Quantum Computing
(Fall 2013)
Instructor: Shengyu Zhang.
Lecture 8 Quantum communication complexity: A recent protocol
The original plan was to talk about lower bounds on quantum communication complexity, but
it may again well fall into the category of being “too difficult”. So after some reflections, I
decided to change to something lighter but probably with a broader range of applications.
1. Fourier analysis over {𝟎, 𝟏}𝒏
In computer science, we often run into real-valued functions defined on {0,1}𝑛 . (Name some
examples yourself; you’d find that actually it’s even hard to think of any common function
we studied that doesn’t fall into this category.) These functions form a linear space
{𝑓: {0,1}𝑛 → ℝ}
It is easily seen that the addition of any two real-valued functions on the same domain gives
another real-valued function on the domain.
A standard set of basis of this space contains the following ones:
{𝑓𝑠 : 𝑠 ∈ {0,1}𝑛 } where 𝑓𝑠 (𝑥) = {
1 𝑖𝑓 𝑥 = 𝑠
.
0 𝑖𝑓 𝑥 ≠ 𝑠
Actually these basis functions, when scaled up by a factor of √2𝑛 , are orthonormal under the
inner product defined by
⟨𝑓, 𝑔⟩ = 𝐄𝑥∈{0,1}𝑛 [𝑓(𝑥)𝑔(𝑥)]
where all expectations in this note are over uniform distribution unless stated otherwise.
Namely, we have that ⟨𝑓𝑠 , 𝑓𝑡 ⟩ = 𝛿𝑠𝑡 . There is actually another orthonormal basis defined by
characters.
{𝜒𝛼 : 𝛼 ∈ {0,1}𝑛 } where 𝜒𝛼 (𝑥) = (−1)𝛼⋅𝑥
Exercise. Verify that these character functions form an orthonormal basis.
The new basis is usually called the Fourier basis. Any real-valued function can be written in
the Fourier basis:
𝑓 = ∑ 𝑓̂(𝛼)𝜒𝛼 ,
𝛼
and the coefficients 𝑓̂(𝛼) are called Fourier coefficients, computed by
𝑓̂(𝛼) = 𝐄𝑥 [𝑓(𝑥)𝜒𝛼 (𝑥)].
One useful fact about Fourier transform is that the multiplication in one domain becomes the
convolution in the other. For 𝑓, 𝑔: {0,1}𝑛 → ℝ, their convolution 𝑓 ⋆ 𝑔 is defined by
(𝑓 ⋆ 𝑔)(𝑥) = 𝐄𝑦 [𝑓(𝑦)𝑔(𝑥 + 𝑦)] = 𝐄𝑦 [𝑓(𝑥 + 𝑦)𝑔(𝑦)]
In the Fourier domain, the definition is similar except for a normalization factor.
(𝑓̂ ⋆ 𝑔̂)(𝛼) = ∑ 𝑓̂(𝛽)𝑔̂(𝛼 + 𝛽) = ∑ 𝑓̂(𝛼 + 𝛽)𝑔̂(𝛽).
𝛽
𝛽
̂ = 𝑓̂ ⋆ 𝑔̂, 𝑓̂
Fact. 𝑓𝑔
⋆ 𝑔 = 𝑓̂𝑔̂.
One can define the ℓ𝑝 -norm for 𝑓̂ as usual:
1/𝑝
𝑝
‖𝑓̂‖ℓ = (∑|𝑓̂(𝛼)| )
𝑝
𝛼
When 𝑝 = 0, the quantity is the Fourier sparsity of 𝑓, i.e. the number of nonzero Fourier
coefficients.
Similarly, we can define an 𝐿𝑝 -norm for 𝑓 but note that it’s more convenient to use the
expectation instead of summation.
‖𝑓‖𝐿𝑝 = (𝐄𝑥 [|𝑓(𝑥)|𝑝 ])1/𝑝
An elegant fact is that
‖𝑓‖𝐿2 = ‖𝑓̂‖ .
ℓ
2
In computer science, we run into Boolean functions a lot. There are two ways of writing
expressing a Boolean value---{0,1} or {+1, −1}. Note that the group ℤ2 ≅ ({0,1}, ⊕) ≅
({+1, −1}, ⋅ ), where ⊕ and ⋅ are the XOR and multiplication, respectively. These two
ways of representing a Boolean value can be exchanged easily as follows. Suppose that we
use 𝑓 for {0,1}-valued function and 𝑔 for the same function but with range {+1, −1}, then
𝑓 = (1 − 𝑔)/2, and 𝑔 = 1 − 2𝑓.
For Boolean functions with range {+1, −1}, we have the following fact.
Parseval Identity. For any function 𝑓: {0,1}𝑛 → {+1, −1}, we have ∑𝛼 𝑓̂(𝛼)2 = 1.
So one can view 𝑓̂(𝛼)2 as a distribution over all 𝛼 ∈ {0,1}𝑛 .
2. 𝔽𝟐 -degree, discrete derivatives
For Boolean functions 𝑓: {0,1}𝑛 → {0,1}, we can either view them as a polynomial over ℝ or a
polynomial over 𝔽2 .
Exercise. For the Parity function of 2 bits, write down it as a polynomial 𝑓 ∈ ℝ[𝑥1 , 𝑥2 ] and
𝑓 ∈ 𝔽2 [𝑥1 , 𝑥2 ].
Whenever we have a polynomial, we can talk about its degree. Note from the above example
that the degree of 𝑓 as a polynomial over ℝ or that over 𝔽2 are different. We denote by
deg(𝑓) and deg 2 (𝑓) the degrees 𝑓 as a polynomial over ℝ and that over 𝔽2 ,
respectively.
One interesting operator on functions is the derivative. For any 𝑓: {0,1}𝑛 → {0,1} and any
direction vector 𝑡 ∈ {0,1}𝑛 , the discrete derivative of 𝑓 along the direction 𝑡 is another
function
𝛥𝑡 𝑓: {0,1}𝑛 → {0,1} defined by 𝛥𝑡 𝑓(𝑥) = 𝑓(𝑥) + 𝑓(𝑥 + 𝑡)
Note that when we talk about polynomials over 𝔽2 , all additions in the above equation are
also over 𝔽2 .
Just as derivatives for polynomials over ℝ, the discrete derivative also decrease the degree of
a polynomial. That is,
deg 2 (Δ𝑡 𝑓) ≤ deg 2 (𝑓) − 1.
When we use {+1, −1} as the range, then the derivative changes to 𝛥𝑡 𝑔(𝑥) = 𝑔(𝑥)𝑔(𝑥 + 𝑡).
3. Quantum Fourier transform
For a function 𝑓: {0,1}𝑛 → {+1, −1}, one can define an n-qubit quantum state
1
√𝑁
∑ 𝑓(𝑥)|𝑥⟩
𝑥
If one applies Hadamard gates, one on each qubit, then the state becomes
1
∑ 𝑓̂(𝛼)|𝛼⟩
√𝑁 𝛼
Verify this!
4. XOR functions
XOR functions are the class of functions 𝐹(𝑥, 𝑦) = 𝑓(𝑥 + 𝑦) for some 𝑛-bit function 𝑓. We
sometimes denote such function 𝐹 by 𝑓 ∘⊕ The class contains interesting functions such as
Equality and Hamming Distance. It also has an intimate connection to Fourier analysis because the
rank of the communication matrix 𝑀𝐹 is nothing but the Fourier sparsity of 𝑓, i.e. the number of
nonzero Fourier coefficients.
̂‖ .
Exercise. rank(𝑀𝑓∘⊕ ) = ‖𝑓
0
A lower bound for the quantum communication complexity for XOR functions is the following. First,
let’s define a variant of the Fourier 1-norm.
‖𝑓‖1,𝜖 = min{‖𝑔‖1 : |𝑓(𝑥) − 𝑔(𝑥)| ≤ 𝜖, ∀𝑥 ∈ {0,1}𝑛 }
Theorem ([LS09]). 𝑄𝜖 (𝑓 ∘⊕) = Ω(log‖𝑓‖1,𝜖 ).
It was considered to be pretty tight, but it has lacked rigorous argument. Recently, the following
bound was shown, which implies that the above lower bound is tight for low degree polynomials.
Theorem ([Z14]) 𝑄𝜖 (𝑓 ∘⊕) = 𝑂(2𝑑 log‖𝑓‖1,𝜖 ) where 𝑑 = deg 2 (𝑓).
Next we give a protocol for a simpler case of exact communication, namely protocols without error.
The communication cost is at most 2𝑑 log‖𝑓‖0. The main idea is degree reduction. You’ll see how
log‖𝑓‖0 naturally comes into the picture when using quantum protocols.
The analysis of the protocol needs the following fact.
̂
̂
Fact. For any 𝑡 ∈ {0,1}𝑛 , supp(𝛥̂
𝑡 𝑓 ) ⊆ supp(𝑓 ) + supp(𝑓 ).
Proof. Let’s see what the Fourier coefficients of 𝛥𝑡 𝑓 are. Define 𝑓𝑡 (𝑥) = 𝑓(𝑥 + 𝑡). Then
𝑓̂𝑡 (𝛼) = 𝐄𝑥 [𝑓(𝑥 + 𝑡)𝜒𝛼 (𝑥)] = 𝐄𝑦 [𝑓(𝑦)𝜒𝛼 (𝑦 + 𝑡)] = 𝐄𝑦 [𝑓(𝑦)𝜒𝛼 (𝑦)𝜒𝛼 (𝑡)] = 𝑓̂(𝛼)𝜒𝛼 (𝑡).
Thus,
̂
̂
̂
̂
̂
𝛥̂
𝑡 𝑓 (𝛼) = 𝑓𝑓𝑡 (𝛼) = ∑ 𝑓(𝛼 + 𝛽)𝑓𝑡 (𝛽) = ∑ 𝑓(𝛼 + 𝛽)𝑓(𝛽) 𝜒𝛽 (𝑡)
𝛽
𝛽
Therefore, if 𝛼 ∉ supp(𝑓̂) + supp(𝑓̂), then there isn’ t 𝛽 with both 𝑓̂(𝛼 + 𝛽) and 𝑓̂(𝛽) being
nonzero. So 𝛥̂
𝑡 𝑓 (𝛼) = 0 for those 𝛼.
𝑘
Notation in the protocol: 𝑓: {0,1}𝑛 → {+1, −1}, 𝐴 = supp(𝑓̂), 𝐸𝑘 : {0,1}𝑛 → [|𝐴|2 ], 𝑧 = 𝑥 + 𝑦.
To see why the protocol works, consider the second round. If 𝛥𝑡1 𝑓 is a constant, then 𝑓(𝑧) =
𝑓(𝑧 + 𝑡1 ) + 𝛥𝑡1 𝑓 is known; else, repeat the above procedure for 𝑓2 = 𝛥𝑡1 𝑓. Let 𝑘 = 2, and use 𝐸𝑘
to encode Fourier coefficients of 𝛥𝑡1 𝑓. Note that though Alice doesn’t know 𝑡1 and thus the Fourier
coefficients of 𝛥𝑡1 𝑓, she can still do that because for any 𝑡1 , the Fourier support of 𝛥𝑡1 𝑓 is within
𝐴 + 𝐴.
𝑘
Complexity: Stage 𝑘 takes log|A|2 = 2𝑘 log |𝐴| communication qubits, and there is at most 𝑑
stages. Thus the total complexity is 2𝑑 log |supp(𝑓̂)|.
Protocol:
Alice
Bob
|𝜓⟩ = 2−1/2 (|0⟩𝐶 |0⟩𝑀
|𝜓⟩
+|1⟩𝐶 ∑𝑠∈𝐴 𝑓̂(𝑠)𝜒𝑠 (𝑦)|𝐸1 (𝑠)⟩𝑀 )
On |1⟩𝐶 :
|𝐸1 (𝑠)⟩𝑀 → 𝜒𝑠 (𝑥)|𝐸1 (𝑠)⟩𝑀
|𝜓⟩′ = 2−1/2 (|0⟩𝐶 |0⟩𝑀
+|1⟩𝐶 ∑𝑠∈𝐴 𝑓̂(𝑠)𝜒𝑠 (𝑧)|𝐸1 (𝑠)⟩𝑀 )
|𝜓⟩′
On |1⟩𝐶 : |𝐸1 (𝑠)⟩𝑀 → |𝑠⟩𝑀
|𝜓⟩′ = 2−1/2 (|0⟩𝐶 |0⟩𝑀
+|1⟩𝐶 ∑𝑠∈𝐴 𝑓̂(𝑠)𝜒𝑠 (𝑧)|𝑠⟩𝑀 )
Apply FT on M: |𝑠⟩𝑀 → 𝑁 −1/2 𝜒𝑠 (𝑡)|𝑡⟩𝑀
|𝜓⟩′ = (2𝑁)−1/2 (|0⟩𝐶 ∑𝑡|𝑡⟩𝑀
+|1⟩𝐶 ∑𝑡 ∑𝑠∈𝐴 𝑓̂(𝑠)𝜒𝑠 (𝑧 + 𝑡)|𝑡⟩𝑀 )
= (2𝑁)−1/2 (|0⟩𝐶 ∑𝑡|𝑡⟩𝑀 + |1⟩𝐶 ∑𝑡 𝑓(𝑧 + 𝑡)|𝑡⟩𝑀 )
= 𝑁 −1/2 ∑𝑡
|0⟩𝐶 +𝑓(𝑧+𝑡)|1⟩𝐶
√2
|𝑡⟩𝑀
Apply Hadamard on C, then measure M
A random 𝑡 ∈ {0,1}𝑛 and the corresponding 𝑓(𝑧 + 𝑡)
Note
See [LS09] for a survey of classical and quantum lower bounds using norm-based methods. The
protocol is from [Z14].
Reference
[LS09] Troy Lee and Adi Shraibman, Lower bounds on communication complexity, Foundations and
Trends in Theoretical Computer Science, 2009.
[Z14] Shengyu Zhang, Efficient quantum protocols for XOR functions, Proceedings of the 25th
Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2014, to appear.