adaptiveshadows

Locally-biased classical shadows with an adaptive update

How do I run this?

Install requirements, and navigate to folder adaptiveshadows/python. Then call python sim.py <name> where <name> is the molecule and encoding joined by an underscore. Molecules are: h2, lih, beh2, h2o, nh3 and encodings are jw, parity, bk. So one example of <name> is h2_jw.

Current results:

Take a Hamiltonian, and allow access to the ground state 1000 times. Record the energy difference between the estimated energy and the true ground energy.

| Molecule | Encoding | Derand | Adaptive | Adaptive RSME | Diagonalize | Simulation | |----------|----------|--------|--------------------------------------------------------------------|---------------|-------------|------------| | H2 | JW | 0.06 | 0.003, 0.04, 0.08, 0.09, 0.12, 0.15, 0.09, 0.004, 0.08, 0.03, 0.046| 0.08 | 1sec | 10sec | | | Parity | 0.03 | 0.02, 0.047, 0.054, 0.058, 0.036, 0.085, 0.029, 0.031, 0.01, 0.053 | 0.05 | | | | | BK | 0.06 | 0.12, 0.03, 0.01, 0.03, 0.05, 0.04, 0.01, 0.02, 0.017, 0.2 | 0.08 | | | | LiH | JW | 0.03 | 0.07, 0.027, 0.07, 0.01, 0.05, 0.02, 0.03, 0.04, 0.06, 0.02, 0.04 | 0.04 | 1sec | 1min50sec | | | Parity | 0.03 | 0.01, 0.05, 0.07, 0.01, 0.05, 0.07, 0.03, 0.04, 0.03, 0.05 | 0.05 | | 1min30sec | | | BK | 0.04 | 0.02, 0.009, 0.1, 0.09, 0.06, 0.1, 0.008, 0.1, 0.02, 0.02 | 0.07 | | 1min40sec | | BeH2 | JW | 0.06 | 0.02, 0.057, 0.05, 0.07, 0.06, 0.04, 0.055, 0.07, 0.07, 0.11 | 0.06 | 7sec | 6min20sec | | | Parity | 0.09 | 0.09, 0.051, 0.06, 0.06, 0.05, 0.05, 0.07, 0.06, 0.1, 0.0009 | 0.06 | | 5min40sec | | | BK | 0.06 | 0.07, 0.06, 0.05, 0.05, 0.08, 0.07, 0.06, 0.07, 0.07, 0.06 | 0.06 | | 6min | | H2O | JW | 0.12 | 0.12, 0.12, 0.13, 0.1, 0.1, 0.12, 0.10, 0.09, 0.09, 0.1 | 0.11 | 15sec | 6min45sec | | | Parity | 0.22 | 0.16, 0.056, 0.12, 0.1, 0.12, 0.1, 0.12, 0.09, 0.11, 0.09, 0.11 | 0.11 | | 6min10sec | | | BK | 0.20 | 0.028, 0.11, 0.11, 0.1, 0.11, 0.09, 0.12, 0.11, 0.1, 0.11 | 0.10 | | 6min10sec | | NH3 | JW | 0.18 | 0.11, 0.076, 0.0507, 0.12, 0.14, 0.18, 0.16, 0.14, 0.14, 0.14 | 0.13 | 5min30 | 28min | | | Parity | 0.21 | 0.079, 0.086, 0.17, 0.14, 0.14, 0.14, 0.18, 0.15, 0.16, 0.16 | 0.14 | | 26min | | | BK | 0.12 | 0.06, 0.09, 0.13, 0.13, 0.13, 0.14, 0.06, 0.15, 0.03, 0.10 | 0.11 | | 27min |

How do we generate a measurement basis?

One qubit at a time update idea

Consider $H = \sum_P \alpha_P P$ on $n$ qubits. Set $\mathcal{B}={X,Y,Z}$.

Let $i:[n]\to[n]$ be a random bijection. This gives an ordering of the qubits $i(1), \dots, i(n)$. Write $i(j)$ for $j\in[n]$.

We will chose $\beta_{i(j)}:\mathcal{B}\to\mathbb{R}^+$ in an adaptive style.

Let's do this slowly:

• $j=1$.
  • Set $A = { P \in H | P_{i(1)} \in \mathcal{B} }$.
  • Minimise $\Delta(\beta_{i(1)}) = \sum_{P\in A} \alpha_P^2 / \beta_{i(1)}(P_{i(1)})$.
  • Pick $B_{i(1)}$ from $\beta_{i(1)}$.
• $j>1$.
  • Set $A = { P \in H | P_{i(j)} \in \mathcal{B} \textrm{ and } P_{i(j')} \in {I, B_{i(j')}} \textrm{ for } j'<j }$.
  • Minimise $\Delta(\beta_{i(j)}) = \sum_{P\in A} \alpha_P^2 / \beta_{i(j)}(P_{i(j)})$.
  • Pick $B_{i(j)}$ from $\beta_{i(j)}$.

Now set $B = \otimes_{i\in[n]} B_i$

Actually, the minimisation procedure can be done analytically. Given $j$ and $A$ do the following:

• Set $A = A_X \cup A_Y \cup A_Z$ where $A_B = { P\in A | P_{i(j)}=B }$
• Set $c_B = \sum_{P \in A_B} \alpha_P^2$
• Set $c = \sum_B \sqrt{c_B}$
• If $c=0$ set $\beta_{i(j)}(B)=1/3$ else set $\beta_{i(j)}(B) = \sqrt{c_B} / c$