Distance Measures over distribution
Given two distribution { p x } , { q x } \{p_x\},\{q_x\} { p x } , { q x }
Trace distance (Total variance): D ( p x , q x ) = 1 2 ∑ x ∈ X ∣ p x − q x ∣ D(p_x,q_x) = \frac{1}{2}\sum_{x\in X}|p_x-q_x| D ( p x , q x ) = 2 1 ∑ x ∈ X ∣ p x − q x ∣
Fidelity: F ( p x , q x ) = ∑ x ∈ X p x q x F(p_x,q_x)=\sum_{x\in X}\sqrt{p_xq_x} F ( p x , q x ) = ∑ x ∈ X p x q x
Properties (distribution)
[ 0 , 1 ] [0,1] [ 0 , 1 ] D ( p x , q x ) = 0 ⟺ F ( p x , q x ) = 1 ⟺ ∀ x p x = q x D(p_x,q_x)=0\iff F(p_x,q_x)=1\iff \forall x p_x=q_x D ( p x , q x ) = 0 ⟺ F ( p x , q x ) = 1 ⟺ ∀ x p x = q x D ( p x , q x ) = 1 ⟺ F ( p x , q x ) = 0 ⟺ ∀ x , supp ( p x ) ∩ supp ( q x ) = ∅ , supp ( p x ) = { x ∣ p x > 0 } D(p_x,q_x)=1\iff F(p_x,q_x)=0\iff \forall x ,\text{supp}(p_x)\cap\text{supp}(q_x)=\emptyset,\text{supp}(p_x)=\{x|p_x>0\} D ( p x , q x ) = 1 ⟺ F ( p x , q x ) = 0 ⟺ ∀ x , supp ( p x ) ∩ supp ( q x ) = ∅ , supp ( p x ) = { x ∣ p x > 0 } D ( p x , q x ) = max S ⊆ X ∣ p x − q x ∣ D(p_x,q_x)=\max_{S\subseteq X}|p_x-q_x| D ( p x , q x ) = max S ⊆ X ∣ p x − q x ∣
Distance Measures over quantum states
Given two quantum states
Trace distance: D ( ρ , σ ) = 1 2 ∥ ρ − σ ∥ 2 D(\rho,\sigma)=\frac{1}{2}\|\rho-\sigma\|_2 D ( ρ , σ ) = 2 1 ∥ ρ − σ ∥ 2
D ( ρ , σ ) = D ( U ρ U † , U σ U † ) , ∀ D(\rho,\sigma)=D(U\rho U^\dagger,U\sigma U^\dagger),\forall D ( ρ , σ ) = D ( U ρ U † , U σ U † ) , ∀ U U U D ( ρ , σ ) = max P T r ( P ( ρ − σ ) ) D(\rho,\sigma)=\max_P Tr(P(\rho-\sigma)) D ( ρ , σ ) = max P T r ( P ( ρ − σ )) Total error = 1 2 − 1 2 ∥ ρ − σ ∥ =\frac{1}{2}-\frac{1}{2}\|\rho-\sigma\| = 2 1 − 2 1 ∥ ρ − σ ∥
Theorem: D ( E ( ρ ) , E ( σ ) ) ≤ D ( ρ , σ ) D(\mathcal{E}(\rho),\mathcal{E}(\sigma))\leq D(\rho,\sigma) D ( E ( ρ ) , E ( σ )) ≤ D ( ρ , σ )
Fidelity: F ( ρ , σ ) = T r ρ 1 2 σ ρ 1 2 F(\rho,\sigma)=Tr\sqrt{\rho^{\frac{1}{2}}\sigma\rho^{\frac{1}{2}}} F ( ρ , σ ) = T r ρ 2 1 σ ρ 2 1
purification: Given a density operator ρ \rho ρ A A A ∣ ψ ⟩ A B |\psi\rangle^{AB} ∣ ψ ⟩ A B ρ \rho ρ T r A ∣ ψ ⟩ ⟨ ψ ∣ = ρ Tr_A|\psi\rangle\langle\psi|=\rho T r A ∣ ψ ⟩ ⟨ ψ ∣ = ρ
existence
ρ = ∑ i λ i ∣ u i ⟩ ⟨ u i ∣ A \rho=\sum_i\lambda_i|u_i\rangle\langle u_i|^A ρ = ∑ i λ i ∣ u i ⟩ ⟨ u i ∣ A ∣ ψ ⟩ = ∑ i λ i ∣ u i u i ⟩ A B |\psi\rangle=\sum_i\sqrt{\lambda_i}|u_i u_i\rangle^{AB} ∣ ψ ⟩ = ∑ i λ i ∣ u i u i ⟩ A B
Uhlman's theorem: Suppose ρ \rho ρ σ \sigma σ Q Q Q R R R Q Q Q F ( ρ , σ ) = max ∣ ψ ⟩ , ∣ ϕ ⟩ ∣ ⟨ ψ ∣ ϕ ⟩ ∣ F(\rho,\sigma)=\max_{|\psi\rangle,|\phi\rangle}|\langle\psi|\phi\rangle| F ( ρ , σ ) = max ∣ ψ ⟩ , ∣ ϕ ⟩ ∣ ⟨ ψ ∣ ϕ ⟩ ∣
Theorem: 1 − F ( ρ , σ ) ≤ D ( ρ , σ ) ≤ 1 − F ( ρ , σ ) 2 1-F(\rho,\sigma)\leq D(\rho,\sigma)\leq\sqrt{1-F(\rho,\sigma)^2} 1 − F ( ρ , σ ) ≤ D ( ρ , σ ) ≤ 1 − F ( ρ , σ ) 2
gate fidelity: F ( U , E ) = min ∣ ψ ⟩ F ( U ∣ ψ ⟩ ⟨ ψ ∣ U † , E ∣ ψ ⟩ ⟨ ψ ⟨ E † ) F(U,\mathcal{E})=\min_{|\psi\rangle}F(U|\psi\rangle\langle\psi|U^\dagger,\mathcal{E}|\psi\rangle\langle\psi\langle\mathcal{E}^\dagger) F ( U , E ) = min ∣ ψ ⟩ F ( U ∣ ψ ⟩ ⟨ ψ ∣ U † , E ∣ ψ ⟩ ⟨ ψ ⟨ E † )
minimum fidelity (for quantum channel E \mathcal{E} E F ( E ) = min ∣ ψ ⟩ F ( ∣ ψ ⟩ , E ( ∣ ψ ⟩ ψ ∣ ) ) F(\mathcal{E})=\min_{|\psi\rangle}F(|\psi\rangle,\mathcal{E}(|\psi\rangle\psi|)) F ( E ) = min ∣ ψ ⟩ F ( ∣ ψ ⟩ , E ( ∣ ψ ⟩ ψ ∣ ))
QEC
Bit flip code: 3 physical bits to encode 1 logical bit ∣ 0 ⟩ → ∣ 000 ⟩ |0\rangle\rightarrow |000\rangle ∣0 ⟩ → ∣000 ⟩
Recovery: majority vote
Cannot correct phase error
Phase flip code: 3 physical bits to encode 1 logical bit ∣ 0 ⟩ → ∣ + + + ⟩ |0\rangle\rightarrow |+++\rangle ∣0 ⟩ → ∣ + + + ⟩
Shor code: Syndrome diagnosis
∣ 0 ⟩ → ∣ 0 L ⟩ = ( ∣ 000 ⟩ + ∣ 111 ⟩ ) ( ∣ 000 ⟩ + ∣ 111 ⟩ ) ( ∣ 000 ⟩ + ∣ 111 ⟩ ) 2 2 |0\rangle\rightarrow|0_L\rangle=\frac{(|000\rangle+|111\rangle)(|000\rangle+|111\rangle)(|000\rangle+|111\rangle)}{2\sqrt{2}} ∣0 ⟩ → ∣ 0 L ⟩ = 2 2 ( ∣000 ⟩ + ∣111 ⟩) ( ∣000 ⟩ + ∣111 ⟩) ( ∣000 ⟩ + ∣111 ⟩) ∣ 1 ⟩ → ∣ 1 L ⟩ = ( ∣ 000 ⟩ − ∣ 111 ⟩ ) ( ∣ 000 ⟩ − ∣ 111 ⟩ ) ( ∣ 000 ⟩ − ∣ 111 ⟩ ) 2 2 |1\rangle\rightarrow|1_L\rangle=\frac{(|000\rangle-|111\rangle)(|000\rangle-|111\rangle)(|000\rangle-|111\rangle)}{2\sqrt{2}} ∣1 ⟩ → ∣ 1 L ⟩ = 2 2 ( ∣000 ⟩ − ∣111 ⟩) ( ∣000 ⟩ − ∣111 ⟩) ( ∣000 ⟩ − ∣111 ⟩) Correct arbitrary one-qubit quantum error
Other quantum error correcting code
Steane codes
Calderbank-Shor-Steane codes
Stabilizer codes
Toric codes
Surface codes
NISQ(John Preskill): noisy intemidiate-scale quantum computing
Quantum Supremacy(John Preskill)