## Strengthening weak measurements for qubit multitime correlators

Justin Dressel

Razieh Mohseninia, Jose Raul Gonzalez Alonso
Mordecai Waegell, Nicole Yunger Halpern

PIMan Workshop 2019

Chapman University

March 20, 2019

## An Unexpected Question

https://quantumfrontiers.com/2018/09/23/doctrine-of-the-measurement-mean/

"Four-point [out-of-time-ordered] correlators ... signal quantum chaos and information scrambling."   Can we measure them?"

Nicole Yunger Halpern

What is an Out-of-Time-Ordered Correlator (OTOC)?
(and why should I care?)

Idea (Kitaev) :

• Poisson brackets of canonically conjugated variables at different times signal classical chaos as Lyapunov exponential divergence

• Promoting brackets to quantum case suggests we examine the (average of the square of) the commutator instead to isolate such behavior

• For commutators of unitary operators, an OTOC is the only nontrivial quantity that remains

• Since persistent information scrambling is a hallmark of classical chaos, we guess that the OTOC witnesses quantum information scrambling
\displaystyle \{ x(t), p(0) \}_{P.B.} \propto e^{\lambda t}
\displaystyle \langle [ \hat{x}(t), \hat{p}(0) ]^\dagger [ \hat{x}(t), \hat{p}(0) ]\rangle_{\rho} \overset{?}{\propto} \hbar^2e^{2\lambda t} \geq 0
\hat{x}(t) = \hat{U}^\dagger_t\,\hat{x}\,\hat{U}_t
\displaystyle C(t) \equiv \langle [ \hat{W}(t), \hat{V} ]^\dagger [ \hat{W}(t), \hat{V} ]\rangle_{\rho} = 2[1 - \text{Re}F(t)]
\displaystyle F(t) \equiv \langle \hat{W}^\dagger(t)\hat{V}^\dagger\hat{W}(t)\hat{V}\rangle_{\rho}
\displaystyle C(t) \geq 0 \implies \text{Re}F(t) \leq 1

Noncommutativity causes decay

### Non-integrable spin chain example

Integrable

Non-Integrable

H = -J \sum_{i=1}^{N-1}Z_iZ_{i+1} - h\sum_{i=1}^N Z_i - g\sum_{i=1}^N X_i
\rho_0 \propto \exp(-H/kT)

V and W of OTOC are spin Z operators at opposite ends of a 5-spin chain

persistent decay

But is such an OTOC measurable in the lab?
(after all, it's non-Hermitian)

• This looks suspiciously similar to the non-Hermitian structure of the Kirkwood-Dirac (KD) quasiprobability distribution

• KD is closely related to the quantum weak value, and can be determined from weak measurements

• The OTOC is similarly measurable using weak measurement sequences, and is determined by an extended KD quasiprobability distribution
\displaystyle F(t) \equiv \langle \hat{W}^\dagger(t)\hat{V}^\dagger\hat{W}(t)\hat{V}\rangle_{\rho}

OTOC:

\displaystyle \langle \hat{B}\hat{A}\rangle_{\rho} = \sum_b b\,P(b)\,A_w(b) = \sum_{b,a}b\,a\,\tilde{P}(b,a)
\displaystyle P(b) = \langle b|\hat{\rho}|b \rangle
\displaystyle A_w(b) = \frac{\langle b|\hat{A}\hat{\rho}|b \rangle}{\langle b |\hat{\rho}|b\rangle}
\displaystyle \tilde{P}(b,a) = \langle b|a\rangle\langle a |\hat{\rho}|b \rangle

(probability of b)

(quantum weak value)

(KD distribution)

\displaystyle F(t) \equiv \sum_{w,v,w',v'}w^*\,v^*\,w'\,v'\,\langle w | v\rangle\langle v | w' \rangle \langle w' | v \rangle\langle v | \hat{\rho} | w \rangle
\displaystyle \tilde{P}(w,v,w',v')

Yunger Halpern, Swingle, JD PRA 9, 042105 (2018)

### Non-integrable spin chain: OTOC revisited

Blue : Ideal              Orange : Weak Measurement

With finite temperature and decoherence added, OTOC as witness is less convincing

Gonzalez Alonso, JD et al. PRL 122, 040404 (2019)

### Non-integrable spin chain: OTOC KD QPD

Timescales in OTOC extended KD QPD Nonclassicality still clearly distinguish two dynamical cases

\mathcal{N}(t) \equiv \sum_{w,v,w',v'}|\tilde{P}(w,v,w',v')| - 1

Gonzalez Alonso, JD et al. PRL 122, 040404 (2019)

## Recall (Weak) Indirect Measurement

Measurement produces a channel-valued measure (instrument), not just POVM

This procedure can be repeated many times in a row

Easy circuit : prepare ancilla, entangling gate, measure ancilla

Pang, Dressel, Brun, PRL 113, 030401 (2014)

The most interesting measurement in the world

\hat{M} = \langle \psi_f | \exp(-i (g'/2) \hat{A}\otimes \hat{F} ) |\psi_i \rangle \approx \sqrt{p}\,\hat{1} + (g/2)\,\hat{A} + O(g^2)

Crude idea : measurement operator approximately linearizes in weak measurement regime (linear response regime)

Post-interaction system state couples to anticommutator to linear order in coupling:

\displaystyle \text{Tr}(\hat{M}\hat{\rho}\hat{M}^\dagger) \approx p + g\sqrt{p}\,\text{Tr}\left[\frac{\hat{A}\hat{\rho}+\hat{\rho}\hat{A}}{2}\right] + O(g^2)

Sequences of measurements create nested anticommutator terms to lowest order in the joint couplings

\displaystyle \text{Tr}(\hat{N}\hat{M}\hat{\rho}\hat{M}^\dagger\hat{N}^\dagger) \approx p_1p_2 + g_1\sqrt{p_1}p_2\,\text{Tr}\left[\frac{\{\hat{A}_1,\hat{\rho}\}}{2}\right] + g_2\sqrt{p_2}p_1\,\text{Tr}\left[\frac{\{\hat{A}_2,\hat{\rho}\}}{2}\right] + g_1g_2\sqrt{p_1p_2}\,\text{Tr}\left[\frac{\{\hat{A}_2,\{\hat{A}_1,\hat{\rho}\}\}}{4}\right] + \cdots

Anticommutator for measurement

"Non-Hermitian" term:
real part of KD distribution

Background terms, can be subtracted away once measured independently

"small"

Note: unitary transformation yields commutator instead

\displaystyle \text{Tr}(\hat{N}\hat{M}\hat{\rho}\hat{M}^\dagger\hat{N}^\dagger) \approx p_1p_2 + g_1\sqrt{p_1}p_2\,\text{Tr}\left[\frac{\{\hat{A}_1,\hat{\rho}\}}{2}\right] + g_2\sqrt{p_2}p_1\,\text{Tr}\left[\frac{\{\hat{A}_2,\hat{\rho}\}}{2}\right] + g_1g_2\sqrt{p_1p_2}\,\text{Tr}\left[\frac{\{\hat{A}_2,\{\hat{A}_1,\hat{\rho}\}\}}{4}\right] + \cdots
• Benefit : Isolating such a nested commutator term in measurement sequence gives access to interesting non-Hermitian terms like those that appear in OTOCs and the weak value

• Annoyance : Isolating this term means subtracting away unwanted terms, including the calibrated background and any higher-order terms, which is difficult in practice

• Miracle : For qubits, the anticommutator terms can be isolated exactly using any measurement strengths g

JD, et al. PRA 98, 012032 (2018)

\displaystyle \frac{\text{Tr}[\{\hat{A}_2,\{\hat{A}_1,\hat{\rho}\}\}]}{4} = \text{Re}\langle \hat{A}_1\hat{A}_2\rangle_\rho

Arbitrary Strength Qubit Measurements

Informative Measurement:

Non-Informative Measurement:

\displaystyle \hat{M}_{\phi,a}^{(A)} = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} -(-1)^a \sin\frac{\phi}{2}\hat{A}\right]
\displaystyle \hat{N}_{\phi,a}^{(A)} = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} +i(-1)^a \sin\frac{\phi}{2}\hat{A}\right] = \frac{\exp[i\phi(-1)^a\,\hat{A}]}{\sqrt{2}}

JD, et al. PRA 98, 012032 (2018)

\displaystyle \sum_{a=0,1} \alpha_{\phi,a}\,\hat{M}_{\phi,a}^{\dagger(A)}\hat{M}_{\phi,a}^{(A)} = \hat{A}
\displaystyle \alpha_{\phi,a} = \frac{(-1)^{a+1}}{\sin\phi}

Generalized spectral decomposition

(Works for observables A that square to the identity)

\displaystyle \sum_{a=0,1} \alpha_{\phi,a}\,\hat{N}_{\phi,a}^{\dagger(A)}\hat{N}_{\phi,a}^{(A)} = \hat{0}

Tested in the Laboratory!

Informative Measurement:

White, Mutus, JD, et al. npj Quantum Information 2, 15022 (2016)

\displaystyle \hat{M}_{\phi,a}^{(A)} = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} -(-1)^a \sin\frac{\phi}{2}\hat{A}\right]
\displaystyle \sum_{a=0,1} \alpha_{\phi,a}\,\hat{M}_{\phi,a}^{\dagger(A)}\hat{M}_{\phi,a}^{(A)} = \hat{A}
\displaystyle \alpha_{\phi,a} = \frac{(-1)^{a+1}}{\sin\phi}

Generalized spectral decomposition
"rescales response curves"

Generalized measurement method used experimentally with superconducting qubits (Google)

### Bell-Leggett-Garg Inequality violation with weak measurements

Informative Measurement:

White, Mutus, JD, et al. npj Quantum Information 2, 15022 (2016)

• Avoids "clumsiness loophole" by only correlating nonlocal measurements
• Avoids "fair sampling" loophole by using one experimental ensemble
• Requires weak measurements to see violation (preserve entanglement)

Informative Qubit Measurements

\displaystyle \hat{M}_{\phi,a}^{(A)} = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} -(-1)^a \sin\frac{\phi}{2}\hat{A}\right]

JD, et al. PRA 98, 012032 (2018)

Useful corollaries:

\displaystyle \sum_{a=0,1} \alpha_{\phi,a}\,\hat{M}_{\phi,a}^{\dagger(A)}\hat{M}_{\phi,a}^{(A)} = \hat{A}
\displaystyle \alpha_{\phi,a} = \frac{(-1)^{a+1}}{\sin\phi}
\displaystyle \sum_{a=0,1} \alpha_{\phi,a}\,\hat{M}_{\phi,a}^{(A)}\hat{\rho}\hat{M}_{\phi,a}^{\dagger(A)} = \frac{\{\hat{A},\hat{\rho}\}}{2}

Anticommutators isolated perfectly for any coupling strength

\displaystyle \sum_{a_1,\ldots,a_m=0,1} \alpha_{\phi_1,a_1}\cdots\alpha_{\phi_m,a_m}\,P(a_1,\ldots,a_m) = \left\langle\frac{\{\cdots\{\{\hat{A}_m,\hat{A}_{m-1}\},\hat{A}_{m-2}\}\cdots,\hat{A}_1\}}{2^{m-1}}\right\rangle_\rho

Nested anticommutator averages can be measured directly for any coupling strength

Non-Informative Qubit Measurements

JD, et al. PRA 98, 012032 (2018)

Useful corollaries:

\displaystyle \alpha_{\phi,a} = \frac{(-1)^{a+1}}{\sin\phi}
\displaystyle \sum_{a=0,1} \alpha_{\phi,a}\,\hat{N}_{\phi,a}^{(A)}\hat{\rho}\hat{N}_{\phi,a}^{\dagger(A)} = \frac{[\hat{A},\hat{\rho}]}{2i}

Commutators also isolated perfectly for any coupling strength

\displaystyle \sum_{\tilde{a}_1,\ldots,a_m=0,1} \alpha_{\phi_1,\tilde{a}_1}\cdots\alpha_{\phi_m,a_m}\,P(\tilde{a}_1,\ldots,a_m) = \left\langle\frac{[\cdots\{\{\hat{A}_m,\hat{A}_{m-1}\},\hat{A}_{m-2}\}\cdots,\hat{A}_1]}{2^{m-2}(2i)}\right\rangle_\rho

Nested commutator averages can also be measured directly for any coupling strength

\displaystyle \hat{N}_{\phi,a}^{(A)} = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} +i(-1)^a \sin\frac{\phi}{2}\hat{A}\right]

Any strength? What happened to state collapse?

JD, et al. PRA 98, 012032 (2018)

\displaystyle \alpha_{\phi,a} = \frac{(-1)^{a+1}}{\sin\phi}
\displaystyle \sum_{a=0,1} \alpha_{\phi,a}\,\hat{N}_{\phi,a}^{(A)}\hat{\rho}\hat{N}_{\phi,a}^{\dagger(A)} = \frac{[\hat{A},\hat{\rho}]}{2i}
\displaystyle \hat{N}_{\phi,a}^{(A)} = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} +i(-1)^a \sin\frac{\phi}{2}\hat{A}\right]
\displaystyle \hat{M}_{\phi,a}^{(A)} = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} -(-1)^a \sin\frac{\phi}{2}\hat{A}\right]
\displaystyle \sum_{a=0,1} \alpha_{\phi,a}\,\hat{M}_{\phi,a}^{(A)}\hat{\rho}\hat{M}_{\phi,a}^{\dagger(A)} = \frac{\{\hat{A},\hat{\rho}\}}{2}

Why does this work?   Examine full transformation:

(Lindblad) decoherence terms from the measurement cancel in Pauli-like averages

However, marginalization of results will reveal the collapse-based disturbance

Application: Strongly Measured 2-Point Correlator

JD, et al. PRA 98, 012032 (2018)

\displaystyle \hat{U}_t^\dagger\hat{M}_{\phi,b}^{(B)}\hat{U}_t = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} -(-1)^b \sin\frac{\phi}{2}\hat{B}(t)\right]

2-point temporal qubit correlators may be indirectly measured using any coupling strength

These are common theoretical objects that are usually considered inaccessible to measurement

Bracketing measurement by unitary evolutions probes a Heisenberg-evolved operator

The final unitary may be omitted as inconsequential to the measured average

\displaystyle \sum_{a,b=0,1} \alpha_{\phi_a,a}\cdots\alpha_{\phi_b,b}\,P(a,b) = \left\langle\frac{\{\hat{B}(t),\hat{A}\}}{2}\right\rangle_\rho = \text{Re}\langle\hat{B}(t)\hat{A}\rangle_\rho
\displaystyle \sum_{a,b=0,1} \alpha_{\phi_a,\tilde{a}}\cdots\alpha_{\phi_b,b}\,P(\tilde{a},b) = \left\langle\frac{[\hat{B}(t),\hat{A}]}{2i}\right\rangle_\rho = \text{Im}\langle\hat{B}(t)\hat{A}\rangle_\rho

Application: Strongly Measured 4-Point Correlator

JD, et al. PRA 98, 012032 (2018)

4-point temporal qubit out-of-time-ordered correlators may be directly measured using any measurement strength

This directly recovers the desired OTOC and squared commutator connected with information scrambling and chaos

\displaystyle \sum_{a,b,a',b'=0,1} \alpha_{\phi_a,a}\alpha_{\phi_b,b}\alpha_{\phi_{a'},a'}\alpha_{\phi_{b'},b'}\,P(a,b,a',b') = \left\langle\frac{\{\{\{\hat{B}(t),\hat{A}\},\hat{B}(t)\},\hat{A}\}}{2^3}\right\rangle_\rho = \frac{1 + \text{Re}\langle\hat{B}(t)\hat{A}\hat{B}(t)\hat{A}\rangle_\rho}{2} = \frac{1 + \text{Re}F(t)}{2} = 1 - \frac{C(t)}{4}
\displaystyle \sum_{\tilde{a},b,a',b'=0,1} \alpha_{\phi_a,\tilde{a}}\alpha_{\phi_b,b}\alpha_{\phi_{a'},a'}\alpha_{\phi_{b'},b'}\,P(\tilde{a},b,a',b') = \left\langle\frac{[\{\{\hat{B}(t),\hat{A}\},\hat{B}(t)\},\hat{A}]}{2^2(2i)}\right\rangle_\rho = \frac{\text{Im}\langle\hat{B}(t)\hat{A}\hat{B}(t)\hat{A}\rangle_\rho}{2} = \frac{\text{Im}F(t)}{2}

(one reversed evolution needed)

# Conclusions

• Out-of-time-ordered correlators roughly witness information scrambling
• Nested (anti)commutators are measurable for qubits with any strength
• 2- and 4-point qubit correlators can be directly measured

• Follow-up research in preparation:
• Measuring qubit quasiprobabilities strongly with several techniques
• Measuring qutrit correlators with arbitrary strength
• Investigation of out-of-time-ordered correlators experimentally

Thank you!

## Superconducting Qubits

\hbar\omega_q
\hbar(\omega_q - \delta_q)
\hbar(\omega_q - 2\delta_q)

Mesoscopic quantum coherence of collective charge motion at $$\mu$$m scale

EM Fields produced by charge motion described by Circuit QED

Lowest levels of anharmonic oscillator potentials treated as artificial atoms

## Typical Transmon Parameters

\displaystyle \hat{H} = \hbar\hat{\Omega} = \frac{\hbar\omega_q}{2}\hat{\sigma}_z
\hbar\omega_q
\hbar(\omega_q - \delta_q)
\hbar(\omega_q - 2\delta_q)
\omega_q/2\pi \sim \text{4--7GHz}
\delta_q/2\pi \sim \text{100--300MHz}

## "3D Transmon"

\hbar\omega_q
\hbar(\omega_q - \delta_q)
\hbar(\omega_q - 2\delta_q)
\displaystyle \hat{H}_r = \hbar\omega_r\hat{a}^\dagger\hat{a}

Cavity mode:

\omega_r/2\pi \sim \text{5--7GHz}
|\Delta|\equiv|\omega_q-\omega_r| \gg g
\displaystyle \hat{H}_{c} = \hbar g(\hat{a}+\hat{a}^\dagger)\hat{\sigma}_x
g/2\pi \sim \text{100MHz}

Detuned (dispersive) regime (RWA):

\displaystyle \hat{H}_c \sim \hbar \chi\,\hat{a}^\dagger\hat{a}\,\hat{\sigma}_z

X-X Coupling:

\displaystyle \chi \sim \left[\frac{g}{\Delta}\right]^2\!\delta_q \sim 1\text{MHz}
\displaystyle {\small \chi \neq \frac{g^2}{|\Delta|}}

Korotkov group, Phys. Rev. A 92, 012325 (2015)

Martinis group, Phys. Rev. Lett. 117, 190503 (2016)

## 2D planar chip schematic

Bus acts as Purcell filter, coupled to traveling wave parametric amplifier (TWPA)

\omega_r/2\pi \sim \text{5--7GHz}
g/2\pi \sim \text{100MHz}
\displaystyle \chi \sim \left[\frac{g}{\Delta}\right]^2\!\delta_q \sim 1\text{MHz}
\omega_q/2\pi \sim \text{4--7GHz}
\delta_q/2\pi \sim \text{100--300MHz}

Similar parameters:

v1

v3

Coming soon: two-layer design of 20+ qubits
separated from control circuitry
(similar to Google Bristlecone + IBM Q)

Now on v8+

T_2 \sim 100\mu\text{s}
T_1 \sim 60\mu\text{s}

## UCB 2D planar chip

Multiplexed 10 qubit control and readout