Stabilizing quantum simulations of lattice gauge theories by dissipation
Tobias Schmale Hendrik Weimer








tobias.schmale@itp.uni-hannover.de
Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany

weimer@tu-berlin.de
Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36 EW 7-1, 10623 Berlin, Germany
Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany



  Simulations of lattice gauge theories on noisy quantum hardware
  inherently suffer from violations of the gauge symmetry due to
  coherent and incoherent errors of the underlying physical system
  that implements the simulation. These gauge violations cause the
  simulations to become unphysical requiring the result of the simulation
  to be discarded. We investigate an active correction scheme
  that relies on detecting gauge violations locally and subsequently
  correcting them by dissipatively driving the system back into the
  physical gauge sector. We show that the correction
  scheme not only ensures the protection of the gauge symmetry, but it also
  leads to a longer validity of the simulation results even within the
  gauge-invariant sector. Finally, we discuss further applications of
  the scheme such as preparation of the many-body ground state of the
  simulated system.
  

Stabilizing quantum simulations of lattice gauge theories by dissipation
    Hendrik Weimer
    April 29, 2024
========================================================================

The quantum simulation of lattice gauge theories is
one of the most important applications of quantum computers, given
that the fundamental field theories that govern modern physics are
gauge theories. However, error processes in noisy intermediate-scale
(NISQ) quantum devices may violate the gauge symmetry that is
essential for the viability of the simulation. Here, we show that
engineered dissipation can overcome these limitations and protect the
validity of the results for much longer simulation times.

The main challenge when implementing a quantum simulation of a lattice
gauge theory is that the gauge symmetry has to be explicitly
programmed into the device <cit.>. For current NISQ
devices, this means that unavoidable errors will generically lead to
violations of the gauge symmetry and result in the creation of quantum
states that are unphysical simulation results. To solve this problem,
several solutions have been offered. One solution is to integrate out
the redundant degrees of freedom. This however typically results in
non-local interactions <cit.>, increasing the
complexity of the simulation. Other approaches include modifications
to the Hamiltonian to attribute an energy penalty to a gauge
violation <cit.>, adding (random) gauge
transformations during time evolution in order to stochastically
cancel gauge errors <cit.> or
detecting gauge violations using oracles and rejecting simulations
with gauge violations <cit.>.

In our work, we build on the vast work on the realization of
engineered dissipation channels for quantum many-body systems
<cit.>. Specifically, we detect gauge violations during the time
evolution and apply local correction operations in real time to fix
any gauge violations that might occur. We investigate the effects of
our scheme on the accuracy of simulations and show control over the
temperature of the system as an additional benefit of our approach. We
exemplify our result for a Z_2 lattice gauge theory, however,
generalization of our work to other lattice gauge theories is
straightforward.




 §.§ Model gauge theory

For concreteness, we focus on a paradigmatic Z_2 lattice gauge theory as one of the simplest, yet non-trivial lattice gauge theories <cit.>.
The Hamiltonian is of the form

    H_0 = ∑_j=1^N [J_a(σ^+_jj^zσ^-_j+1 + h.c.) - J_f j^x].

acts on N matter sites, each of which can contain either the vacuum or a hard-core boson. These are described by the Pauli ladder-operators σ^+/-. In between the matter sites j and j+1 sit gauge link variables j^x/z, represented by Pauli x/z matrices. Therefore, the system can be mapped to 2N qubits. Following <cit.>, we use periodic boundary conditions and set the matter-field coupling J_a = 1 and the electric field energy J_f = 0.54.
Gauge-invariance of this Hamiltonian is defined by the Gauge operators

    G_j =1 - (-1)^jτ_j-1, j^xσ^z_jj^x,

which satisfy [H_0, G_j] = [G_j, G_l] = 0 ∀ j, l. Once initialized in an eigenvalue g_0 of the gauge operators, the time-evolution should therefore not change this eigenvalue. 

However, during a simulation of such a lattice gauge theory on a quantum simulator, unitary and non-unitary errors can break gauge invariance, leading to gauge eigenvalues g deviating from the initial g_0. We incorporate these unitary effects by adding a small gauge variant perturbation 

    H_1    = ∑_j=1^N [(σ_j^+ (c_1 j^- +c_2 j^+)σ^-_j+1 + h.c.)]
        + ∑_j=1^N σ^+_jσ^-_j(c_3j^z + c_4τ^z_j-1, j)

to the Hamiltonian, resulting in the complete Hamiltonian

    H = H_0 + λ H_1.

The precise values of the dimensionless coupling constants c_i are not important, in line with previous work we set them to c_1 = 0.51, c_2 = -0.49, c_3 = 0.77, c_4=0.21 <cit.>.
Incoherent errors are represented by single-site Lindbladian jump operators L_i acting on all matter and link sites, resulting in the model dynamics being described by the Lindblad master equation

    ρ̇ = i[ρ, H] + ∑_i [L_iρ L_i^† - 1/2ρ L_i^† L_i - 1/2 L_i^† L_iρ].

As L_i we investigate bit-flips √(γ)σ^x and phase-flips √(γ)σ^z on matter and link sites, and alternatively spontaneous emission √(γ)σ^- = √(γ)(σ^x - i σ^zσ^x) on all sites.

We quantify the gauge violation by 

    ε (t)    = 1/N|∑_j=1^NG_j(t) - ∑_j=1^NG_j(0)| =  1/N∑_j=1^NG_j(t),

where the last equation holds by the choice of the initial gauge G_j(0) = 0 ∀ j.



 §.§ Active gauge correction scheme


We imagine the above lattice gauge theory to be simulated in a Trotterized manner on a digital quantum computer, where time-evolution is split into small time-steps, which are implemented by unitary operations <cit.>. After a variable number of time-steps, mid-circuit measurements of all gauge operators can be performed, e.g. using the circuit from Fig. <ref>, making use of the fact that the gauge operators and the Hamiltonian all mutually commute. <cit.> shows how to do this for general lattice gauge theories. This yields a sequence of gauge eigenvalues corresponding the each gauge operator (G_1, ..., G_N). We can now react to this measured gauge syndrome and apply unitary operations to restore the original gauge (G_1, ..., G_N) = (0, ..., 0). 

In order find a set of possible correction operators, we first study the action of bit-flips and phase flips on the gauge eigenvalue. The gauge operators G_j have two eigenvalues 0 and 2. A bit-flip on matter site j will flip the eigenvalue of G_j. Similarly, a phase-flip on link site j will flip the eigenvalue of the operators G_j and G_j+1. On the other hand, bit-flips on link sites and phase-flips on matter sites do not result in a change of the gauge eigenvalue. This is summarized in Table <ref>. It therefore makes sense to separate the errors into gauge-variant, i.e. gauge violating errors and the remaining gauge-invariant errors. 



Since not all errors are visible to the gauge operators, one can not hope to achieve full error correction from the gauge theory alone. It is possible to extend the lattice gauge theory to a full quantum error correcting code by adding more degrees of freedom and formulating constraints on these <cit.>. Instead we here investigate phenomena that to not require extra qubit resources.



The gauge syndromes in Table <ref> suggests the following correction scheme:

    
  * If a sequence (G_j-i, G_j, G_j+1) = (0, 2, 0) is measured, a correcting bit-flip is applied to matter site j. 
    
  * If a sequence (G_j, G_j+1) = (2, 2) is measured, a correcting phase-flip is applied to link site j. 

In both cases, all gauge eigenvalues are subsequently restored to the physical sector after correction.
This scheme could now be applied stroboscopically after a fixed number n of Trotter steps, each evolving the system by time dt, using mid-circuit measurements and real-time feedback <cit.>.

For our numerical simulations, we instead choose a different but equivalent formulation based on jump operators.
We define P^0_j and P^2_j as the projectors that project into the two eigenvalues 0 and 2 of G_j. In fact, here this means that P^2_j = G_j/2 and P^0_j = 1 - P^2_j.
We then define the correction jump operators as 

    C^x_j    = √(γ_c)σ^x_j P^0_j-1P^2_jP^0_j+1    and
    
        C^z_j    = √(γ_c)j^z P^2_jP^2_j+1,

which correct the two gauge syndromes given above and annihilate any
other sequence of gauge eigenvalues. γ_c = 1/n· dt
is the correction rate and specifies how many correction
operations are applied per unit time.  We then simulate the dynamics
of the correction scheme by using these operators as additional jump
operators in the master equation. This is equivalent to the
stroboscopic application of the correction circuit since the latter
corresponds to a Trotterized form of the master equation dynamics
<cit.>.



The result of this correction scheme in action can be seen in Figure <ref>. Without gauge correction (γ_c = 0), the gauge violation ε increases linearly until a steady-state of maximal gauge violation is reached. Turning on the correction (γ_c > 0) results in this steady-state violation to be suppressed inversely proportional to the correction rate γ_c. This shows that this scheme is effective in suppressing gauge errors and therefore prevents the simulation of the lattice gauge theory to turn unphysical.  We note that since the gauge operators mutually commute, our scheme implements a form of stabilizer pumping  <cit.>.



 §.§ Sympathetic cooling during gauge correction
 

We now make a slight modification to the Hamiltonian by adding the gauge violation as a new term

    H → H + 1/N∑_j=1^N G_j.

This associates an energy penalty  to a gauge violation. We choose
g=1 and show a dependence of our results on this choice in the
appendix. While such a term can be used to reduce gauge violations
stemming from coherent errors such as in H_1
<cit.>, they cannot directly suppress
incoherent errors such as the ones in the master equation
(<ref>). However, the combination with our dissipative
gauge correction scheme realizes a setup corresponding to a sympathetic cooling of the gauge-invariant sector <cit.>. Here, the gauge-invariant sector is the system to be cooled, while the gauge degrees of freedom act as a bath. The gauge degrees of freedom are rapidly cooled into the ground state of having no gauge violations, while the coherent gauge errors in Eq. (<ref>) lead to a system-bath coupling and allow energy to be dissipated out of the gauge-invariant sector.

This is similar to the cooling schemes shown in
<cit.> and
<cit.>, with the crucial difference
that these schemes require extra bath degrees of freedom to be
artificially added to achieve cooling, therefore increasing the cost
of the simulation. Here, the bath is implemented by the gauge degrees
of freedom and does not require any additional resources.



We show this in Figure <ref>, where we start in a high-energy product state and demonstrate how the dissipative time-evolution drives the system to lower energy states. 
To ensure that any energy difference is not a direct consequence of the gauge penalty (<ref>), we only measure the energy of the system in the physical sector as

    H_g(t) = Tr[PH_0P ρ(t)]/Tr[Pρ(t)],

where P=∏_j P^0_j is the projector into the physical gauge sector.
Depending on the magnitude of the competing heating caused by the dissipative errors, an energy close to the ground state can be reached. 
This final state is then stable under further time evolution, i.e. is protected from further heating. We show this by directly solving for the steady state of the time evolution by computing the eigenvector of the Liouvillian corresponding to the zero-eigenvalue. The results in Figure <ref> (lower left) show three distinct phenomena: (i) For low correction rates γ_c, energy is not removed fast enough to compete with heating, hence no significant cooling is achieved. (ii) For too high correction rates, gauge errors are removed too quickly and therefore do not have time to interact with the physical gauge sector by means of the unitary errors. Therefore cooling is also not observed in this ”quantum Zeno” regime. (iii) In between these two extremes lies a regime where optimal cooling is achieved. If a gauge error occurs, it has time to interact with the physical sector and a subsequent correction removes its high energy contributions.
The steady-state gauge violation ε_ss that is achieved scales mostly as ε_ss∼γ/γ_c, as evident from Fig. <ref> (upper left). Exceptions to this rule are regimes of weak correction where the gauge violation saturates as well as intermediate regimes with low incoherent error rates where coherent errors dominate the gauge violation.
Remarkably, these results are quite insensitive to the exact nature of the incoherent errors. In Figure <ref> (right) we show the same results for spontaneous emission σ^- acting on all matter and link sites and the results still hold.
The dependence of these results on the magnitude of coherent errors and the gauge penalty is shown in the appendix.







 §.§ Stabilizing quantum simulations
 

Next, we apply our scheme to increase the accuracy of observables
during a simulated time evolution. To this extent, we start the
system in a J_a = 0 eigenstate and then quench to the previously
used J_a > 0 and track the dynamics of observables over
time. Figure <ref> show the
trajectory of the observable

    O = ∑_j τ^x_j-1, jτ^x_j, j+1,

i.e. a link-link correlator. In the appendix, we show results for more observables. The lower panels show the deviation of the observable from the exact, noise-free dynamics. This deviation is time-averaged over a running window of 10 time units for improved legibility.

We first focus on gauge-variant errors, i.e., bit-flips on matter sites and phase-flips on link sites. Since bit and phase errors can be exchanged by a local unitary transformation, this scenario is equivalent to the case where only bit or phase errors occur on all sites. Experimental platforms where one type of error dominates over the other are quite common, ranging from trapped ions <cit.> to solid-state spin qubits <cit.>.

As incoherent gauge-variant errors are localized single-qubit errors,
their presence causes in correlations to quickly decay to zero if no
correction is present, while the noise free evolution shows
interesting dynamics even for long times. As evident from
Fig. <ref> (lower left), turning on
the gauge corrections restores these dynamics and prevents a decay of
the correlators. In this scenario with only gauge-invariant errors,
the gauge corrections represent a full error correction of individual
errors, as every error is visible as a gauge violation and can
uniquely be decoded and corrected. We note that this works most
reliably if the correction rate is faster than the timescale of the
Hamiltonian to be simulated. However, this is guaranteed in digital
quantum simulation approaches where the individual parts of the
Hamiltonian are implemented in a Trotterized form and the correction
is carried out after each Trotter step.

If we also turn on the gauge-invariant errors, the protective effect of the correction is reduced, but does not vanish entirely. The corrected dynamics still show reduced errors as compared to the uncorrected ones. Since only the gauge-variant errors can be corrected, one may assume the simulation to be similar to an uncorrected simulation with only gauge-invariant errors. The effect of the correction is therefore to reduce the magnitude of the incoherent errors.

While sympathetic cooling requires moderate coherent errors λ and comparatively small correction rates γ_c, stabilizing time evolution does not require coherent errors but higher correction rates.
Hence λ and γ_c are the crucial parameters that govern which of the two regimes are explored. 
Depending of the desired use-case, the coherent errors are either a result of simulation errors, or can be artificially added to engineer the cooling effect. Hence it is possible to tune both parameters to achieve the desired application.





 §.§ Discussion


We have demonstrated that an active error correction scheme can successfully suppress gauge errors in lattice gauge theory simulations, inversely proportional to the correction rate. We showed how this gauge correction scheme can be interpreted as a sympathetic cooling setup and showed that this allows for tuning the parameters to the point where even ground states can be prepared and stabilized. This cooling scheme is very robust and its effects can be seen no matter the exact structure of the coherent errors, incoherent errors or the nature of the correction scheme. Our results can be directly applied to efforts to simulate lattice gauge theory on present NISQ devices <cit.>.
Finally we showed that this scheme can increase the accuracy of observables during simulation and therefore acts like a precursor to error correction.  It is particularly suitable in situations where errors are anisotropic as the scheme turns into full error correction in this case. However even without assumptions about the structure of the error, the scheme allows steady states of observables to be estimated more accurately.





  Master-equation time-evolution trajectories and steady-state computations were computed using the Qutip library <cit.>. This work was funded by the Quantum Valley Lower Saxony (QVLS) through the Volkswagen foundation and the ministry for science and culture of Lower Saxony and by Germany’s Excellence Strategy – EXC-2123 QuantumFrontiers – 390837967.

  



§ COMPETING INTERESTS

The authors declare no competing interests.



 


§ PARAMETER DEPENDENCIES OF STEADY STATE RESULTS



Fig. <ref> shows the dependence of the achieved cooling on the magnitude of the coherent errors. Stronger decoherence requires more coherent errors, at the cost of a larger steady-state gauge violation. For smaller decoherence rates, the system is not sensitive to the exact value of the coherent error magnitude. 
In Figure <ref> we also show the dependence of the cooling results on the gauge penalty g. 
The best cooling is achieved when the gauge penalty lies in a region with many energy transitions of H_0 <cit.>.
The steady-state gauge violations here recover the known result that the gauge penalty alone can yield a gauge-correcting effect <cit.> if only unitary errors are present.



§ MATTER-MATTER CORRELATIONS


In special cases, the steady-error of observables may even be reduced by the correction scheme as shown in Fig. <ref>,  where a matter-matter correlator is tracked over time. As with the previous results, gauge-variant errors can be fully corrected. In contrast to the result in the main text, the long-term behavior of the uncorrected results starts to significantly deviate from the true solution, while the errors of the corrected simulations remain bounded.







