2001). Encoding a qubit in an oscillator.Phys. Rev. A, 64:012310.

[Gu et al., 2009] Gu, M., Weedbrook, C., Menicucci, N. C., Ralph, T. C., and van Loock, P. (2009). Quantumcomputing with continuous-variable clusters. Phys. Rev. A, 79:062318.

123

Page 11: Fast Quantum Algorithms - Chalmers school... · 2019. 11. 15. · Figure 5.3: An overview circuit gure of the phase estimation circuit. (From Nielsen & Chuang), Fig. 5.3. probability

[Hamilton et al., 2017] Hamilton, C. S., Kruse, R., Sansoni, L., Barkhofen, S., Silberhorn, C., and Jex, I.(2017). Gaussian boson sampling. Physical review letters, 119(17):170501.

[Harrow and Montanaro, 2017] Harrow, A. W. and Montanaro, A. (2017). Quantum computationalsupremacy. Nature, 549(7671):203.

[Horodecki et al., 2006] Horodecki, P., Bruß, D., and Leuchs, G. (2006). Lectures on quantum information.

[Kirchmair et al., 2013] Kirchmair, G., Vlastakis, B., Leghtas, Z., Nigg, S. E., Paik, H., Ginossar, E., Mir-rahimi, M., Frunzio, L., Girvin, S. M., and Schoelkopf, R. J. (2013). Observation of quantum state collapseand revival due to the single-photon kerr effect. Nature, 495(7440):205.

[Kockum and Nori, 2019] Kockum, A. F. and Nori, F. (2019). Quantum Bits with Josephson Junctions. InTafuri, F., editor, Fundamentals and Frontiers of the Josephson Effect, pages 703–741. Springer.

[Kuperberg, 2015] Kuperberg, G. (2015). How hard is it to approximate the jones polynomial? Theory ofComputing, 11:183.

[Lechner et al., 2015] Lechner, W., Hauke, P., and Zoller, P. (2015). A quantum annealing architecture withall-to-all connectivity from local interactions. Science advances, 1(9):e1500838.

[Leonhardt, 1997] Leonhardt, U. (1997). Measuring the Quantum State of Light. Cambridge UniversityPress, New York, NY, USA, 1st edition.

[Leonhardt and Paul, 1993] Leonhardt, U. and Paul, H. (1993). Realistic optical homodyne measurementsand quasiprobability distributions. Phys. Rev. A, 48:4598.

[Lucas, 2014] Lucas, A. (2014). Ising formulations of many np problems. Frontiers in Physics, 2:5.

[Lund et al., 2017] Lund, A., Bremner, M. J., and Ralph, T. (2017). Quantum sampling problems, boson-sampling and quantum supremacy. npj Quantum Information, 3(1):15.

[Lund et al., 2014] Lund, A. P., Rahimi-Keshari, S., Rudolph, T., OÕBrien, J. L., and Ralph, T. C. (2014).Boson sampling from a gaussian state. Phys. Rev. Lett., 113:100502.

[Mari and Eisert, 2012] Mari, A. and Eisert, J. (2012). Positive wigner functions render classical simulationof quantum computation efficient. Phys. Rev. Lett., 109:230503.

[Meaney et al., 2014] Meaney, C. H., Nha, H., Duty, T., and Milburn, G. J. (2014). Quantum and classicalnonlinear dynamics in a microwave cavity. EPJ Quantum Technology, 1(1):7.

[Menicucci, 2014] Menicucci, N. C. (2014). Fault-tolerant measurement-based quantum computing withcontinuous-variable cluster states. Phys. Rev. Lett., 112:120504.

[Menicucci et al., 2011] Menicucci, N. C., Flammia, S. T., and van Loock, P. (2011). Graphical calculus forgaussian pure states. Physical Review A, 83(4):042335.

[Meystre and Sargent, 2007] Meystre, P. and Sargent, M. (2007). Elements of quantum optics. SpringerScience & Business Media.

[Nielsen and Chuang, 2000] Nielsen, M. A. and Chuang, I. L. (2000). Quantum Computation and QuantumInformation. Cambridge University Press.

[Nielsen and Chuang, 2011] Nielsen, M. A. and Chuang, I. L. (2011). Quantum Computation and QuantumInformation: 10th Anniversary Edition. Cambridge University Press, New York, NY, USA, 10th edition.

124

Page 12: Fast Quantum Algorithms - Chalmers school... · 2019. 11. 15. · Figure 5.3: An overview circuit gure of the phase estimation circuit. (From Nielsen & Chuang), Fig. 5.3. probability

[Nigg et al., 2017] Nigg, S. E., Lörch, N., and Tiwari, R. P. (2017). Robust quantum optimizer with fullconnectivity. Science advances, 3(4):e1602273.

[Paris et al., 2003] Paris, M. G. A., Cola, M., and Bonifacio, R. (2003). Quantum-state engineering assistedby entanglement. Phys. Rev. A, 67:042104.

[Pednault et al., 2019] Pednault, E., Gunnels, J. A., Nannicini, G., Horesh, L., and Wisnieff, R. (2019).Leveraging secondary storage to simulate deep 54-qubit sycamore circuits.

[Puri et al., 2017] Puri, S., Andersen, C. K., Grimsmo, A. L., and Blais, A. (2017). Quantum annealingwith all-to-all connected nonlinear oscillators. Nature communications, 8:15785.

[Rahimi-Keshari et al., 2016] Rahimi-Keshari, S., Ralph, T. C., and Caves, C. M. (2016). Sufficient condi-tions for efficient classical simulation of quantum opticss. Phys. Rev. X, 6:021039.

[Raussendorf and Briegel, 2001] Raussendorf, R. and Briegel, H. J. (2001). A One-Way Quantum Computer.Phys. Rev. Lett., 86:5188.

[Raussendorf et al., 2003] Raussendorf, R., Browne, D. E., and Briegel, H. J. (2003). Measurement-basedquantum computation on cluster states. Phys. Rev. A, 68:022312.

[Rodríguez-Laguna and Santalla, 2018] Rodríguez-Laguna, J. and Santalla, S. N. (2018). Building an adia-batic quantum computer simulation in the classroom. American Journal of Physics, 86(5):360–367.

[Scheel, 2004] Scheel, S. (2004). Permanents in linear optical networks. arXiv preprint quant-ph/0406127.

[Spring et al., 2013] Spring, J. B., Metcalf, B. J., Humphreys, P. C., Kolthammer, W. S., Jin, X.-M., Bar-bieri, M., Datta, A., Thomas-Peter, N., Langford, N. K., Kundys, D., Gates, J. C., Smith, B. J., Smith,P. G. R., and Walmsley, I. A. (2013). Boson sampling on a photonic chip. Science, 339:798.

[Stollenwerk et al., 2019] Stollenwerk, T., Lobe, E., and Jung, M. (2019). Flight gate assignment with aquantum annealer. In International Workshop on Quantum Technology and Optimization Problems, pages99–110. Springer.

[Tinkham, 2004] Tinkham, M. (2004). Introduction to superconductivity. Courier Corporation.

[Ukai et al., 2010] Ukai, R., Yoshikawa, J.-i., Iwata, N., van Loock, P., and Furusawa, A. (2010). Uni-versal linear bogoliubov transformations through one-way quantum computation. Physical review A,81(3):032315.

[Vikstål, 2018] Vikstål, P. (2018). Continuous-variable quantum annealing with superconducting circuits.Master Thesis, Chalmers.

[Walls and Milburn, 2007] Walls, D. F. and Milburn, G. J. (2007). Quantum optics. Springer Science &Business Media.

[Walther et al., 2005] Walther, P., Resch, K. J., Rudolph, T., Schenck, E., Weinfurter, H., Vedral, V.,Aspelmeyer, M., and Zeilinger, A. (2005). Experimental one-way quantum computing. Nature, 434:169.

[Watrous, 2009] Watrous, J. (2009). Encyclopedia of Complexity and Systems Science, chapter QuantumComputational Complexity, pages 7174–7201. Springer New York, New York, NY.

[Wendin, 2017] Wendin, G. (2017). Quantum information processing with superconducting circuits: a re-view. Reports Prog. Phys., 80:106001.

125