QuaRTZ: Quartz and Quantum Circuit Superoptimizer

• QuaRTZ is a polysemous label referring to quartz-based platforms—characterized by trigonal symmetry, chirality, and anisotropic responses—and a distinct quantum circuit superoptimizer.
• Quartz-centered applications exploit piezoelectric, electro-optic, and phononic properties for THz nonlinear optics, precision measurements, nanodevices, and X-ray optics.
• The Quartz superoptimizer uses symbolic equivalence and cost-based backtracking to automatically generate optimized circuit rewrites, achieving significant gate count reductions.

QuaRTZ is a polysemous label in the research literature. In most uses it denotes quartz-centered platforms built around $\alpha$-quartz, the stable low-temperature polymorph of $\mathrm{SiO_2}$, exploited as a piezoelectric, electro-optic, phononic, X-ray-optical, or mechanically robust medium. In a separate computer-systems usage, “Quartz” denotes a quantum circuit superoptimizer rather than a material platform. Across the quartz-centered works, the recurring technical themes are trigonal symmetry, chirality, absence of inversion symmetry, anisotropic elastic and optical response, mature hydrothermal growth, and low-loss resonant behavior over wide frequency ranges (Sutter et al., 2016).

1. Terminological scope and major usages

The term does not denote a single standardized framework. Instead, it appears across several research programs whose common element is either quartz as an active functional material or, in one case, a software system named Quartz.

Usage	Technical focus	Representative paper
THz nonlinear optics	Optical rectification and electro-optic sampling in z-cut $\alpha$-quartz	(Balos et al., 2022)
Precision tests with phonons	Rotating quartz BAW oscillators for Lorentz-invariance tests	(Goryachev et al., 2018)
Cryogenic phononics	Minimally diffracting SAW resonators and thin-film quartz PCRs	(Emser et al., 2022, Emser et al., 2024)
Nanodevices and hybrid quantum systems	Angled-etched quartz nanomechanics/nanophotonics and quartz–superconductor electromechanics	(Sohn et al., 2017, Woolley et al., 2016)
X-ray optics, defects, and extreme conditions	Monochromators, radiation damage, defect electronics, fracture, and shock response	(Sutter et al., 2016, Wang et al., 2015, Pandya et al., 25 Apr 2025, Tracy et al., 2020, Erhard et al., 2024, Hu, 2024)
Quantum compilation	Automatic generation and verification of circuit rewrites	(Xu et al., 2022)

Most quartz-centered usages are grounded in $\alpha$-quartz as a noncentrosymmetric crystal with cut-dependent electromechanical and optical behavior. A notable exception is the circuit-optimization system, where “Quartz” is a compiler framework and not a quartz-material technology (Xu et al., 2022).

2. Nonlinear, electro-optic, and thermomechanical quartz functionality

In THz photonics, z-cut $\alpha$-quartz has been shown to support both THz optical rectification and electro-optic sampling. The reported optical-rectification emission spectrum and electro-optic-sampling response are broad, extending to about $8~\mathrm{THz}$, with pronounced features near $\sim 4~\mathrm{THz}$ and $\sim 8~\mathrm{THz}$ attributed to phonon-related resonances. Thin quartz plates measure intense THz pulses with echo-free waveforms, and the work reports no saturation for fields of a few $\mathrm{MV/cm}$, specifically concluding that quartz can measure intense THz electric fields up to about $2~\mathrm{MV/cm}$ without saturation. The effective nonlinearity is reduced in the stated geometry to

$\mathrm{SiO_2}$0

which produces the observed six-fold rotational symmetry in THz emission and EOS. The same study reports that both OR and EOS amplitudes are roughly independent of quartz thickness over the range studied, presented as a plausible indication of strong surface contribution to the total second-order nonlinear susceptibility of quartz (Balos et al., 2022).

The phase-mismatch analysis in the same work uses

$\mathrm{SiO_2}$1

and the approximation

$\mathrm{SiO_2}$2

with coherence length

$\mathrm{SiO_2}$3

The reported coherence length decreases from about $\mathrm{SiO_2}$4 at $\mathrm{SiO_2}$5 to about $\mathrm{SiO_2}$6 at $\mathrm{SiO_2}$7, explaining why thin plates are especially useful for faithful waveform retrieval and why thicker plates can yield front-side and back-side contributions separated by optical/THz group-velocity mismatch (Balos et al., 2022).

A distinct quartz-based electromechanical use appears in on-chip detection of anisotropic thermopolarization. There, local Joule heating generates a temperature gradient, the gradient induces thermal expansion and stress, and the stress is converted into polarization and measurable charge through the piezoelectric tensor. The coupling is expressed as

$\mathrm{SiO_2}$8

and the measured current is described phenomenologically by

$\mathrm{SiO_2}$9

Because Joule heating scales as $\alpha$0, the induced polarization and current appear at the second harmonic $\alpha$1. The on-chip heater–detector geometry reveals the anisotropy of the quartz piezoelectric tensor: Z-cut quartz exhibits a threefold angular response, while X-cut quartz exhibits a twofold angular response, and the response is detected in both current and voltage modes (Iwakiri et al., 17 May 2026).

For Z-cut quartz, the effective polarization near the surface is written as

$\alpha$2

assuming the experimentally relevant shear contribution is negligible. This symmetry fingerprint, together with finite-element simulations showing dominant in-plane stresses near the heater, supports the interpretation that the measured electrical signal arises from thermally generated stress acting through electromechanical coupling rather than from ordinary thermoelectric transport in a conductor (Iwakiri et al., 17 May 2026).

3. Acoustic resonators, phononics, and quantum electromechanics

In precision measurement, quartz BAW resonators have been developed as room-temperature phonon-sector tests of Lorentz invariance. The central observable is the fractional frequency difference $\alpha$3 between two orthogonally oriented quartz oscillators on a rotating platform. The signal model is

$\alpha$4

where $\alpha$5 is the turntable rotation frequency and $\alpha$6 is local sidereal time. The second-generation setup uses two ultra-stable $\alpha$7 quartz OCXO/BAW oscillators, rotation at roughly $\alpha$8, PLL and interferometric comparison channels, a high-speed digitizer sampling at $\alpha$9 with averaging over $\alpha$0 samples, and Demodulated Least Squares to handle year-scale data sets of $\alpha$1–$\alpha$2 points. The projected sensitivity after one year of data is of order $\alpha$3 in SME matter-sector coefficients, described as a two-orders-of-magnitude improvement over the prior acoustic phonon-sector experiment (Goryachev et al., 2018).

For cryogenic surface acoustic wave devices, quartz anisotropy has been used to suppress diffraction loss rather than merely compensate it geometrically. The beam-steering angle is defined as

$\alpha$4

and the diffraction parameter as

$\alpha$5

The diffraction-limited quality factor is then

$\alpha$6

Minimal diffraction occurs when $\alpha$7. Finite-element simulations at $\alpha$8 identified the cut $\alpha$9 as a cryogenically optimized low-diffraction orientation, named COLD quartz. The simulations give $\alpha$0 for COLD quartz and $\alpha$1 for ST quartz. Experimentally, one-port SAW resonators fabricated on this cut and measured at $\alpha$2 reached internal quality factors greater than $\alpha$3, including $\alpha$4 at $\alpha$5 and about a $\alpha$6 improvement over equivalent resonators on ST quartz for a representative narrow-aperture device (Emser et al., 2022).

Thin-film quartz has also emerged as a high-coherence platform for piezoelectric phononic crystal resonators. The reported devices are fabricated from $\alpha$7-thick ST-cut $\alpha$8-quartz on silicon, with aluminum electrodes for microwave coupling and a $\alpha$9 phononic crystal defining a localized defect mode in a simulated complete bandgap of $8~\mathrm{THz}$0–$8~\mathrm{THz}$1. In dilution-refrigerator measurements, a representative resonance near $8~\mathrm{THz}$2 at mean occupancy $8~\mathrm{THz}$3 yielded

$8~\mathrm{THz}$4

while high-power ringdown at $8~\mathrm{THz}$5 yielded $8~\mathrm{THz}$6, $8~\mathrm{THz}$7, and

$8~\mathrm{THz}$8

Loss modeling attributed the low-power limitation to TLS physics, with extracted intrinsic loss tangents $8~\mathrm{THz}$9 for quartz and $\sim 4~\mathrm{THz}$0 for aluminum, indicating that a significant portion of the TLS bath is associated with the coupling electrodes rather than the quartz itself (Emser et al., 2024).

A related hybrid-quantum proposal couples a monolithic quartz BAW oscillator to a superconducting transmon through an intermediate $\sim 4~\mathrm{THz}$1 resonator. The relevant modes are a mechanical quartz BAW mode, an electrical $\sim 4~\mathrm{THz}$2 circuit mode, and a transmon qubit. Under resonant electromechanical coupling, the interaction is reduced in the RWA to

$\sim 4~\mathrm{THz}$3

With representative parameters $\sim 4~\mathrm{THz}$4, $\sim 4~\mathrm{THz}$5, $\sim 4~\mathrm{THz}$6, and $\sim 4~\mathrm{THz}$7, the analysis finds ground-state cooling of the quartz mode to be feasible through sideband cooling of the electrical mode and sympathetic cooling of the mechanics. The qubit fluorescence spectrum contains motional sidebands, so the qubit functions both as a cooling resource and as a transducer of the electromechanical steady state (Woolley et al., 2016).

4. Nanodevices, crystal optics, and quartz as an engineered platform

Single-crystal $\sim 4~\mathrm{THz}$8-quartz has been adapted to nanoscale electromechanical and photonic structures through Faraday cage angled etching. This approach undercuts bulk quartz to create suspended cantilevers, tuning-fork-like structures, and optical microrings without requiring a thin-film quartz platform. The reported demonstrations include quartz nanomechanical cantilevers and ring resonators with measured quality factors of $\sim 4~\mathrm{THz}$9 and $\sim 8~\mathrm{THz}$0, respectively. For mechanical characterization, $\sim 8~\mathrm{THz}$1 cantilevers on one Z-cut substrate, with common cross-section and lengths spanning $\sim 8~\mathrm{THz}$2–$\sim 8~\mathrm{THz}$3, were measured by optical interferometric displacement detection. The loss model

$\sim 8~\mathrm{THz}$4

yielded

$\sim 8~\mathrm{THz}$5

at $\sim 8~\mathrm{THz}$6 confidence, interpreted as likely surface loss. For a $\sim 8~\mathrm{THz}$7 cantilever, the measured first-order temperature coefficient of frequency was $\sim 8~\mathrm{THz}$8, with deviations within $\sim 8~\mathrm{THz}$9 over $\mathrm{MV/cm}$0 to $\mathrm{MV/cm}$1 (Sohn et al., 2017).

The same work demonstrated quartz microring resonators fabricated using a sputtered titanium mask to reduce scattering loss relative to lift-off masks. Using tapered-fiber coupling in the telecom band, the best measured loaded optical quality factor was

$\mathrm{MV/cm}$2

with estimated

$\mathrm{MV/cm}$3

Surface roughness was identified as the likely optical loss mechanism, paralleling the surface-loss interpretation for the mechanical devices (Sohn et al., 2017).

At larger length scales, $\mathrm{MV/cm}$4-quartz has been proposed as a high-energy-resolution crystal optic for X-ray monochromators and analyzers. The motivation is crystallographic rather than merely chemical: silicon, germanium, and diamond share the high-symmetry space group $\mathrm{MV/cm}$5, which yields a low density of unique backscattering Bragg reflections per unit photon-energy interval. Trigonal $\mathrm{MV/cm}$6-quartz provides many more inequivalent reflections, improving the chance of finding a backscattering condition near a desired RIXS or NRS resonance. The backscattering argument is governed by Bragg’s law,

$\mathrm{MV/cm}$7

and the bandwidth relation

$\mathrm{MV/cm}$8

As $\mathrm{MV/cm}$9, meV-class resolution becomes accessible (Sutter et al., 2016).

The same quartz-optics review clarifies longstanding convention problems in the crystallographic literature, including handedness, space-group labeling, rhombohedral settings, and piezoelectric orientation. It also compiles material constants relevant to precision optics. At $2~\mathrm{MV/cm}$0, the recommended linear thermal-expansion coefficients are $2~\mathrm{MV/cm}$1 along the $2~\mathrm{MV/cm}$2 axis and $2~\mathrm{MV/cm}$3 along the $2~\mathrm{MV/cm}$4 axis. At $2~\mathrm{MV/cm}$5, the recommended thermal conductivities are $2~\mathrm{MV/cm}$6 along $2~\mathrm{MV/cm}$7 and $2~\mathrm{MV/cm}$8 along $2~\mathrm{MV/cm}$9. Practical fabrication relies on hydrothermal growth rather than melt growth; the review describes nutrient transport in aqueous $\mathrm{SiO_2}$00 or $\mathrm{SiO_2}$01 from a hotter zone around $\mathrm{SiO_2}$02 to a cooler seed zone around $\mathrm{SiO_2}$03 under pressures up to about $\mathrm{SiO_2}$04 for NaOH growth. Demonstrated X-ray results include backscattering from $\mathrm{SiO_2}$05 at $\mathrm{SiO_2}$06 with energy width below $\mathrm{SiO_2}$07, from $\mathrm{SiO_2}$08 at $\mathrm{SiO_2}$09 with $\mathrm{SiO_2}$10 bandwidth, and a curved diced quartz analyzer on the $\mathrm{SiO_2}$11 plane with $\mathrm{SiO_2}$12 resolution at $\mathrm{SiO_2}$13 (Sutter et al., 2016).

5. Defects, radiation damage, and defect-enabled spectroscopy

Atomistic simulations of radiation damage in crystalline $\mathrm{SiO_2}$14-quartz show that the dominant ballistic defects are not oxygen vacancies. Using molecular dynamics with a BKS potential smoothly connected to ZBL at short range, cascades initiated by $\mathrm{SiO_2}$15, $\mathrm{SiO_2}$16, and $\mathrm{SiO_2}$17 primary knock-on atoms produced mainly over-coordinated Si and O, under-coordinated O and Si, and network-connectivity defects such as small Si–O rings and edge-sharing Si tetrahedra. The defect count follows

$\mathrm{SiO_2}$18

with fitted exponents $\mathrm{SiO_2}$19 for $\mathrm{SiO_2}$20, $\mathrm{SiO_2}$21 for $\mathrm{SiO_2}$22, $\mathrm{SiO_2}$23 for $\mathrm{SiO_2}$24, and $\mathrm{SiO_2}$25 for $\mathrm{SiO_2}$26. Persistent oxygen vacancies are rare: less than $\mathrm{SiO_2}$27 of oxygen PKAs ultimately leave a persistent oxygen vacancy at the original site, and the vacancy fraction among all defects is reported as $\mathrm{SiO_2}$28 at $\mathrm{SiO_2}$29, $\mathrm{SiO_2}$30 at $\mathrm{SiO_2}$31, and $\mathrm{SiO_2}$32 at $\mathrm{SiO_2}$33, decreasing further upon extrapolation to higher energies. The orientation-averaged threshold displacement energies are $\mathrm{SiO_2}$34 for oxygen and $\mathrm{SiO_2}$35 for silicon, with minimum values of $\mathrm{SiO_2}$36 along $\mathrm{SiO_2}$37 for oxygen and $\mathrm{SiO_2}$38 along $\mathrm{SiO_2}$39 for silicon (Wang et al., 2015).

These MD results directly challenge the widespread practice of treating charged oxygen vacancies or $\mathrm{SiO_2}$40 centers as a complete proxy for radiation damage. The reported defect ensemble is dominated by over-coordination and topological remodeling of the Si–O network, not by surviving oxygen vacancies. A plausible implication is that optical or ESR protocols keyed only to vacancy-related centers inevitably undersample the full structural damage state (Wang et al., 2015).

A complementary DFT literature models quartz defects as electron traps, hole traps, or recombination centers relevant to luminescence and ESR dosimetry. For pure quartz, the calculated band gap from the tail-state criterion is about $\mathrm{SiO_2}$41, with Si–O distances around $\mathrm{SiO_2}$42–$\mathrm{SiO_2}$43. A neutral oxygen vacancy produces a DOS peak about $\mathrm{SiO_2}$44 below the conduction-band edge and is interpreted as a shallow electron trap. The $\mathrm{SiO_2}$45 charged oxygen vacancy produces a peak near $\mathrm{SiO_2}$46 above the valence-band maximum and is linked to a hole-trapping or ESR-active $\mathrm{SiO_2}$47-type center, whereas the $\mathrm{SiO_2}$48 charged vacancy yields no distinct in-gap defect states. Hydrogen and hydroxyl passivation strongly modify or erase vacancy-related gap states: a fully hydrogenated oxygen vacancy and a fully hydrolyzed silanol-like configuration both yield DOS close to pure quartz, and the paper interprets this as reduced luminescence sensitivity in H-rich or OH-rich quartz (Pandya et al., 25 Apr 2025).

The same DFT study distinguishes several other defect chemistries. Silicon vacancies produce peaks near $\mathrm{SiO_2}$49, $\mathrm{SiO_2}$50, and $\mathrm{SiO_2}$51 above the valence band and behave primarily as hole traps. Excess-oxygen peroxy linkages create both valence-side and conduction-side states, suggesting paired hole and electron traps with possible tunneling-related fading. A peroxy intrinsic defect produces a sharp peak about $\mathrm{SiO_2}$52 above the valence-band maximum and is proposed as a possible source of the $\mathrm{SiO_2}$53 TL peak. Aluminum substitution produces negligible change in DOS, whereas iron substitution creates strong states at about $\mathrm{SiO_2}$54 and $\mathrm{SiO_2}$55, interpreted as hole-trapping and recombination-related states. The paper also argues that the often-cited $\mathrm{SiO_2}$56 center is not supported as a clear trap state by its DOS calculations (Pandya et al., 25 Apr 2025).

6. Mechanical strength, fracture, and dynamic high-pressure response

A recent structure-based fracture model treats quartz as a crystalline material built from a characteristic $\mathrm{SiO_2}$57-ring crystal unit of size

$\mathrm{SiO_2}$58

Using a theoretical tensile strength of $\mathrm{SiO_2}$59–$\mathrm{SiO_2}$60 for $\mathrm{SiO_2}$61 and an intrinsic molecular defect scale $\mathrm{SiO_2}$62, the model predicts an intrinsic quartz strength of $\mathrm{SiO_2}$63–$\mathrm{SiO_2}$64 and fracture toughness of $\mathrm{SiO_2}$65–$\mathrm{SiO_2}$66. The central toughness relation is

$\mathrm{SiO_2}$67

The paper compares these values with a reduced experimental quartz-toughness range of roughly $\mathrm{SiO_2}$68–$\mathrm{SiO_2}$69 and states that the agreement is good (Hu, 2024).

Under shock compression, quartz exhibits markedly path-dependent behavior. Time-resolved gas-gun experiments with in-situ synchrotron X-ray diffraction interrogated alpha-quartz up to $\mathrm{SiO_2}$70 and found a mixed-phase region at about $\mathrm{SiO_2}$71–$\mathrm{SiO_2}$72 and a fully transformed state above roughly $\mathrm{SiO_2}$73. The transformed high-pressure state is reported to be crystalline but disordered: not crystalline stishovite and not fully amorphous. The strongest evidence against stishovite is the absence of the diagnostic stishovite $\mathrm{SiO_2}$74 peak across Z-cut, X-cut, Y-cut, and polycrystalline starting materials. The diffraction instead shows broad low-angle features near $\mathrm{SiO_2}$75 together with sharper peaks, persistent Debye-Scherrer rings, and texture, leading to interpretation in terms of a dense metastable phase related to an oxygen close-packed framework with disordered or partially ordered Si occupancy (Tracy et al., 2020).

Machine-learning-driven atomistic simulations provide an atomistic route for this interpretation. For uniaxial shock compression of $\mathrm{SiO_2}$76-quartz along the $\mathrm{SiO_2}$77 axis to a peak pressure of about $\mathrm{SiO_2}$78 in the detailed run, the reported pathway is

$\mathrm{SiO_2}$79

Within about $\mathrm{SiO_2}$80, essentially no quartz remains and the amorphous fraction rises to about $\mathrm{SiO_2}$81; over tens of nanoseconds, the system recrystallizes into d-NiAs-structured silica with an hcp oxygen sublattice and partially ordered silicon domains. Local motifs classified as seifertite, NaTiF$\mathrm{SiO_2}$82-type, SnO$\mathrm{SiO_2}$83-type, and $\mathrm{SiO_2}$84-type silica appear as domains of partial silicon order. The same study maps a separate non-hydrostatic pathway to rosiaite-structured silica: rosiaite appears only when there is substantial compression both along and perpendicular to the $\mathrm{SiO_2}$85 axis, roughly $\mathrm{SiO_2}$86 and $\mathrm{SiO_2}$87, and solid-state NEB calculations give a barrier of about $\mathrm{SiO_2}$88 for direct diffusionless transformation under appropriate strain conditions (Erhard et al., 2024).

Taken together, the experimental and simulation results show that shocked quartz is controlled not only by pressure but by the full stress–strain path and the kinetics of silicon ordering. This explains why the literature can report amorphization, d-NiAs-like diffraction, or rosiaite depending on loading geometry and timescale (Tracy et al., 2020).

7. Quartz as a quantum circuit superoptimizer

In quantum compilation, Quartz is a software system unrelated to $\mathrm{SiO_2}$89-quartz as a material. It is a quantum circuit superoptimizer that automatically generates and verifies circuit transformations for arbitrary gate sets instead of relying on expert-designed rewrite libraries. The formal basis is symbolic circuit equivalence up to global phase for all parameter assignments, compactly represented by equivalent circuit classes (ECCs). An ECC with $\mathrm{SiO_2}$90 circuits implicitly contains $\mathrm{SiO_2}$91 transformations, and Quartz targets $\mathrm{SiO_2}$92-complete ECC sets that subsume all valid transformations over circuits with at most $\mathrm{SiO_2}$93 gates and $\mathrm{SiO_2}$94 qubits (Xu et al., 2022).

Transformation generation uses RepGen, a representative-based exploration algorithm. Rather than enumerating all circuits of size up to $\mathrm{SiO_2}$95, RepGen extends only representative circuits from previously identified ECCs and groups candidates by fingerprints. Verification is then delegated to an automated theorem-proving pipeline: symbolic matrices are converted to quantifier-free nonlinear real arithmetic by eliminating explicit global phase, expanding exponentials and trigonometric functions, and replacing $\mathrm{SiO_2}$96 and $\mathrm{SiO_2}$97 with fresh variables constrained by

$\mathrm{SiO_2}$98

The resulting formulas are checked with Z3. Quartz then optimizes input circuits through cost-based backtracking search over verified rewrites, using gate count as the experimental cost model (Xu et al., 2022).

The system was evaluated on three gate sets—Nam, IBM, and Rigetti—and on $\mathrm{SiO_2}$99 benchmark circuits including adders, controlled-$\alpha$00 constructions, Galois-field multipliers, modular arithmetic circuits, and Fourier-related circuits. Reported average gate-count reductions were $\alpha$01 for Nam, $\alpha$02 for IBM, and $\alpha$03 for Rigetti, compared with $\alpha$04, $\alpha$05, and $\alpha$06 for existing optimizers in the stated comparisons. Transformation generation and verification for each gate set completed in under $\alpha$07 minutes on $\alpha$08 cores. In this usage, “Quartz” denotes a correctness-aware, retargetable compiler framework rather than a quartz-material platform (Xu et al., 2022).