A robust quantum advantage scheme using fermionic linear optics and magic input states

Fermion Sampling

PRX Quantum 3, 020328 (2022)

Michał Oszmaniec, ND, Mauro E. Morales, Zoltán Zimborás

Presented at Physics Research Seminar, CU

15 Sep 2022

Computational complexity seeks to characterize the difficulty of a mathematical problem based on the scaling of the resource required to solve the problem w.r.t the problem size \(n\)

  • A polynomial scaling \(\propto n^c\) is said to be efficient
  • \(O(f(n))\) complexity gives an upper bound (worst-case complexity)

Computational Complexity

hackerdashery, Youtube

P vs NP

Traversing the graph without crossing the same edge twice

Traversing the graph without visiting the same vertex twice

  • Generalize to \(\mathrm{Euler}(G)\) and \(\mathrm{Hamilton}(G)\) where a graph \(G\) defines an instance of the problem
  • If an Eulerian cycle exists, it can be found in \(\propto n\) steps. Meanwhile no efficient way to find a Hamiltonian path is known despite the similarity of the two problems!

Eulerian cycle

Hamiltonian cycle

P vs NP

Map of Computational Complexity

Difficulty

P: problems solvable in polynomial time

  • Eulerian cycle
  • Multiplication
  • Primality testing

NP: problems checkable in polynomial time 

  • Hamiltonian cycle
  • Traveling salesman
  • Packing
  • Graph coloring

#P: counting the number of solutions to an NP problem

BQP: problems solvable in polynomial time on a quantum computer

  • Factoring
  • Discrete log
  • Jones polynomial
  • Introduction - What is a simulation?
  • Fermionic linear optics
  • Summary

Outline

Defining the Problem

{\bf x} \in \{0,1\}^n
01001011\cdots 0
00110001\cdots 1
\underbrace{10101010\cdots 0}_n

Intro to Java  Programming, Y. Daniel Liang

String of \(n\) bits

Output distribution over \(2^n\) bitstrings \(p({\bf x}) = p_{\bf x}\)

Simulating a probability distribution?

\implies

Strong vs Weak Simulation

Strong simulation

Weak simulation

Can compute \(p_{\bf x}\) 

Can sample from \(p\)

|p_{\bf x} - q_{\bf x}| \le \eta p_{\bf x}
\sum_{\bf x} |p_{\bf x} - q_{\bf x}| \le \epsilon

In practice, there need to be some notions of errors

In this talk, I will use these definitions of strong and weak simulation unless stated otherwise

Relative error

Additive error

Quantum Advantage?

Can a classical computer weakly simulates a quantum computer?

  • Weak simulation is a natural problem for quantum computers
  • Strong simulation is very hard (#P)
  • Weak simulation is easy
  • Strong simulation is a bit harder than NP (\(\Delta_3^{\mathrm{P}}\)) (Stockmeyer theorem)

Quantum Advantage

Aaronson and Arkhipov (STOC'11) argued that one can prove that weak classical simulation of a family of quantum circuits is hard from

  • A sufficiently robust #P-hardness of strong simulation
  • Anti-concentration of output probabilities

Rio Grande, 2017

  • Introduction - What is a simulation?
  • Fermionic linear optics
  • Summary

Outline

\mathrm{fSim}(\theta,\varphi) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\theta & -i\sin\theta & 0 \\ 0 & -i\sin\theta & \cos\theta & 0\\ 0 & 0 & 0 & e^{-i\varphi} \end{pmatrix}

Fermionic Linear Optics (FLO)

Google AI Blog

  • fSim gates are realized with high fidelity in the demonstration of random circuit sampling on the 53-qubit Sycamore quantum processor (Arute et al., Nature 2019, Foxen et al., PRL 2020)
  • The same processor is used for proof-of-principle quantum chemistry calculation (Arute et al., Science 2020), non-planar QAOA (Harrigan et al., Nat Phys 2021), preparing the toric code ground state (Satzinger et al., Science 2021)
=\begin{pmatrix} R^{\dagger}_Z(\varphi)_{11} & & R_Z^{\dagger}(\varphi)_{12} \\ & \!\!\!\!\!\Large{R_X(\theta)}\!\!\!\! \\ R_Z^{\dagger}(\varphi)_{21} & & R_Z^{\dagger}(\varphi)_{22} \end{pmatrix}

Fermionic Linear Optics (FLO)

\mathrm{fSim}(\theta,\varphi) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\theta & -i\sin\theta & 0 \\ 0 & -i\sin\theta & \cos\theta & 0\\ 0 & 0 & 0 & e^{-i\varphi} \end{pmatrix}

Even parity

Odd parity

M(U^+,U^-) =\begin{pmatrix} U^+_{11} & & U^+_{12} \\ & \Large{U^-} \\ U^+_{21} & & U^+_{22} \end{pmatrix}
\det U^+ = \det U^-

FLO gates AKA Matchgates

(Valiant, SIAM J Comput 2002)

\mathrm{SWAP} = \begin{pmatrix} 1 & & 0 \\ & \!\!\!\!\Large{X}\!\!\! \\ 0 & & 1 \end{pmatrix}

Non-example

Fermionic Linear Optics (FLO)

X = \begin{pmatrix}0&1\\1&0 \end{pmatrix}
Y = \begin{pmatrix}0&\!\!\!-i\\i&0 \end{pmatrix}
Z = \begin{pmatrix}1&0\\0&\!\!\!-1 \end{pmatrix}

Reminder

\sigma^{\dagger} = X-iY = \begin{pmatrix}0&0\\1&0 \end{pmatrix}
\sigma^{\dagger}|0\rangle = |1\rangle

Identify computational basis states with occupation number states

Jordan-Wigner

f_j f^{\dagger}_k +f^{\dagger}_k f_j = \delta_{jk}\mathbb{I}
(f^{\dagger}_j)^2 = 0

CAR

f^{\dagger}_j \coloneqq \sigma_j^{\dagger} \displaystyle{\prod_{k=1}^{j-1}} Z_k

\(f_j^{\dagger}\) creates a fermion at site \(j\)

Pauli exclusion

Fermionic Linear Optics (FLO)

m_j^X \coloneqq X_j \displaystyle{\prod_{k=1}^{j-1}} Z_k
m_j^Y \coloneqq Y_j \displaystyle{\prod_{k=1}^{j-1}} Z_k

cf. Dirac's gamma matrices

m_{\alpha}m_{\beta} + m_{\beta}m_{\alpha} = \delta_{\alpha\beta} \mathbb{I}
\alpha = (j,X\,\mathrm{or}\,Y)

Majoranas

\begin{pmatrix} 1 & & 0 \\ & \!\!\!\!\!\Large{-iX}\!\!\!\! \\ 0 & & 1 \end{pmatrix}

Non-example

\mathrm{SWAP}= \mathbb{I}\!\otimes\!\mathbb{I}\!+\!X\!\otimes\! X \!+\! Y\!\otimes \!Y \!+\! Z\!\otimes \!Z

\(Z\!\otimes\! Z\) interaction is not FLO

FLO Hamiltonians are quadratic

Antisymmetric

H_{\mathrm{FLO}} = \displaystyle{\frac{i}{4}} \sum_{\substack{\alpha,\beta \\ \alpha>\beta}} h_{\alpha\beta} [m_{\alpha},m_{\beta}]

Fermionic Linear Optics (FLO)

\vec{m}(t) = V^{\dagger} \vec{m} V = e^{{\bf h}t} \vec{m}

Generators of rotations SO(\(2n\))

V_{\mathrm{FLO}} = e^{iH_{\mathrm{FLO}}t}

(Defining representation)

Antisymmetric

FLO unitaries occupy an exponentially small corner of the space of unitaries!

Fermionic Linear Optics (FLO)

\rho = (|0\rangle\langle 0|)^{\otimes n} = \displaystyle{\prod_{j=1}^n} \left( \displaystyle\frac{\mathbb{I}_j+Z_j}{2} \right) = \displaystyle{\prod_{j=1}^n} \left( \displaystyle\frac{\mathbb{I}_j-i m^X_j m^Y_j}{2} \right)
\propto \exp\left( \displaystyle{\frac{\beta}{2} }\sum_{j=1}^n im^X_j m^Y_j \right)
\beta\to\infty
V\rho V^{\dagger} = \exp^{V (\cdots) V^{\dagger}}

(Adjoint representation)

Induce a transformation of the state \(\rho = (|0\rangle\langle 0|)^{\otimes n}\)

\(2n\times 2n\) matrices instead of \(2^n\times 2^n\) matrices!

  • More convenient to work with correlation matrix (2nd moments)
  • All higher moments can be extracted from \(\Sigma\) via Wick's theorem
\Sigma_{\alpha\beta} \coloneqq \frac{i}{2}\mathrm{Tr}(\rho[m_{\alpha},m_{\beta}])
\langle\prod_{k\in K} Z_k\rangle = i^{-|K|}\mathrm{Pf} (\Sigma\bigr|_K)

Fermion Sampling

  • FLO with computational-basis input state admits strong simulation (compute \(p_{\bf x}\) up to logarithmic number of decimals)
  • Not the case with magic input state; strong simulation is #P-hard (cf. Ivanov, PRA 2017)
  • Our work shows that the weak simulation is also hard!
V_{FLO}
|\Psi_4\rangle = \displaystyle{\frac{|0011\rangle + |1100\rangle}{\sqrt{2}}}

Hardness Guarantees

Quantum advantage scheme Robustness
(additive error)		 Anti-concentration
Boson Sampling
Random circuit sampling (Google layout)
Fermion Sampling
2^{-\tilde{O}(n)}
2^{-\tilde{O}(n^{1.5})}
2^{-\tilde{O}(n^2)}

Bouland et al., FOCS 2021

  • Introduction - What is a simulation?
  • Fermionic linear optics
  • Summary

Outline

Summary

  • There are several notions of simulation of randomness:
    • Strong simulation refers to the ability to compute \(p_{\bf{x}}\)
    • Weak simulation refers to the ability to sample from \(p\)
  • The notion of strong simulation is often too strong;  a (possibly quantum) computer cannot strongly simulate itself
  • However, (#P-) hardness of strong simulation is a necessary ingredient to show hardness of classical weak simulation of quantum computers (quantum advantage)
  • FLO gates, which can be realized with high fidelity in near-term superconducting architectures, can be used to demonstrate such quantum advantage when supplemented by magic input states

What we want to prove is this:

If I pick a quantum circuit \(U\) from a family \(\mathcal{C}\) at random, then there is a high chance that a weak simulation of the output probability \(p_{\bf x}(U)\) is infeasible

Quantum Advantage

(Don't average the results over random \(U\)! Too much scrambling)

Weak simulation of Fermion Sampling

Result:

Anti-concentration

Strong simulation on average (over Haar)

in \(\Delta_3^{\mathrm{P}}\)

Result:

Average-case hardness of approximating up to additive error \(2^{-\tilde{O}(n^2)}\)

Collapse of the polynomial hierarchy

(conjectured to be false)

p_{\mathbf x_0}(V,\Psi_{in}) = |\langle{\mathbf x_0}|V|\Psi_{in}\rangle|^2

Conjecture: average-case hardness of approximating up to additive error

\sum_{\mathbf{x}} |p_{\mathbf{x}}-q_{\mathbf{x}}| \le \epsilon

Quantum Advantage Conjecture

\frac{1}{2^n \mathrm{poly}(n)}

Anti-concentration

  • \(l^2\)-distance to the uniform distribution \(\Vert  p-p_{\mathrm{unif}}\Vert^2 = Z - \displaystyle{\frac{1}{2^n}}\)
  • implies \(\Pr\left(p_{{\bf x}} \ge \displaystyle{\frac{\beta}{2^n}}\right) \ge \alpha(1-\beta)^2 \) via the Paley-Zygmund inequality
\Pr\left( X\ge \beta\mathbb{E}[X] \right) \ge (1-\beta)^2 \displaystyle{\frac{\mathbb{E}[X]^2}{\mathbb{E}[X^2]}}

To turn relative error to additive error, we need most probabilities to be large

|p_{\bf x} - q_{\bf x}| \le \eta p_{\bf x} \longleftrightarrow \sum_{\bf x} |p_{\bf x} - q_{\bf x}| \le \epsilon
\Pr\left(p_{{\bf x}} \ge \displaystyle{\frac{C_1}{2^n}}\right) \ge C_2

Collision probability

Z = \mathbb{E}_U \mathbb{E}_{\bf x}\, [ p^2_{\bf x}(U)]
Z \le \displaystyle{\frac{1}{\alpha 2^n}}

\(p_{\bf{x}}(U)\) anti-concentrates if

Z_{\mathrm{unif}} = \displaystyle{\frac{1}{2^n}}

Modified from Dalzell et al., PRX Quantum 2022

  • Uniform \(Z=\displaystyle{\frac{1}{2^n}}\)
  • Global Haar \(Z =\displaystyle{\frac{2}{2^n+1}}\)
  • Local Haar \(Z=\displaystyle{\frac{(4/3)^n}{2^n}}\)
2
2
2

Anti-concentration

\mathbb{E}_V [p^2] = \int dV\, \mathrm{Tr}(V|\Psi\rangle \langle \Psi| V^{\dagger}|{\bf x}\rangle\langle {\bf x}|)^2
= \mathrm{Tr}(|\Psi\rangle \langle \Psi| \int dV\, V^{\dagger}|{\bf x}\rangle\langle {\bf x}|V)^2
= \mathrm{Tr}\left[(|\Psi\rangle \langle \Psi|)^{\otimes 2} \underbrace{\int dV\, (V^{\dagger})^{\otimes 2}(|{\bf x}\rangle\langle {\bf x}|)^{\otimes 2}V^{\otimes 2}}\right]

Sum of projectors onto irreps under nice conditions

  • SO(\(2n\)) for FLO
  • U(\(n\)) for passive (number-preserving) FLO

Anti-concentration

Worst-to-average-case reduction

  • Average-case hardness of strong simulation of Boson Sampling originates from the argument for average-case hardness of the computation of the permanent via polynomial interpolation (Lipton 1991).
  • Bouland et al., Nat Phys (2018) and Movassagh 2019 import the argument to Random Circuit Sampling
  • There exists a "worst case circuit" \(V_0\) for which \(p_0 = |\langle \mathbf{x}_0|V_0|\Psi_{in}\rangle|^2\) is #P-hard to approximate
  • Deform to average-case (Haar random) circuits where \(p_0 (\theta)= |\langle \mathbf{x}_0|V(\theta)|\Psi_{in}\rangle|^2\) is a polynomial of low-degree \(q\)
  • Sample \(\gtrsim q\) points (some can be errorneous\(\iff\)robustness) and recover \(p_0(\theta)\) via Berlekamp–Welch
  • Plug in \(\theta = 0\) to solve a #P-hard problem! 

\(-\) Actual polynomial

\(-\) Noisy polynomial

Unitary Paths

First idea: truncated exponential path

U(\theta) = U_0 e^{i\theta H}

Movassage 2019

= U_0 \left(\mathbb{I} + i\theta H + \frac{(i \theta H)^2}{2!} + \cdots \right)

Truncation gives rise to a non-unitary operator

f(X) = (\mathbb{I}-X)(\mathbb{I}+X)^{-1}
g_{\theta} = g_0 \displaystyle{ \frac{(1-\theta) \mathbb{I} +(1+\theta) g}{ (1+\theta) \mathbb{I} +(1-\theta)g }}

Better idea: Cayley path

Unitary

Hermitian

Hardness Guarantees

Quantum advantage scheme Robustness
(additive error)		 Anti-concentration
Boson Sampling
Random circuit sampling (Google layout)
Fermion Sampling
2^{-\tilde{O}(n)}
2^{-\tilde{O}(n^{1.5})}
2^{-\tilde{O}(n^2)}

Bouland et al., FOCS 2021

Other Results

  • Numerics suggest that random FLO circuits anti-concentrate in sub-linear depth
|+_X^p\rangle \\ \text{or} \\ |+_Y^p\rangle
p,p' = 1,2,\dots,d
V_{FLO}
|+_X^p\rangle = |0\rangle^{\otimes (p-1)} \otimes |+_X\rangle \otimes |0\rangle^{\otimes (n-p)}
|+_Y^p\rangle = |0\rangle^{\otimes (p-1)} \otimes |+_Y\rangle \otimes |0\rangle^{\otimes (n-p)}
  • Assuming that the circuit is FLO, circuit tomography can be performed efficiently utilizing single-mode input states and product measurements