Observation of separated dynamics of charge and spin in the Fermi-Hubbard model

Google AI Quantum and collaborators∗(Dated: October 19, 2020)

Strongly correlated quantum systems give rise to many exotic physical phenomena, includinghigh-temperature superconductivity. Simulating these systems on quantum computers may avoidthe prohibitively high computational cost incurred in classical approaches. However, systematicerrors and decoherence effects presented in current quantum devices make it difficult to achieve this.Here, we simulate the dynamics of the one-dimensional Fermi-Hubbard model using 16 qubits ona digital superconducting quantum processor. We observe separations in the spreading velocitiesof charge and spin densities in the highly excited regime, a regime that is beyond the conventionalquasiparticle picture. To minimize systematic errors, we introduce an accurate gate calibrationprocedure that is fast enough to capture temporal drifts of the gate parameters. We also employ asequence of error-mitigation techniques to reduce decoherence effects and residual systematic errors.These procedures allow us to simulate the time evolution of the model faithfully despite having over600 two-qubit gates in our circuits. Our experiment charts a path to practical quantum simulationof strongly correlated phenomena using available quantum devices.

The deceivingly simple Fermi-Hubbard model hasgreatly advanced our understanding of superconductiv-ity, superfluidity, and quantum magnetism in correlatedmaterials [1, 2]. The model is extremely hard to solve onclassical computers in certain regimes, and it is widelyused to benchmark numerical methods for strongly cor-related systems [3]. A remarkable property of the one-dimensional (1D) Fermi-Hubbard model is spin-chargeseparation, i.e., spin and charge excitations travel at dif-ferent speeds due to interparticle interactions [4–6]. Sig-natures of spin-charge separation were observed in solidstate systems using angle-resolved photoemission spec-troscopy [7–9] and tunneling spectroscopy [10–12] as wellas in cold-atom systems [13, 14] using site-resolved quan-tum gas microscopy [15, 16].

Here, we simulate the dynamics of an 8-site 1D Fermi-Hubbard model on a programmable superconductingquantum processor by suddenly removing the trappingpotentials and turning on the on-site interactions. Weobserve separations in the spreading velocities of spinand charge in a regime beyond the low-energy physicsdescribed by the Luttinger liquid theory [17, 18]. Ourplatform enjoys the flexibility that analog devices do not,such as the ability to measure arbitrary observables, re-verse the time evolution, and prepare for various initialstates including BCS-type states [19]. It also has highrepetition rates compared to some other platforms andavoids finite temperature effects.

The key technical advancement that enabled this ex-periment is a calibration protocol that we recently devel-oped for entangling gates, which we call Floquet calibra-tion. It allows for gate parameters to be characterizedrapidly and precisely, which unlocks the ability to com-pensate for systematic errors caused by drifts and fluctu-ations. Floquet calibration is based on the idea that anentangling gate can be uniquely determined by the eigen-

∗ Corresponding author (Z. Jiang): qzj@google.com;Corresponding author (V. Smelyanskiy): smelyan@google.com

values of the composite gates consisting of the entanglinggate and several different sets of single-qubit gates; onegets different snapshots of the entangling gate by chang-ing the parameters in the single-qubit gates. It is also ro-bust to state preparation and measurement errors, shar-ing many common traits with robust phase estimationfor calibrating single-qubit gates [20, 21]. We also showthat decoherence effects can be drastically suppressed byusing a combination of error mitigation schemes. Alongwith our calibration tool, these techniques allow us toincrease the circuit depths (evolution times) by an orderof magnitude. Our work paves the way toward simulat-ing strongly correlated quantum systems [19, 22–32] onexisting digital quantum computers.

I. THE MODEL

Consider the 1D Fermi-Hubbard model on L latticesites with open-boundary conditions,

H = −JL−1∑j=1

∑ν=↑,↓

c†j,νcj+1,ν + h.c.

+ U

L∑j=1

nj,↑nj,↓ +

L∑j=1

∑ν=↑,↓

εj,ν nj,ν , (1)

where cj,ν (c†j,ν) are the fermionic annihilation (creation)operators associated to site number j and spin state ν,and nj,ν = c†j,νcj,ν are the number operators. The hop-ping term with coefficient J in Eq. (1) describes particlestunneling between neighboring sites, the onsite interac-tion term with coefficient U introduces an energy differ-ence for doubly occupied sites, and the term εj,ν repre-sents spin-dependent local potentials. The charge andspin densities are defined as the sum and difference ofthe spin-up and -down particle densities, respectively,

ρ±j = 〈nj,↑ 〉 ± 〈nj,↓ 〉 . (2)

arX

iv:2

010.

0796

5v1

[qu

ant-

ph]

15

Oct

202

0

2

a 1: J odd

1↑2↑

3↑4↑

5↑6↑

7↑8↑1↓

2↓3↓

4↓5↓

6↓7↓

8↓

2: U odd

1↑2↑

3↑4↑

5↑6↑

7↑8↑1↓

2↓3↓

4↓5↓

6↓7↓

8↓

3: iSWAP 5: J even & iSWAP†

1↑2↑

3↑4↑

5↑6↑

7↑8↑1↓

2↓3↓

4↓5↓

6↓7↓

8↓

4: U even

1↑3↑

2↑5↑

4↑7↑

6↑8↑1↓

3↓2↓

5↓4↓

7↓6↓

8↓

b

K =

1 0 0 00 cos θ −i sin θ 00 −i sin θ cos θ 00 0 0 1

G =

1 0 0 00 cos θ − sin θ 00 sin θ cos θ 00 0 0 1

φ=

1 0 0 00 1 0 00 0 1 00 0 0 e−iφ

iSWAP

=

1 0 0 00 0 i 00 i 0 00 0 0 1

c

1↑...

8↑1↓...

8↓

Initial

Initial Trotter

...

...

η times

dInitial state preparation

X

XG

G

G

G

G

G

G

G

G

G

G

G

e One Trotter step1↑2↑3↑4↑5↑6↑7↑8↑

1↓2↓3↓4↓5↓6↓7↓8↓

K

K

K

K

K

K

K

K

φ

φ

φ

φ

iSWAP

iSWAP

iSWAP

iSWAP

iSWAP

iSWAP

φ

φ

φ

φ

K

K

K

K

K

K

J odd U odd iSWAP U even J eveniSWAP†

1

FIG. 1. Qubit layouts and quantum circuits. a. The 5 stages in a Trotter step, where the orange and green qubitsrepresent spin-up and -down fermionic sites, respectively. In stage 1 and 2, the blue and red edges represent the hopping andinteraction terms, respectively. In stage 3, we change the positions of the odd and even sites by applying the iSWAP gatesacross the blue edges, which allows for implementing U even in the next stage. In stage 5, we swap the sites back to theiroriginal positions using the iSWAP† gates, which are combined with the J even terms. b. The matrix representations of thetwo-qubit gates. The Givens rotation gate G (yellow) is used to prepare the initial state. The iSWAP-like gate K (blue) andthe CPHASE gate (red) are used to implement the time evolution under the hopping and interaction terms, respectively. TheiSWAP gate (green) is a special case of K(θ), with θ = −π/2. In Supplementary Fig. S1, we show that any of the four gates canbe decomposed into two K(π/4) gates and several single-qubit gates. c. The entire quantum circuit includes an initializationpart, η Trotter steps, and measurements in the Pauli-Z basis. d. The circuit to prepare the ground state of a noninteractingHamiltonian with two particles (excitations), where the angles of the Givens rotations can be determined using an efficientclassical algorithm. e. The quantum circuit to implement one Trotter step of the model.

We map the fermionic operators to qubit operators us-ing the Jordan-Wigner transformation (JWT) for eachspin state, cj,ν 7→ 1

2 (Xj,ν + iYj,ν)Z1,ν · · ·Zj−1,ν , whereXj,ν , Yj,ν , and Zj,ν are the Pauli operators. Under theJWT, the unoccupied and occupied spin orbitals are rep-resented by the qubit states | 0 〉 and | 1 〉, respectively.We use the product formula [33], i.e., Trotter steps, tosimulate the time evolution of the system, where eachterm in the Hamiltonian (1) is implemented separately.A single Trotter step is implemented with the 5 stagesdepicted in Fig. 1a, where each spin state of the model ismapped to a zigzag chain of 8 qubits; this optimizes thecircuit depths under the geometric constraints.

Under the JWT, the hopping term c†j,νcj+1,ν + h.c. ismapped to 1

2 (Xj,νXj+1,ν + Yj,νYj+1,ν). Its time evolu-tion can be implemented using the two-qubit gate K(θ)in Fig. 1b. This gate is used in the first and last stagesin the circuit depicted in Fig. 1e. In the first stage, weset θ = −τ J/h to implement a time step of length τ .In the last stage, we set θ = −τJ/h + π/2, where theextra angle π/2 is used to undo the iSWAP gates in thethird stage to change the positions of the fermionic sites.The gate K(θ) with arbitrary θ can be decomposed intotwo K(π/4) =

√iSWAP

† gates and several single-qubitZ rotations, see Supplementary Fig. S1c. Our hardwarenative two-qubit gate takes the form K(ϑ) CPHASE(ϕ),

where ϑ ≈ π/4 and the parasitic controlled phase ϕ <∼π/20. At the time we took the data, the means andstandard deviations of the two parameters across dif-ferent pairs of qubits were ϑ = 0.783± 0.012 rad andϕ = 0.138± 0.015 rad. The CPHASE(ϕ) term introducesan interaction term between neighboring fermionic sitesV nj,ν nj+1,ν with V = 2hϕ/τ . It has sizable effects forlonger evolution times, and we include it in our numericalsimulations to compare to experimental results. The en-tanglement part in our native two-qubit gate takes about12 ns and is preceded by single-qubit Z rotations, whichtake 10 ns with a 5 ns padding on each side. Therefore,one hopping term takes about 2×32 ns = 64 ns to imple-ment on the hardware.

The time evolution of the on-site interaction termnj,↑nj,↓ can be implemented using CPHASE(φ) gate withφ = τU/h. It can be decomposed exactly into two na-tive two-qubit gates and single-qubit X and Z rotations,see Supplementary Information A. There are three layersof X rotations (microwave gates) in the composite gate,each taking about 25 ns. Therefore, the entire compos-ite CPHASE gate takes about 139 ns to implement. Asshown in Fig. 1a, we implement the CPHASE gate onthe odd and even sites separately due to geometry con-straints. Idling qubits are susceptible to crosstalk andlow-frequency noises in the Z basis, and we mitigate themby applying spin echos consisting of pairs of X gates (not

3

0 1 2 3

6

8

10

Cha

rge

spre

adκ

+

u = 0

Charge

Spin

0 1 2 3

Time (h/J)

0

2

Spr

ead

rate

(J/h

) u = 0

Charge

Spin

0 1 2 3

u = 1

0 1 2 3

Time (h/J)

u = 1

0 1 2 3

u = 2

0 1 2 3

Time (h/J)

u = 2

0 1 2 3

u = 3

-2

0

2

Spi

nsp

read

κ−

0 1 2 3

Time (h/J)

u = 3

0.2

0.5

0.8

1.0C

harg

ede

nsit

yρ

+t = 0 t = 1.2h/J

1 2 3 4 5 6 7 8

Site index

0.2

0.5

0.8

1.0

Cha

rge

dens

ityρ

+

t = 1.8h/J

1 2 3 4 5 6 7 8

Site index

t = 3h/J

-0.2

0.0

0.2

0.5

Spi

nde

nsit

yρ−

-0.2

0.0

0.2

0.5

Spi

nde

nsit

yρ−Charge

Spin

a b

c

FIG. 2. Separation in charge and spin densities. We initialize the quantum state with N↑ = N↓ = 2 (quarter filling),where the charge and spin densities ρ±j = 〈nj,↑ 〉 ± 〈nj,↓ 〉 are peaked around the middle sites. We then evolve the stateunder the Fermi-Hubbard Hamiltonian (1) with Trotter step length τ = 0.3h/J . Points and solid lines represent experimentaland numerical (exact) results, respectively. a. Time evolved charge (blue) and spin (red) densities for u ≡ U/J = 3 withtJ/h = 0, 1.2, 1.8, 3 (corresponding to Trotter numbers η = 0, 4, 6, 10), where the error bars represent the standard error of themean over 16 simulations with different choices of qubits and their arrangements, see Supplementary Fig. S8; the uncertaintiesdue to finite sample sizes are much smaller (omitted on the plots). The charge density spreads faster than the spin density andreaches the boundaries earlier. b. The charge and spin spreads κ± =

∑j

∣∣j − (L+ 1)/2∣∣ ρ±j as functions of the evolution time.

For u = 0, they almost lay on top of each other; the small discrepancy is due to the parasitic CPHASE in our native gate. Incomparison, they are well separated for larger interaction strengths u ≥ 1. c. The numerical derivatives of κ± with respect tothe evolution time.

shown in Fig. 1).We initialize the system into the ground state of a

non-interacting fermionic Hamiltonian using networksof Givens rotations [25], i.e., two-mode fermionic basistransformations. The Givens rotation takes the matrixform G in Fig. 1b when acting on neighboring qubits.It can also be decomposed into two K(π/4) gates andsingle-qubit Z rotations, see Supplementary Fig. S1b.By parallelizing the Givens rotations, the ground state ofan arbitrary L-mode non-interacting Hamiltonian can beprepared in circuit depth O(L) [19, 27]. Recently, theGivens rotation network was successfully used to vari-ationally construct a chemically-accurate Hartree-Fockstate [34]. Here we use the OpenFermion code [35] basedon the scheme in [19], which requires ∼L2/4 Givens rota-tions with circuit depth ∼L near half filling. In Fig. 1d,we plot the initialization circuit for two fermions.

II. SEPARATION OF SPIN AND CHARGEVELOCITIES

In the Luttinger liquid description of the 1D Fermi-Hubbard model [36–38], low-energy charge and spin ex-citations propagate at different characteristic velocities;see Supplementary Information B. It is based on the as-sumption that the system is close to its ground state.Here we observe separations in the dynamics of chargeand spin densities in a highly excited regime, where theLuttinger liquid theory does not formally apply.

Consider an 8-site 1D Fermi-Hubbard system with Nν

particles in the spin state ν. We prepare the initial state|ψ0 〉 as the ground state of an non-interacting Hamil-tonian H0 by setting U = 0 in Eq. (1). The localpotentials in H0 are chosen to have a Gaussian formεj,ν = −λν e−

12 (j−mν)2/σ2

ν , where λν , mν , and σν set themagnitude, center, and width of the potentials, respec-tively. We set the parameters of the spin-up Gaussianpotential to λ↑ = 4, m↑ = 4.5, and σ↑ = 1 while leavingthe spin-down potential to zero. This generates initialcharge and spin density peaks in the middle of the chain,see subplot t = 0 in Fig. 2a.

We then evolve the system under the Hamiltonian (1)with the Trotter step described in Fig. 1e by setting thetime step length to τ = 0.3h/J and the local potentialsto εj,ν = 0. In Fig. 2a, we plot the distributions of thecharge and spin densities at several evolution times forN↑ = N↓ = 2 and u ≡ U/J = 3, where the dynam-ics of the charge and spin degrees of freedom are sepa-rated. We leave more detailed results for this case andthe N↑ = N↓ = 3 case in Supplementary Figs. S12 andS13, respectively.

To quantify the degree that charge and spin densitiesspread from the middle of the chain, we introduce

κ±η =

L∑j=1

∣∣j − (L+ 1)/2∣∣ ρ±j,η , (3)

where ρ±j,η = 〈nj,↑ 〉 ± 〈nj,↓ 〉 are the charge and spindensities after η Trotter steps. In Fig. 2b, we plot κ±as functions of the evolution time for several interactionstrengths u. When u = 0, they nearly coincide with each

4

0 3 6 9 12

Time (h/J)

1

3

5

7

Ave

rage

pos

itio

n

0 3 6 9 12

Time (h/J)0 3 6 9 12 15 18

Time (h/J)

Up meanDown mean

0 3 6 9 12 15 18

Time (h/J)

Final

2 4 6 8

Site index

0

3

6

9

12

15

18T

ime

(h/J

)Experiment ↑

2 4 6 8

Site index

Numerics ↑

2 4 6 8

Site index

Experiment ↓

2 4 6 8

Site index

Numerics ↓

0.0

0.4

0.8

Up

dens

ity

0.0

0.4

0.8

Dow

nde

nsit

y

ba Post selection Floquet calibration Rescaling

FIG. 3. Noninteracting time evolution and error mitigation. a. Time evolution of the particle densities 〈nj,ν 〉, wherej = 1, . . . , 8 and ν =↑, ↓. The experiment results match well with the numerics (exact) for t <∼ 16.5h/J , corresponding to acircuit depth of 228 layers of two-qubit gates with execution time 7.3µs. b. Demonstrating error mitigation schemes usingthe average positions

∑j〈nj,ν 〉 j, where the solid lines represent numerical results for the spin-up (yellow) and -down (green)

particles. The triangles and the shaded areas represent the means and sample standard deviations over 16 simulations withdifferent choices of qubits and their arrangements, see Supplementary Fig. S8. This procedure, which we call qubit assignmentaveraging, removes inhomogeneous effects in the system. Decoherence due to loss of excitations (T1 errors) can be removed bypostselecting the measurement results with the correct numbers of excitations. Floquet calibration improves both the meansand standard deviations of the average positions. Crucially, the particle density distributions obtained after the calibrationare similar to the exact solutions with damped amplitudes. The damping factor is a function of the evolution time t; its valueis approximately 1 at t = 0 and 0.16 at t = 16.5h/J . This allows us to get excellent agreement with theory predictions byrescaling.

other; the small separation is the result of the nearest-neighbor interaction term V nj,ν nj+1,ν , caused by theparasitic CPHASE in our native gate. The gaps betweenκ+ and κ− widen as u increases, demonstrating increasedseparations as the system goes into the strongly interact-ing regime. The charge spread κ+ reaches the maximumvalue after it has fully hit the boundaries. In comparison,κ− increases at significantly lower rates for all u 6= 0 andnever fully reaches the boundaries within the maximumevolution time.

In Fig. 2c, we plot the numerical derivatives of thecharge and spin spreads κ±. Because our initial wave-function is real, both charge and spin currents equal zeroat t = 0. The small nonzero values of the observed ini-tial rates of charge and spin spreads are due to the finiteTrotter step length and the parasitic CPHASE in the ini-tialization circuit. The charge spreading rate graduallyincreases until the particles starts to hit the boundaries.As the interaction strength u increases, the maximumspin spread rate decreases. In comparison, the maximumcharge spread rate roughly keeps the same.

III. ERROR MITIGATION AND CALIBRATION

To reach the desired circuit depths, we employ a com-bination of error mitigation and calibration schemes. Toillustrate that, we consider the case where there is exactlyone particle in each spin state N↑ = N↓ = 1. We initial-ize the spin-up (down) particle into a left (right)-movingGaussian wavepacket. This can be achieved by first creat-ing a real Gaussian wavepacket with the Givens rotationsand then generating a phase gradient using single-qubit Zrotations. We evolve the system under the hopping termsby including only the first and last stages in Fig. 1e with

time step length τ = 0.3h/J . In Fig. 3a, we comparethe numerics with experimental results after applying allthe error mitigation schemes. They match well with eachother up to t ≈ 16.5h/J (or 55 Trotter steps), and theclear interference patterns at larger times indicate thatthe evolution is coherent instead of diffusive. The twospin states evolve independently here, i.e., there is nogate between the corresponding sets of qubits. However,we can account for the effects of crosstalks more properlyby including the two spin states. The impact of each stepin our error mitigation procedure is illustrated in Fig. 3b,which we describe below one by one.

The quantum circuit for the Fermi-Hubbard modelconserves the total number of excitations for each spinstate. However, excitations can leave the system dueto interactions with the environment. In SupplementaryFig. S10, we plot the T1 map of the device at the timethat the final data were taken. This kind of error can beremoved by postselecting the measurement results withthe correct numbers of excitations. It also reduces statepreparation and measurement (SPAM) errors to some ex-tent. The postselection success rates of the noninteract-ing instance shown in Fig. 3 are about 0.5 for η = 0 and0.2 for η = 55, where η is the number of Trotter steps.In Fig. 4a, we plot the success rates as functions of theevolution time. We take 20,000 samples for each circuitto make sure that the uncertainties of the expectationvalues are low after postselection.

Systematic errors are especially detrimental to quan-tum computation as their effects can add up coherently.In addition to our routine calibration, we also use Flo-quet calibration, a fast calibration method that we re-cently developed for characterization of entangling gates.To capture crosstalks, we calibrate the two-qubit gatesin each configuration in Fig. 1a by applying them si-

5

1 2 3 4 5

u

0.9

1.0

1.1

b

0.05

0.1

a

0 1 2 3

Time (h/J)

0.0

0.1

0.2

0.3

0.4

0.5S

ucce

ssra

te

u = 0

u = 1, 2, 3, 4

a b

FIG. 4. Postselection success rate and rescaling pa-rameters. a. The post selection success rates as functionsof the evolution time. The squares denote the results for thenoninteracting case in Fig. 3, which decays much slower dueto the reduced number of gates in each Trotter step. Thehorizontal and vertical bars denote results for the interactingcase in Fig. 2 with different values of u; the cross formationof the horizontal and vertical bars shows that the success ratedoes not depend on u. b. The rescaling parameters a and bin Eq. (4) for the case in Fig. 2. Their values barely dependon the interaction strength u, allowing one to infer them bycomparing numerical and experimental results in the weaklyinteracting regime.

multaneously; the results are then used to correct thetwo-qubit gates in the quantum circuit within the sameconfiguration. Floquet calibration can determine mostparameters of the two-qubit gates to an uncertainty ofless than 10−3 rad under one minute, sufficiently fast tocharacterize errors due to drifts and fluctuations in thecontrol fields and qubit frequencies. It is also robustto state preparation and measurement (SPAM) errors ingeneral. In Supplementary Fig. S2, we plot the parame-ters in our native two-qubit gate obtained using Floquetcalibration during a period of several hours. We alsoplot the two-qubit gate fidelity map from our routinecalibration using cross-entropy benchmarking (XEB) inSupplementary Fig. S11; more about XEB and our cali-bration process can be found in Supplementary Informa-tion in Ref. [39]. We leave detailed discussions on theimplementations and properties of Floquet calibration inSupplementary Information C.

Inhomogeneities in gate parameters and decoherencerates are common in quantum computing devices. Theymake the experiment results unpredictable and imple-mentation dependent. We solve this issue using qubitassignment averaging, where experiment results are av-eraged over 16 different realizations, see SupplementaryFig. S8. In each realization, we either choose a differ-ent set of qubits, arrange the qubits differently, or doboth. In Fig. 3b, we show that the averaged results aresmooth even if the outcomes from individual implemen-tations fluctuate significantly. More importantly, averag-ing makes it possible to describe the simulation resultsusing models with randomized parameters. The valuesof the observables are often damped in a predictable wayin these models, making it possible for further mitigationof the errors.

Finally, we rescale the damped expectation values〈nj,ν 〉, leading to excellent agreement with theoretical

Case tevol(h/J)

tcircuit(µs)

Circuitdepths

2-qubitcounts

µ-wavecounts

RZcounts

U 6= 0NP = 4,6

1.5 2.8 159 328 364 5663.0 5.2 289 608 724 1056

U = 0NP = 2

9.0 4.1 257 434×2 2 836×2

16.5 7.3 457 784×2 2 1511×2

TABLE I. Circuit statistics. Circuit statistics for the in-teracting case U 6= 0 and the noninteracting case U = 0 withdifferent numbers of particles NP , where tevol and tcircuit arethe Hamiltonian evolution time and circuit execution time, re-spectively. The circuit depths include the contributions fromthe two-qubit gates, microwave gates (µ-wave), and single-qubit Z rotations. We also count the total numbers of theconstituent gates. The microwave gates are single-qubit rota-tions along axes on the X-Y plane, which are used in the in-teraction terms and the initial state preparations. The single-qubit Z rotations (RZ) are always bundled with our two-qubitgates and do not require extra time to implement.

predictions. We choose the fiducial point for rescalingto be nν = Nν/L, i.e., the averaged particle density forthe spin state ν. We observe that the damping factor isapproximately linear in the number of Trotter steps η,

〈nj,ν 〉exp − nν〈nj,ν 〉num − nν

≈ b− aη , (4)

where exp and num stand for experimental and numericalresults, respectively. The parameter a (b) describes thedamping effect of the Trotter steps (initial state prepa-ration circuit). The linear relation fits the experimentalresults well when the damping factor is >∼ 0.2 for bothnoninteracting and interacting cases, see SupplementaryFig. S9. The fitted values of a and b hardly depend onthe interaction strength U , see Fig. 4b. Therefore, wecan estimate their values by comparing the experimen-tal and numeric results in a regime that is easy to solveclassically, e.g., the weak interaction regime. The linearrelation (4) is not essential to our rescaling procedure.However, the weak dependence of the damping factor onthe interaction strength U is crucial.

IV. CONCLUSION AND OUTLOOK

Using a combination of the error mitigation and cali-bration schemes, we have extended our quantum circuitsto unprecedented depths, see statistics in Table I. Thisopens the possibility of simulating strongly correlatedsystems on current quantum computing devices, suchas the classically hard 2D Fermi-Hubbard model. Therecipe for error mitigation that we use here can also beuseful to many other applications, including the varia-tional quantum eigensolver (VQE) [40] and quantum ap-proximate optimization algorithm (QAOA) [41, 42]. Wealso expect our calibration technique to play a centralrole in quantum device characterization and Hamiltonianlearning.

6

V. AUTHOR CONTRIBUTIONS

Z. Jiang and V. Smelyanskiy designed the experi-ment; Z. Jiang and W. Mruczkiewicz developed the code,collected the data, and wrote the paper; L. Ioffe, K.Kechedzhi, and V. Smelyanskiy assisted with the phys-ical theory of the model; R. Babbush, S. Boixo, J. Mc-Clean, and N. Rubin contributed to the algorithmicpart of the project; Z. Jiang and V. Smelyanskiy de-veloped the theory for Floquet calibration; Y. Chen, Z.Jiang, W. Mruczkiewicz, C. Neill, M. Niu, and XiaoMi implemented the Floquet calibration; S. J. Cotton,C. Mejuto-Zaera, P. Schmitteckert, N. Tubman, and N.Vogt helped with numerical simulations of the model; J.Gross, J. Martinis, P. Roushan, and X. Mi helped forimproving the presentations of the results. Experimentswere performed—through Google’s Quantum ComputingService—using a quantum processor that was recentlydeveloped and fabricated by a large effort involving theentire Google AI Quantum team.

VI. ACKNOWLEDGEMENTS

Dave Bacon is a CIFAR Associate Fellow in the Quan-tum Information Science Program. ZJ would like to

thank Philipp Hauke for his helpful comments on themanuscript. Funding: This work was supported byGoogle LLC. N.M.T and S.J.C are grateful for supportfrom NASA Ames Research Center as well as supportfrom the AFRL Information Directorate under grantF4HBKC4162G001. Some calculations were performedas part of the XSEDE computational Project No. TG-MCA93S030. Competing Interests: The authors de-clare no competing interests. Supplementary Informa-tion is available for this paper. Data and materialsavailability: The code used for this experiment and atutorial for running it can be found in the open sourcelibrary ReCirq, located at https://doi.org/10.5281/zenodo.4091471. All data needed to evaluate the con-clusions in the paper are present in the paper or theSupplementary Information. Data presented in the fig-ures can be found in the Dryad repository located athttps://doi.org/10.5061/dryad.crjdfn32v.

Google AI Quantum and Collaborators

Frank Arute1, Kunal Arya1, Ryan Babbush1, Dave Bacon1, Joseph C. Bardin1, 2, Rami Barends1, AndreasBengtsson1, Sergio Boixo1, Michael Broughton1, Bob B. Buckley1, David A. Buell1, Brian Burkett1, NicholasBushnell1, Yu Chen1, Zijun Chen1, Yu-An Chen1, 3, Ben Chiaro1, 4, Roberto Collins1, Stephen J. Cotton5, 6,William Courtney1, Sean Demura1, Alan Derk1, Andrew Dunsworth1, Daniel Eppens1, Thomas Eckl7, CatherineErickson7, Edward Farhi1, Austin Fowler1, Brooks Foxen1, Craig Gidney1, Marissa Giustina1, Rob Graff1,Jonathan A. Gross1, 8, Steve Habegger1, Matthew P. Harrigan1, Alan Ho1, Sabrina Hong1, Trent Huang1, WilliamHuggins1, Lev B. Ioffe1, Sergei V. Isakov1, Evan Jeffrey1, Zhang Jiang1, Cody Jones1, Dvir Kafri1, KostyantynKechedzhi1, Julian Kelly1, Seon Kim1, Paul V. Klimov1, Alexander N. Korotkov1, 9, Fedor Kostritsa1, DavidLandhuis1, Pavel Laptev1, Mike Lindmark1, Erik Lucero1, Michael Marthaler10, Orion Martin1, JohnM. Martinis1, 4, Anika Marusczyk7, Sam McArdle1, 11, Jarrod R. McClean1, Trevor McCourt1, Matt McEwen1, 4,Anthony Megrant1, Carlos Mejuto-Zaera12, Xiao Mi1, Masoud Mohseni1, Wojciech Mruczkiewicz1, Josh Mutus1,Ofer Naaman1, Matthew Neeley1, Charles Neill1, Hartmut Neven1, Michael Newman1, Murphy Yuezhen Niu1,Thomas E. O’Brien1, Eric Ostby1, Bálint Pató1, Andre Petukhov1, Harald Putterman1, Chris Quintana1,Jan-Michael Reiner10, Pedram Roushan1, Nicholas C. Rubin1, Daniel Sank1, Kevin J. Satzinger1, VadimSmelyanskiy1, Doug Strain1, Kevin J. Sung1, 13, Peter Schmitteckert10, Marco Szalay1, Norm M. Tubman5, AmitVainsencher1, Theodore White1, Nicolas Vogt10, Z. Jamie Yao1, Ping Yeh1, Adam Zalcman1, Sebastian Zanker10

1 Google Research2 Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA3 Department of Physics, California Institute of Technology, Pasadena, CA4 Department of Physics, University of California, Santa Barbara, CA5 Quantum Artificial Intelligence Laboratory (QuAIL), NASA Ames Research Center, Moffett Field, CA6 KBR, 601 Jefferson St., Houston, TX 770027 Robert Bosch GmbH, Robert-Bosch-Campus 1, 71272 Renningen, Germany8 Institut quantique and Départment de Physique, Université de Sherbrooke, Québec J1K 2R1, Canada9 Department of Electrical and Computer Engineering, University of California, Riverside, CA10 HQS Quantum Simulations GmbH, Haid-und-Neu-Straße 7, 76131 Karlsruhe, Germany11 Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom

7

12 Department of Chemistry, University of California, Berkeley, CA13 Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI

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9

Supplementary Information

A GATE DECOMPOSITIONS

In Fig. S1, we provide decompositions of the parame-terized two-qubit gates used in this work into the stan-dard

√iSWAP

† gate, i.e., K(π/4). Our native two-qubitgate is close to K(π/4) and can be better described byK(ϑ) CPHASE(ϕ), where ϑ ≈ π/4 and ϕ ≈ π/23. Herewe provide a derivation on decomposing the CPHASE gateinto two native two-qubit gates and several single-qubitgates. To simplify notations, we introduce the gate

F (ϑ, ϕ) = e−iϑ(X⊗X+Y⊗Y )/2−iϕZ⊗Z/4 , (1)

which is related to our native gate by Z rotations

K(ϑ) CPHASE(ϕ) = e−iϕ/4 eiϕ(Z1+Z2)/4 F (ϑ, ϕ) , (2)

where Z1 = Z ⊗ I and Z2 = I ⊗Z. The sign of ϑ can beflipped by using single-qubit Z gates,

Z1 F (ϑ, ϕ)Z1 = Z2 F (ϑ, ϕ)Z2 = F (−ϑ, ϕ) . (3)

Sandwiching a microwave gate with two two-qubit gatesof opposite ϑ, we have

F (−ϑ, ϕ) eiαX1F (ϑ, ϕ) = Γ1 ⊗ I − iZ ⊗ Γ2 . (4)

where X1 = X ⊗ I and the Schmidt operators are

Γ1(α) = cosα cos(ϕ/2) I + i sinα cosϑX , (5)

Γ2(α) = cosα sin(ϕ/2)Z − sinα sinϑY , (6)

with the Schmidt coefficients ‖Γ1‖ and ‖Γ2‖. The uni-tary (4) is equivalent to a CPHASE gate up to single-qubitgates,

CPHASE(φ) = e−iφ(I−Z)⊗(I−Z)/4 , (7)

which has two non-zero Schmidt coefficients |cos(φ/4)|and |sin(φ/4)|. We require ‖Γ2(α)‖ = |sin(φ/4)| to matchthe Schmidt coefficients of the two unitaries, which yields

sinα =

√sin(φ/4)2 − sin(ϕ/2)2

sin(ϑ)2 − sin(ϕ/2)2. (8)

This equation can be solved when one of the followingtwo conditions is satisfied

|sinϑ| ≤ |sin(φ/4)| ≤ |sin(ϕ/2)| , (9)

|sin(ϕ/2)| ≤ |sin(φ/4)| ≤ |sinϑ| . (10)

For the parameters in our native gate, these conditionscan be simplified to |φ| ≥ 2|ϕ|. To match the Schmidtoperators on the first qubit, we introduce twoX rotationswith the same angle

RX(ξ1) Γ1(α)RX(ξ1) = cos(φ/4) I , (11)

RX(ξ1)Z RX(ξ1) = Z , (12)

where RX(ξ) = e−iξX/2 and

ξ1 = tan−1

(tanα cosϑ

cos(ϕ/2)

)+π

2

(1− sgn

(cos

ϕ

2

)). (13)

To match the Schmidt operators on the second qubit, weintroduce two X rotations with opposite angles

RX(−ξ2) Γ2(α)RX(ξ2) = sin(φ/4)Z , (14)

where

ξ2 = tan−1

(tanα sinϑ

sin(ϕ/2)

)+π

2

(1− sgn

(sin

ϕ

2

)). (15)

Put everything together, we have

RX(ξ1,−ξ2)F (−ϑ, ϕ) eiαX1 F (ϑ, ϕ)RX(ξ1, ξ2)

= cos(φ/4) I ⊗ I − i sin(φ/4)Z ⊗ Z= eiφ/4 e−iφ(Z1+Z2)/4 CPHASE(φ) , (16)

where RX(ξ1, ξ2) = RX(ξ1) ⊗ RX(ξ2). This implementsthe desired CPHASE gate up to single-qubit Z rotations.

B SPIN-CHARGE SEPARATION BYBOSONIZATION

The bosonization theory discussed here only appliesto low-energy and long-wavelength excitations in the 1DFermi-Hubbard model, whereas the quenched dynamicspresented in the main text involves highly excited stateswith short wavelengths. As a result, the theory can onlybe used to explain the findings in the main text qualita-tively.

The dispersion relation of a particle in a 1D homo-geneous quantum liquid is linear εk = ±vfk, and boththe spin and charge excitations travel at the Fermi ve-locity vc = vs = vf . For nonzero interaction strengths,the spin and charge-wave packages propagate at differ-ent velocities. As was proposed by Haldane [43, 44], the1D spin-1/2 Fermi gas can be mapped to an effectivehydrodynamic Hamiltonian, which describes the originalsystem faithfully at wavelengths much larger than the in-terparticle spacing. This theory of noninteracting bosonsis called Luttinger liquid, where all correlation functionscan be exactly calculated. This Hamiltonian takes theform

H =∑α=c,s

∫dx

hvα2

[KαΠ2

α +1

Kα

(∂xφα

)2], (17)

where φα is a bosonic field operator and Πα its conjugatemomentum operator. The low-energy physics is com-pletely characterized by the phenomenological parame-ters: the density-wave velocity vα and Luttinger param-eter Kα, which depends on the interaction [45]. Thesingle particle spectral function of 1D Fermi liquid hastwo power-law singularities for the spin and charge exci-tations respectively [46, 47].

10

a

√iSWAP

=Z

√iSWAP

†

Z

b

G(θ) =

√iSWAP RZ(−θ)

RZ(θ) √iSWAP

†

c

K(θ) =RZ

(−π4

)

RZ(π4

)

√iSWAP RZ(θ)

RZ(−θ) √iSWAP

† RZ(π4

)

RZ(−π4

)

d

CPHASE(φ) =RZ

(−φ2

)

RZ(−φ2

)RX(ξ1)

RX(ξ2) √iSWAP

† RX(−2α)

√iSWAP RX(ξ1)

RX(−ξ2)

1

FIG. S1. Gate decomposition. Decomposition of various two-qubit gates into single-qubit gates and two√

iSWAP† ≡

K(π/4) gates, an approximate description of our native two-qubit gate. a. The two gates√

iSWAP and its inverse (Hermitianconjugate) are equivalent up to single-qubit Pauli-Z gates. b. Decomposition of the Givens rotation G(θ) for initial statepreparations, where RZ(θ) = exp(−iθZ/2). c. Decomposition of the K(θ) = e−iθ(X⊗X+Y⊗Y )/2 gate. d. Decomposition of theCPHASE (φ) = diag(1, 1, 1, e−iφ) gate, where RX(ξ) = exp(−iξX/2) and the parameters ξ1, ξ2, and α are functions of φ.

C FLOQUET CALIBRATION

Calibration of quantum gates is one of the most cru-cial steps in achieving high-fidelity quantum computationand its large-scale deployment [48–50]. Temporal insta-bilities, including drifts and fluctuations in the controlfields and qubit frequencies [51–58], can propagate andaccumulate coherently in large quantum circuits. There-fore, it is crucial to develop fast and accurate calibra-tion methods to characterize and mitigate these errors.However, common calibration tools, such as randomizedbenchmarking [59, 60], compressed sensing [61, 62], gateset tomography [63, 64], and cross-entropy benchmark-ing [65] are often too slow to capture drifts and fluctua-tions in the hardware.

Quantum metrology [66, 67] offers a route to this goal.The Heisenberg limit O(1/n) sets a fundamental lowererror bound in phase estimation with n photons [68–71], whereas the standard quantum limit O(1/

√n) refers

to the minimum uncertainty allowed by using semi-classical states. Modified versions of the quantum phase-estimation algorithm [72–74] have been shown to reachHeisenberg scaling theoretically [75–80] and experimen-tally [81, 82]. Based on this idea, Kimmel et al. [83, 84]proposed a protocol to characterize the axis and angle ofa single-qubit rotation. It achieves uncertainty O(1/n)by repeating identical operations O(n) times.

Here we describe a fast and accurate calibration pro-tocol for entangling gates. It is based on the idea thatan entangling gate can be uniquely determined by theeigenvalues of the composite gates making up the entan-gling gate and different sets of single-qubit gates. Byrepeating the composite gate n times in a quantum cir-cuit, small changes in the gate parameters are amplifiedn times. Repeating the cycle unitary for many times alsomakes the protocol robust to state preparation and mea-surement (SPAM) errors. In these respects, our protocolresembles gate set tomography, with the composite gatesplaying the role of “germs” therein, though by prioritiz-ing the errors we wish to calibrate and leveraging wellthe form of single-qubit gates we require far fewer re-sources than is typical for gate set tomography. We show

that this protocol can be implemented both adaptivelyand non-adaptively. In Sec. C1, we introduce the proce-dure of characterizing two-qubit gates that conserves thenumbers of excitations. In Subsec. C2, we discuss theeffects of decoherence on the ultimate precision of theprocedure. In Subsec. C3, we study how to best choosethe cycle repetition numbers. In Subsec. C4, we showthat in principle our procedure can be applied to generalmulti-qubit gates.

C1 Excitation-number-conserving gates

The most general excitation-number-conserving two-qubit gate takes the following form with the basis statesin the order | 00 〉, | 01 〉, | 10 〉, and | 11 〉,

U(θ, ζ, χ, γ, φ) =1 0 0 0

0 e−i(γ+ζ) cos θ −i e−i(γ−χ) sin θ 0

0 −i e−i(γ+χ) sin θ e−i(γ−ζ) cos θ 0

0 0 0 e−i(2γ+φ)

, (18)

where 0 ≤ θ ≤ π/2 is the iSWAP angle, φ is thecontrolled-phase angle, and ζ, χ, and γ are single-qubitphase angles. In Fig. S2, we plot the values of theparameters of our hardware native gate (except for χ)obtained by Floquet calibration over a time period ofseveral hours. We denote the single-qubit Z rotationas RZ(z) = diag(1, eiz), equivalent to the definitionRZ(z) = diag(e−iz/2, eiz/2) in Cirq [85] up to an overallphase. Single-qubit Z rotations acting on two qubits takethe form

RZ(z1, z2) = diag(1, eiz2 , eiz1 , ei(z1+z2)

)(19)

= U(0, z−, 0,−z+, 0) , (20)

where z± = (z1 ± z2)/2. The general number-conservinggate defined in Eq. (18) can be decomposed into the se-

11

0.76

0.78

0.80

θ(r

ad)

0.02

0.04

0.06

ζ(r

ad)

0 60 120 180 240

Time (min)

0.12

0.14

0.16

φ(r

ad)

0 60 120 180 240

Time (min)

−0.12

−0.10

−0.08

γ(r

ad)

FIG. S2. The values of the four (out of five) parameters ofour hardware native gate obtained by Floquet calibration.The mechanism of the oscillations in the parameters ζ andγ is still not completely clear, but likely due to fluctuationsin temperature. Since the changes of these two parametersare significant, they must be corrected in real time to obtaindesired results.

quence

U(θ, ζ, χ, γ, φ) =

RZ(−γ,−γ)RZ(β,−β)U(θ, 0, 0, 0, φ)RZ(α,−α) , (21)

where α = (ζ + χ)/2, β = (ζ − χ)/2. It also takes theblock diagonalized form,

U = diag(

1, e−iγu(θ, ζ, χ), e−i(2γ+φ)), (22)

where the 2× 2 matrix u reads

u(θ, ζ, χ) =

(e−iζ cos θ −i eiχ sin θ

−i e−iχ sin θ eiζ cos θ

)(23)

= I cos Ω(θ, ζ)− i σ(θ, ζ, χ) sin Ω(θ, ζ) , (24)

where Ω(θ, ζ) = arccos(cos θ cos ζ) ∈ [0, π] is the Rabiangle and the idempotent matrix σ reads

σ(θ, ζ, χ) =((X cosχ− Y sinχ) sin θ + Z cos θ sin ζ

)/sin Ω . (25)

The eigenstates of σ(θ, ζ, χ) with eigenvalues ±1 are

|ψ+ 〉 = cos(s/2) | 0 〉+ sin(s/2) e−iχ | 1 〉 , (26)

|ψ− 〉 = sin(s/2) | 0 〉 − cos(s/2) e−iχ | 1 〉 , (27)

where s = arccot(cot θ sin ζ) ∈ [0, π]. The n-th power ofthe number-conserving gate reads

Un = diag(1, e−inγu(θ, ζ, χ)n, e−in(2γ+φ)

), (28)

where un can be solved using the representation (24),

un = I cos(nΩ)− i σ(θ, ζ, χ) sin(nΩ) , (29)

−π/2 −π/4 0 π/4 π/2

z− (rad)

0.8

1.0

1.2

1.4

1.6

Ωc

(rad

)

Theory

Experiment

FIG. S3. Experiment data for Ωc as a function of z−. Theycompare well with the fitted curve based on the analytic ex-pression (38). This indicates that gate bleeding between RZand the two-qubit gate U is negligible, i.e., their control pulsesdo not interleave with each other.

which takes the matrix form(cos(nΩ)− iΛn cos θ sin ζ −iΛneiχ sin θ

−iΛne−iχ sin θ cos(nΩ) + iΛn cos θ sin ζ

),

(30)

where Λn = sin(nΩ)/ sin Ω.To calibrate the gate parameters, we introduce the cy-

cle unitary made up of U and singe-qubit Z rotations

Uc ≡ U(θ, ζ, χ, γ, φ)RZ(z1, z2) (31)

= U(θ, ζc, χc, γc, φ) , (32)

where the parameters of Uc are related to the originalones via the linear relations

ζc = ζ + z− , χc = χ+ z− , γc = γ − z+ . (33)

The two parameters z± = (z1±z2)/2 can be controlled byadjusting the single-qubit pulses. The hidden assumptionhere is: the two-qubit gate U does not depend on z1

and z2, i.e., the control pulses do not interleave (no gatebleeding). The cycle unitary has two trivial eigenstates| 00 〉 and | 11 〉,

Uc| 00 〉 = | 00 〉 , Uc| 11 〉 = e−i(2γc+φ)| 11 〉 (34)

and two nontrivial eigenstates

|Ψ+c 〉 = cos(sc/2) | 01 〉+ sin(sc/2) e−iχc | 10 〉 , (35)

|Ψ−c 〉 = sin(sc/2) | 01 〉 − cos(sc/2) e−iχc | 10 〉 , (36)

where sc = arccot(cot θ sin ζc). The corresponding eigen-value equations are

Uc|Ψ±c 〉 = e−i(γc±Ωc) |Ψ±c 〉 , (37)

where Ωc ∈ [θ, π − θ] is the Rabi angle

Ωc = arccos(cos θ cos(ζ + z−)

). (38)

Knowing the eigenvalues of Uc allows one to learn γc, Ωc,and φ, from which one can learn γ using the last identity

12

|0〉

|0〉 X

RZ

(z1)

RZ

(z2) U(θ, ζ, χ, γ, φ)

n times

FIG. S4. The circuit to calibrate the parameters θ and ζ inthe excitation-number-conserving gate. The cycle unitary isrepeated for n times before we measure the qubits in the Zbasis.

in Eq. (33). To learn θ and ζ using Eq. (38), one needs toknow Ωc for at least two different values of z−. In Fig. S3,we plot the experiment results for Ωc as a function ofz− and compare them with the fitted curve based onEq. (38). They conform extremely well with each other,indicating that gate bleeding between the single-qubit Zrotations and the two-qubit U is negligible.

In the following, we introduce three sets of calibra-tion circuits to learn the five parameters in the number-conserving gate (18). We run each set of circuits withthe cycle repetition numbers from the set

N =drke

∣∣ k = 0, 1, . . . ,K − 1, r > 0 . (39)

This is necessary because the eigenvalues of Uc can onlybe determined up to modulo 2π/n when Uc is repeatedfor n times. To make sure that we search in the correctprincipal region, the true value should be located withinthe (π/n)-neighborhood of the prior estimate with highprobability. We will discuss this issue and how to choosethe real number r in more detail in Section C2.

Calibration circuits 1

This set of calibration circuits are used to learn theparameters θ and ζ in the gate (18), see Fig. S4. Theprobability of measuring the state | 10 〉 is

pn =∣∣〈 10 |Unc RX(0, π)| 00 〉

∣∣2=∣∣〈 1 |unc | 0 〉∣∣2 =

(sin(nΩc) sin θ/ sin Ωc

)2, (40)

where unc ≡ u(θ, ζc, χc)n is given in Eq. (30). By post-

selecting the measurement results with one excitation,i.e., | 01 〉 and | 10 〉, we make the results robust to T1 er-ror and bit-flip errors in the measurement. An unbiasedestimator of pn based on the measurement results is

pn =Number of outcome 10

Number of outcomes 01 and 10, (41)

and its variance decreases as the number of measurementspassing postselection (the denominator) increases.

Using Eqs. (38) and (40), we can relate the measure-ment probability pn to the gate parameters θ and ζ andthe adjustable variable z−. For n = 1, 2, we have

|sin θ| = √p1 , sin(2θ) |cos(ζ + z−)| = √p2 , (42)

which can be used to get initial estimates of θ and ζ.To get robust estimates for larger n, we use aggregatedresults from several previous runs by introducing the costfunction (we use the Hellinger distance, but other metricsmight work as well)

C`n(x, y)

=∑

`≤m≤nm∈N

∑z−∈Z−

m

(∣∣∣ sin(mΩc) sinx

sin Ωc

∣∣∣−√pm,z−)2

, (43)

where Ωc(x, y, z−) = arccos

(cosx cos(y + z−)

)and the

lower bound ` determines the number of terms includedin the cost function. This approach also provides us withthe flexibility of using a different set of z− for each cyclerepetition number m, which we denote as Z−m. When nis small, we use the global minimum of the cost functionas the estimators of θ and ζ. For larger n, the land-scape of the cost function becomes rugged, and we useits local minimum around the prior estimates as the newestimates. We choose ` based on the following rules. Forsmall n, it is easy to get into the wrong principal regions,and we include all the prior runs in the cost function bysetting ` = 1. As n increases, we fix the number of termsin the cost function by increasing `. We then graduallyreduce the number of terms in the cost function to min-imize the variance of the estimates at the end.

The variances of the estimates of θ and ζ diverge wheneither |∂Ωc/∂θ| or |∂Ωc/∂ζ| is close to zero. This can beavoided by adaptively choosing the values of z− based onthe current estimates of θ and ζ. We may choose the twovalues of z− close to π/4 and 3π/4, which are apart byπ/2 for best performance. This choice also leaves a largemargin between ζc (mod π/2) and 0 for small ζ, whereeither |∂Ωc/∂θ| or |∂Ωc/∂ζ| equals to zero. The standarderror in the estimate of Ωc is inversely proportional to thederivative

∂pn∂Ωc

' n sin(2nΩc)(sin θ/ sin Ωc

)2, (44)

where we assume n 1 and neglect terms of orderO(1). We maximize the fast oscillating part sin(2nΩc)in Eq.(44) by choosing z− from the neighborhood of π/4and 3π/4 such that

nΩc = π/4 (mod π/2) , (45)

where Ωc is evaluated using the estimates of θ and ζ.These calibration circuits can also be implemented

non-adaptively at the price of increasing the number ofcircuits. We run the circuit in Fig. S4 for several equidis-tant values of z− in the two intervals π/4 ± w/n and3π/4± w/n, where w = π/(2 cos θ). This choice guaran-tees that Eq. (44) has big values at least for some of theselected values of z−. The values of θ and ζ can then beestimated by fitting the data to Eq. (40).

13

|0〉

|0〉 √X

RZ

(z1)

RZ

(z2) U(θ, ζ, χ, γ, φ)

√X

n times

FIG. S5. The circuit to calibrate the parameters γ and χin the general FSIM gate. The initial

√X gate creates a

superposition between the state | 00 〉 and the single-excitationstate | 01 〉.

Calibration circuits 2

Here we will focus on γ and χ, since θ and ζ can be bet-ter estimated with the former calibration circuits. Thecircuit in Fig. S5 can be used to measure the accumulatedphase nγ + χ, from which one can estimate γ with highprecision. In comparison, χ can only be estimated withlow precision (also susceptible to SPAM errors) due tolack of the scaling factor n. Consider the measurementprobabilities for the circuit in Fig. S5,

pn =∣∣〈 00 |RX(π/2, 0)Unc RX(0, π/2) | 00 〉

∣∣2=

1

4

∣∣einγc − 〈 1 |unc | 0 〉∣∣2 , (46)

and

qn =∣∣〈 01 |RX(π/2, 0)Unc RX(0, π/2) | 00 〉

∣∣2=

1

4|〈 0 |unc | 0 〉|2 . (47)

To estimate the parameter γc = γ−z+ using Eq. (46), wewill need to learn 〈 1 |unc | 0 〉. Its phase can be calculatedusing Eq. (30),

arg(〈 1 |unc | 0 〉) = −χ− z− − π

2sgn Λn , (48)

where Λn = sin(nΩc)/ sin Ωc and its sign can be deter-mined with confidence provided that we have good es-timates of θ and ζ. Using the normalization condition|〈 0 |unc | 0 〉|2 + |〈 1 |unc | 0 〉|2 = 1, we have

|〈 1 |unc | 0 〉| =√

1− 4qn . (49)

The measurement probability pn in Eq. (46) is a functionof |〈 1 |unc | 0 〉| and the relative phase

µn = nγc − arg(〈 1 |unc | 0 〉) (50)

= (nγ + χ)− nz+ + z− +π

2sgn Λn . (51)

Knowing µn for different values of n allows one to esti-mate γ and χ. It is related to the measurement proba-bilities through the relation∣∣eiµn −√1− 4qn

∣∣2 = 4pn , (52)

|0〉

|0〉

√X

X

RZ

(z1)

RZ

(z2) U(θ, ζ, χ, γ, φ) √

X

n times

FIG. S6. The circuit to calibrate the parameter φ. The initialmicrowave gates create a superposition between the state | 11 〉and the single-excitation state | 01 〉.

or equivalently

cosµn =1− 2(pn + qn)√

1− 4qn. (53)

Again, we introduce the cost function

C`n(x, y) =∑

`≤m≤nm∈N

(fn(x, y)− 1− 2(pn + qn)√

1− 4qn

)2

, (54)

where fn(x, y) = cos(nx+y−nz+ + z−+ π2 sgn Λn). The

parameters γ and χ can be estimated by using the argu-ments x and y that minimize the cost function, respec-tively. We follow the same prescription as in the formercase to choose ` and minimize the cost function.

To reduce the variances of estimators, we choose z−such that |〈 1 |unc | 0 〉| is maximized using the estimates ofθ and ζ. We also choose z+ such that µn ' π/2 (mod π),which maximizes |∂pn/∂µn|. It also leaves a big margin(close to π) between µn and other phases sharing thesame cosine value; this reduces the chance of misidenti-fication of the phase. We can also implement this non-adaptively by running two values of z+ apart by π/2,which allows for estimating any phase to the same preci-sion.

Calibration circuits 3

This set of circuits can be used to estimate θ, ζ, andγ + φ with high precision, see Fig. S6. We will focuson γ + φ, from which we can derive the value of thecontrolled phase φ given that γ is known. Consider themeasurement probabilities

pn =∣∣〈 10 |RX(0, π/2)Unc RX(π/2, π) | 00 〉

∣∣2 (55)

=1

4

∣∣e−in(γc+φ) − 〈 1 |unc | 0 〉∣∣2 , (56)

qn =∣∣〈 00 |RX(0, π/2)Unc RX(π/2, π) | 00 〉

∣∣2 (57)

=1

4|〈 0 |unc | 0 〉|2 . (58)

We define the relative phase

µn = −n(γc + φ)− arg(〈 1 |unc | 0 〉) (59)

= −n(γ + φ) + χ+ nz+ + z− +π

2sgn Λ , (60)

which can be estimated using the same procedure as thelast case.

14

C2 Phase estimation under decoherence

Quantum metrology protocols are typically extremelysensitive to noise [86]. For example, the depolarizingnoise—no matter how small—ruins the possibility ofsub-shot-noise performances of a quantum interferome-ter [87]. It was also demonstrated that quantum metrol-ogy can only do better than classical approaches by aconstant factor for any nonzero loss [88, 89]. In this sec-tion, we discuss the decoherence effects on single- andtwo-qubit phase estimation.

Consider a single-qubit system described by the mas-ter equation ρ = − i

h [H, ρ] + L(ρ), where the Lindbladoperator L(ρ) takes the form

L(ρ) =1

T1

(σ−ρσ+ − 1

2(σ+σ−ρ+ ρσ+σ−)

)+

1

2T2(σzρσz − ρ) . (61)

We prepare the initial state |ψin 〉 = (| 0 〉+ | 1 〉)/√

2 andapply the phase gate RZ(ϕ) = diag(1, eiϕ) for n times.We then apply another phase shift diag(1, eis) beforemeasuring the system in the X basis. The probabilitythat an excitation does not decay after n gate cycles ise−nλ1 , where λ1 = gate time/T1. Therefore, the proba-bility of getting the measurement outcome + is

pn(s) = e−nλ1qn(s) +1− e−nλ1

2, (62)

where qn(s) is the probability of getting the outcome +with only T2 error

qn(s) =1 + e−nλ2 cos(nϕ+ s)

2, (63)

where λ2 = gate time/T2. An unbiased estimator of theprobability pn is

pn =Number of outcome +

Mn, (64)

where Mn is the circuit repetition number. The varianceof the estimator pn is

V (pn) =pn(1− pn)

Mn(65)

=e−2nλ1qn(1− qn)

Mn+

1− e−2nλ1

4Mn, (66)

and the variance of the estimator of ϕ can be determinedusing the chain rule

V (ϕn) =V (pn)

(∂pn/∂qn)2 (∂qn/∂ϕ)2(67)

=e2nλ1 − e−2nλ2 cos(nϕ+ s)2

Mnn2e−2nλ2 sin(nϕ+ s)2(68)

≤ e2n(λ1+λ2)

Mnn2 sin(nϕ+ s)2. (69)

Equation (69) diverges when sin(nϕ+ s) = 0, which canbe avoided by adjusting the phase shift s adaptively. Onecan also implement this non-adaptively by running twoexperiments at s = 0 and s = π/2. Using the combinedinformation of the two, we have

V (ϕn) ≤ e2n(λ1+λ2)

Mnn2. (70)

Therefore, the standard deviation of the estimator ϕndecreases as n−1 before it blows exponentially. By setting∂V (ϕn)/∂n = 0, we have

n? =1

λ1 + λ2 1 , V (ϕn?) ≤ e2(λ1 + λ2)2

Mn?

. (71)

The minimum variance that one can achieve is thereforeset by λ1 + λ2. If the estimator ϕn is unbiased, the min-imum variance that one can achieve is bounded by

V (ϕ) ≥(∑n∈N

1

V (ϕn)

)−1

=1

F, (72)

where F is the Fisher information and N is the set of thecycle repetition numbers. The equal sign in Eq. (72) isachieved by using

ϕ =1

F

∑n∈N

ϕnV (ϕn)

. (73)

For the two-qubit case, the impact of decoherence usu-ally depends on the specific shapes of the control pulses.For simplicity, we consider the resonant case Ωc = θ (orequivalently ζc = 0) of the two-qubit number conservinggate, where the decoherence effects are pulse-shape inde-pendent. After removing the T1 error using postselection,the measurement probability in Eq. (40) reads

pn =1− e−2nλ2 cos(2nθ)

2, (74)

where the exponential factor comes from dephasing ofthe two qubits. The variance of the estimator pn is

V (pn) =pn(1− pn)

Mne−nλ1, (75)

where Mne−nλ1 is the number of measurement pass the

post selection. The variance of the estimator of θ can becalculated by the chain rule

V (θn) =V (pn)

(∂pn/∂θ)2(76)

=1− e−4nλ2 cos(2nθ)2

4Mnn2e−n(λ1+4λ2) sin(2nθ)2(77)

≤ en(λ1+4λ2)

4Mnn2 sin(2nθ)2. (78)

Compared to the single-qubit case (69), the effect of λ1

is reduced by a factor of two due to post selection while

15

0 π/2 π 3π/2 2π

Phase (rad)

0.0

0.5

1.0

1.5

Cos

t

0 π/2 π 3π/2 2π

Phase (rad)

0.0

0.5

1.0

1.5

Cos

t

Cost function

True phase

Minima

a b

FIG. S7. Comparison of two ideal cost functions using expo-nential circuit repetition numbers rk. Left: exponent r = 2with 7 runs, i.e., n = 1, 2, 4, 8, 16, 32, 64. Right: exponentr = 1.9 with 7 runs, i.e., n = 1, 2, 4, 7, 14, 25, 48. There aremany deep local minima that are close in value to the globalminimum (red vertical line) in the left plot for r = 2. In com-parison, the local minima are much shallower in the right plotfor r = 1.9. This is partly due to the less spaced repetitionnumbers in r = 1.9. More importantly, it is due to the factthat the repetition numbers for r = 2 are not mutually prime,which allows for minimizing a large number of terms in thecost function simultaneously.

the effect of λ2 is doubled due to the dephasing from twoqubits. By setting ∂V (θn)/∂n = 0, we have

n? =2

λ1 + 4λ2 1 , V (θn?) ≤ e2(λ1 + 4λ2)2

4m. (79)

C3 Cycle repetition numbers

When the SPAM errors are large, the estimates fromprior runs with smaller cycle repetition numbers can failto locate the principal region of the true value. As aresult, one gets estimates with higher and higher resolu-tions, but in a completely wrong region. Here we brieflydiscuss how to overcome this issue by properly choosingthe set of cycle repetition numbers N. Consider the costfunction to estimate a single-qubit phase ϕ,

Cn(x) =∑

m∈N,m≤n

∣∣eimx − eimϕ∣∣2 , (80)

where eimϕ is the estimate of eimϕ using quantum circuitswith cycle repetition number m. A good choice of Nleads to a cost function with a dominant global minimaaround ϕ. This reduces the probability of mistaking oneof the local minimum of Cn(x) as its global minimum,which corresponds to misidentifying the principal regionof the phase. Depending on the error rates and the circuitrepetition numbers, one may choose N = drke | k =0, 1, . . . ,K − 1 with r > 0. In Fig. S7, we plot the idealcost functions by replacing eimϕ with eimϕ in Eq. (80).We found that the local minima in the right panel (r =1.9) are much shallower than the left panel (r = 2). Thisis mainly because the numbers of cycles for r = 2 arenot mutually prime, where a large number of terms in

the cost function can be minimized simultaneously. Wefound that the local minima of the cost functions withlarger exponents 2 < r < 3 are typically shallower thanthose of r = 2. In general, we should avoid using integer rand perturb the elements in N so that they are mutuallyprime.

C4 General multi-qubit unitaries

In this section, we provide a simple argument that anyL-qubit gates U can be determined by knowing the eigen-values of the composite gate of the form

Uc = UR , R =

L⊗`=1

R` , (81)

where R` acts only on the `-th qubit. We show thatU can be uniquely determined by the eigenvalues of Ucfor various R. We formally write down the eigenvalueequation Uc |ψj 〉 = e−iεj |ψj 〉, where j = 1, . . . , 2L andεj is the j-th quasi energy. The probability of measuringthe state |ψout 〉 after applying Uc to the input state |ψin 〉for n times is

pn =

∣∣∣∣ 2L∑j=1

e−inεj 〈ψout |ψj 〉〈ψj |ψin 〉∣∣∣∣2 (82)

=

2L∑j,k=1

e−in(εj−εk)aja∗k , (83)

where aj = 〈ψout |ψj 〉〈ψj |ψin 〉. In principal, the dif-ference εj − εk can be estimated by running the circuitswith different n and |ψin 〉 and |ψout 〉.

To determine U , it suffices to learn tr(UR) for a set ofproduct unitaries R that form a complete operator basis,e.g., the Pauli group P,

U =1

2L

∑R∈P

tr(UR)R . (84)

Without losing generality, we assume that detU =detR = detUc = 1; therefore, we have the condition∑j εj = 0 (mod 2π). Knowing the differences in the

quasi energies allows one to determine εj (mod 2π/2L).The trace trUc = tr(UR) =

∑j e−iεj can therefore be

estimated up to a phase factor eikπ/2L−1

, where k is aninteger. To get the correct phase factor, we introduce acontinuous set of single-qubit unitaries

R(s) =

L⊗`=1

eisσ·v` , (85)

where σ = (σx σy σz) is the vector of Pauli operatorsand the normalized vector v` determines the rotationalaxes of R`. By choosing a sequence values 0 ≤ s ≤ 1, one

16

can keep track the principal region of the overall phaseof tr(UR). The procedure described here is by no means

optimal, but it shows that our method can be generalizedto multi-qubit gates in principle.

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view A 83, 021804 (2011).

18

a Base

1↑2↑

3↑4↑

5↑6↑

7↑8↑1↓

2↓3↓

4↓5↓

6↓7↓

8↓

b Reversed

1↑2↑

3↑4↑

5↑6↑

7↑8↑

1↓2↓

3↓4↓

5↓6↓

7↓8↓

c Exchanged

1↑2↑

3↑4↑

5↑6↑

7↑8↑

1↓2↓

3↓4↓

5↓6↓

7↓8↓

d Flipped

1↑2↑

3↑4↑

5↑6↑

7↑8↑ 1↓

2↓3↓

4↓5↓

6↓7↓

8↓

e Upper f Lower

FIG. S8. Various qubit assignments of the 1D Fermi-Hubbardmodel on a 23-qubit grid. a. The original configuration isused as the base. We generate new configurations by applyingthe following operations and their combinations on the base:b. reversing the sites, c. exchanging the spin states, d.flipping the sites horizontally. Each configuration can eithertake one of the two subsets of the grid: e. the upper part, f.the lower part. This leads to 16 different qubit assignments.The same assignments are used regardless if interaction termsare present (U 6= 0) or not (U = 0).

0.0

0.5

1.0

Sca

le

Scaling

Up

Down

0.0

0.5

1.0

Sca

le

Scaling

Up

Down

0.0 4.5 9.0 13.5 18.0

Time (h/J)

1

3

5

7

9

Pos

itio

n

No scaling

With scaling

0.0 1.0 2.0 3.0

Time (h/J)

2.0

3.0

4.0

5.0

Spr

ead

No scaling

With scaling

a b

FIG. S9. The damping factor (scale): a. the U = 0 andNP =2 case in Fig. 3, b. the U = 2 and NP = 4 case in Fig. 2.The data points are obtained by comparing the numerical andexperimental results at each Trotter step. The shaded areasrepresent the standard errors of regression and the solid linesrepresent the linear fittings of the data points. The bottomplots show the effectiveness of the rescaling procedure usingthe linear relation in Eq. (4): a. center of mass positions ofthe spin-up (yellow and orange) and spin-down (green andpurple) states, b. spreads of the two spin states.

21.2

18.5

15.0

7.8

18.5

12.3

19.2

20.8

23.8

12.9

17.8

15.9

18.0

15.4

12.7

19.7

12.8

12.0

21.9

13.3

11.9

14.1

16.8

FIG. S10. T1 relaxation times (µs) at idle frequencies.

2.68

1.60

1.63

1.09

1.62

1.14

1.50

1.20

1.36

1.00

0.81

1.16

1.61

1.21

1.78

1.49

0.95

1.41

1.29

1.08

0.82

1.17

1.74

1.23

1.43

1.01

0.85

1.14

1.17

1.12

1.03

1.02

FIG. S11. Two-qubit gate percent Pauli error obtained bycross-entropy benchmarking (XEB), where non-conflictingsubsets of two-qubit gates are applied simultaneously. Thesevalues were measured against the standard

√iSWAP

† withoutthe parasitic controlled phases. The two-qubit gates are cali-brated using our routine calibration process without Floquetcalibration, see Section V in the Supplementary informationin [39].

19

2 4 6 8

Site index

0.5

1.0C

harg

ede

ns.

u=

0

t = 0

2 4 6 8

Site index

t = 0.3h/J

2 4 6 8

Site index

t = 0.6h/J

2 4 6 8

Site index

t = 0.9h/J

2 4 6 8

Site index

t = 1.2h/J

2 4 6 8

Site index

t = 1.5h/J

2 4 6 8

Site index

t = 1.8h/J

2 4 6 8

Site index

t = 2.1h/J

2 4 6 8

Site index

t = 2.4h/J

2 4 6 8

Site index

t = 2.7h/J

2 4 6 8

Site index

t = 3h/J

2 4 6 8

Site index

0.5

1.0

Cha

rge

dens

.

u=

1.5

2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index

2 4 6 8

Site index

0.5

1.0

Cha

rge

dens

.

u=

3

2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index

2 4 6 8

Site index

0.5

1.0

Cha

rge

dens

.

u=

4.5

2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index

0 1 2 3

6

8

10

Cha

rge

spre

ad

u = 0

0 1 2 3

u = 1

0 1 2 3

u = 1.5

0 1 2 3

u = 2

0 1 2 3

u = 2.5

0 1 2 3

u = 3

0 1 2 3

u = 3.5

0 1 2 3

u = 4

0 1 2 3

u = 4.5

0 1 2 3

u = 5

0 1 2 3

Time (h/J)

0

2

Spr

ead

rate

(J/h

)

0 1 2 3

Time (h/J)0 1 2 3

Time (h/J)0 1 2 3

Time (h/J)0 1 2 3

Time (h/J)0 1 2 3

Time (h/J)0 1 2 3

Time (h/J)0 1 2 3

Time (h/J)0 1 2 3

Time (h/J)0 1 2 3

Time (h/J)

0.0

0.5

Spi

nde

ns.

0.0

0.5

Spi

nde

ns.

0.0

0.5

Spi

nde

ns.

0.0

0.5

Spi

nde

ns.

-2

0

2

Spi

nsp

read

a

b

FIG. S12. Detailed data for the N↑ = N↓ = 2 case. a. Charge and spin densities ρ±j as functions of the site numbers at differentevolution times for interaction strengths u = 0, 1.5, 3, 4.5. b. Charge and spin spread κ± and their derivatives as functions oftime for different values of u.

20

2 4 6 8

Site index

0.5

1.0

Cha

rge

dens

.

u=

0

t = 0

2 4 6 8

Site index

t = 0.3h/J

2 4 6 8

Site index

t = 0.6h/J

2 4 6 8

Site index

t = 0.9h/J

2 4 6 8

Site index

t = 1.2h/J

2 4 6 8

Site index

t = 1.5h/J

2 4 6 8

Site index

t = 1.8h/J

2 4 6 8

Site index

t = 2.1h/J

2 4 6 8

Site index

t = 2.4h/J

2 4 6 8

Site index

t = 2.7h/J

2 4 6 8

Site index

t = 3h/J

2 4 6 8

Site index

0.5

1.0

Cha

rge

dens

.

u=

1.5

2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index

2 4 6 8

Site index

0.5

1.0

Cha

rge

dens

.

u=

3

2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index

2 4 6 8

Site index

0.5

1.0

Cha

rge

dens

.

u=

4.5

2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index2 4 6 8

Site index

0 1 2 3

10

12

14

Cha

rge

spre

ad

u = 0

0 1 2 3

u = 1

0 1 2 3

u = 1.5

0 1 2 3

u = 2

0 1 2 3

u = 2.5

0 1 2 3

u = 3

0 1 2 3

u = 3.5

0 1 2 3

u = 4

0 1 2 3

u = 4.5

0 1 2 3

u = 5

0 1 2 3

Time (h/J)

-2

0

2

4

Spr

ead

rate

(J/h

)

0 1 2 3

Time (h/J)0 1 2 3

Time (h/J)0 1 2 3

Time (h/J)0 1 2 3

Time (h/J)0 1 2 3

Time (h/J)0 1 2 3

Time (h/J)0 1 2 3

Time (h/J)0 1 2 3

Time (h/J)0 1 2 3

Time (h/J)

-0.5

0.0

0.5

Spi

nde

ns.

-0.5

0.0

0.5

Spi

nde

ns.

-0.5

0.0

0.5

Spi

nde

ns.

-0.5

0.0

0.5

Spi

nde

ns.

-2

0

2

Spi

nsp

read

a

b

FIG. S13. Detailed data for the N↑ = N↓ = 3 case. a. Charge and spin densities ρ±j as functions of the site numbers atdifferent evolution times for interaction strengths u = 0, 1.5, 3, 4.5. b. Charge and spin spread κ± and their derivatives fordifferent values of u.