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arXiv:2305.07720

Catalytic Embeddings of Quantum Circuits

Matthew Amy, Crawford, M., Andrew N. Glaudell, Melissa L. Macasieb, Mendelson, S. S., Neil J. Ross

Published May 12, 2023• 1 citation
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Abstract

If a set $\mathbb{G}$ of quantum gates is countable, then the operators that can be exactly represented by a circuit over $\mathbb{G}$ form a strict subset of the collection of all unitary operators. When $\mathbb{G}$ is universal, one circumvents this limitation by resorting to repeated gate approximations: every occurrence of a gate which cannot be exactly represented over $\mathbb{G}$ is replaced by an approximating circuit. Here, we introduce catalytic embeddings, which provide an alternative to repeated gate approximations. With catalytic embeddings, approximations are relegated to the preparation of a fixed number of reusable resource states called catalysts. Because the catalysts only need to be prepared once, when catalytic embeddings exist they always produce shorter circuits, in the limit of increasing gate count and target precision. In the present paper, we lay the foundations of the theory of catalytic embeddings and we establish several of their structural properties. In addition, we provide methods to design catalytic embeddings, showing that their construction can be reduced to that of a single fixed matrix when the gates involved have entries in well-behaved rings of algebraic numbers. Finally, we showcase some concrete examples and applications. Notably, we show that catalytic embeddings generalize a technique previously used to implement the Quantum Fourier Transform over the Clifford+$T$ gate set with $O(n)$ gate approximations.

DOI
10.48550/arxiv.2305.07720

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If a set G\mathbb{G}G of quantum gates is countable, then the operators that can be exactly represented by a circuit over G\mathbb{G}G form a strict subset of the collection of all unitary operators. When G\mathbb{G}G is universal, one circumvents this limitation by resorting to repeated gate approximations: every occurrence of a gate which cannot be exactly represented over G\mathbb{G}G is replaced by an approximating circuit. Here, we introduce catalytic embeddings, which provide an alternative to repeated gate approximations. With catalytic embeddings, approximations are relegated to the preparation of a fixed number of reusable resource states called catalysts. Because the catalysts only need to be prepared once, when catalytic embeddings exist they always produce shorter circuits, in the limit of increasing gate count and target precision. In the present paper, we lay the foundations of the theory of catalytic embeddings and we establish several of their structural properties. In addition, we provide methods to design catalytic embeddings, showing that their construction can be reduced to that of a single fixed matrix when the gates involved have entries in well-behaved rings of algebraic numbers. Finally, we showcase some concrete examples and applications. Notably, we show that catalytic embeddings generalize a technique previously used to implement the Quantum Fourier Transform over the Clifford+TTT gate set with O(n)O(n)O(n) gate approximations.
We show that the T -depth of any single-qubit z -rotation can be reduced to 3 if a certain catalyst state is available. To achieve an ϵ -approximation, it suffices to have a catalyst state of size polynomial in log ⁡ ( 1 / ϵ ) . This implies that Q N C f 0 / q p o l y admits a finite universal gate set consisting of Clifford+ T . In particular, there are catalytic constant T -depth circuits that approximate multi-qubit Toffoli, adder, and quantum Fourier transform arbitrarily well. We also show that the catalyst state can be prepared in time polynomial in log ⁡ ( 1 / ϵ ) .
We study the resources required to achieve universal quantum computing via the gate sets that provide the fundamental instructions from which quantum algorithms are built. While single-gate universal sets are known, they rely on precisely tuned irrational rotations, making them difficult to realize on near-term devices. We find that the controlled-VVV gate, an elementary two-qubit interaction directly implementable on leading hardware, is universal and capable of simulating standard universal gate sets with minimal overhead. Specifically, we use catalytic embeddings to develop a constant-overhead algorithm that simulates standard universal gate sets, including Clifford+T+T+T and Clifford+++Toffoli. We combine this simulation algorithm with existing synthesis results to yield exact and approximate synthesis algorithms for unitaries with and without number-theoretic restrictions. The results highlight how full quantum computational power, complete with algorithms for synthesis and simulation, can emerge from unexpectedly simple ingredients.
Epsilon-nets and approximate unitaryttt-designs are natural notions that capture properties of unitary operations relevant for numerous applications in quantum information and quantum computing. In this work we study quantitative connections between these two notions. Specifically, we prove that, forddddimensional Hilbert space, unitaries constituting$\delta −approximate-approximate−approximatet−expandersform-expanders form−expandersform\epsilon −netsfor-nets for−netsfort\simeq \frac {d^{5/2}}{ \epsilon }andandand\delta \simeq \left ({\frac { \epsilon ^{3/2}}{d}}\right)^{d^{2}}.Wealsoshowthatforarbitrary. We also show that for arbitrary.Wealsoshowthatforarbitraryt,,,\epsilon −netscanbeusedtoconstruct-nets can be used to construct−netscanbeusedtoconstruct\delta −approximateunitary-approximate unitary−approximateunitaryt−designsfor-designs for−designsfor\delta \simeq \epsilon t,wherethenotionofapproximationisbasedonthediamondnorm.Finally,weprovethatthedegreeofanexactunitary, where the notion of approximation is based on the diamond norm. Finally, we prove that the degree of an exact unitary,wherethenotionofapproximationisbasedonthediamondnorm.Finally,weprovethatthedegreeofanexactunitarytdesignnecessarytoobtainandesign necessary to obtain andesignnecessarytoobtainan\epsilon −netmustgrowatleastasfastas-net must grow at least as fast as−netmustgrowatleastasfastas\frac {1}{ \epsilon }(forfixeddimension)andnotslowerthan(for fixed dimension) and not slower than(forfixeddimension)andnotslowerthand^{2}(forfixed(for fixed(forfixed\epsilon ).Thisshowsnearoptimalityofourresultconnecting). This shows near optimality of our result connecting).Thisshowsnearoptimalityofourresultconnectingt−designsand-designs and−designsand\epsilon $-nets. We apply our findings in the context of quantum computing. First, we show that that approximate t-designs can be generated by shallow random circuits formed from a set of universal two-qudit gates in the parallel and sequential local architectures considered in (Brandão et al., 2016). Importantly, our gate sets need not to be symmetric (i.e., contains gates together with their inverses) or consist of gates with algebraic entries. Second, we consider compilation of quantum gates and prove a non-constructive Solovay-Kitaev theorem for general universal gate sets. Our main technical contribution is a new construction of efficient polynomial approximations to the Dirac delta in the space of quantum channels, which can be of independent interest.
Additional qubits can reduce the depth of a quantum circuit by providing workspace for parallel computation, but standard constructions assume that this workspace is initialized in a known state. In this work we study catalytic implementations, i.e. asking whether dirty qubits can instead be used provided that their joint state including any entanglement with other registers is restored exactly at the end of the computation. We show that every nnn-qubit Clifford circuit has a catalytic implementation of depth O(log⁡n)O(\log n)O(logn) using O(n2/log⁡2n)O(n^2/\log^2 n)O(n2/log2n) catalytic qubits and no clean qubits, matching the asymptotic depth achievable when clean workspace is available. We extend this approach to diagonal elements of any fixed level CkC_kCk​ of the Clifford hierarchy, which admit catalytic implementations of depth O(log⁡(n+1))O(\log(n+1))O(log(n+1)) with O(nk/log⁡(n))O(n^k/\log(n))O(nk/log(n)) gates and O(nk/log⁡2(n))O(n^k/\log^2(n))O(nk/log2(n)) catalytic qubits, as well as to semi-Clifford Gates.
Quantum computers promise transformative speedups, but environmental noise destroys their fragile states. Conventional quantum error correction (QEC) encodes information redundantly across physical qubits, yet fails above a threshold of about 1% and incurs polynomial qubit overhead. A recent theorem from the resource theory of coherence shows that catalytic covariant operations amplify coherence at an unbounded rate, but this result has never been cast as an operational protocol. The challenge is to turn an asymptotic theorem into a recovery scheme that works at any noise strength with realistic resources. Here we show that catalytic coherence amplification can be cast as an error-correction primitive, Catalytic Quantum Error Correction (CQEC), which recovers a known target state from noisy copies without any error magnitude threshold whenever the target's coherent modes are preserved. In an effective model of the recovery map, fidelity exceeds 0.99 across 200 noise configurations spanning d = 4-64; the catalyst cost drops from the constructive bound n* ~ d^4 e^(2 gamma) to 32 copies at matched fidelity (10^4- to 10^9-fold) via a pipeline of dynamical decoupling, Clifford twirling, and recursive swap-test purification. A first explicitly CPTP, exactly covariant joint-channel implementation validates genuine recovery under dephasing at small dimension (0.54 -> 0.77) while showing that shallow circuits consume the catalyst, quantifying the model-vs-channel gap. These results turn an abstract resource-theoretic statement into a concrete protocol candidate complementary to stabilizer- and purification-based QEC; an open-source package reproducing the benchmarks accompanies this work (arXiv:2603.25774, https://github.com/deeptell-inc/cqec).
Kliuchnikov, Maslov, and Mosca proved in 2012 that a 2×2 unitary matrix V can be exactly represented by a single-qubit Clifford+ T circuit if and only if the entries of V belong to the ring Z[1/2,i] . Later that year, Giles and Selinger showed that the same restriction applies to matrices that can be exactly represented by a multi-qubit Clifford+ T circuit. These number-theoretic characterizations shed new light upon the structure of Clifford+ T circuits and led to remarkable developments in the field of quantum compiling. In the present paper, we provide number-theoretic characterizations for certain restricted Clifford+ T circuits by considering unitary matrices over subrings of Z[1/2,i] . We focus on the subrings Z[1/2] , Z[1/2] , Z[1/i2] , and Z[1/2,i] , and we prove that unitary matrices with entries in these rings correspond to circuits over well-known universal gate sets. In each case, the desired gate set is obtained by extending the set of classical reversible gates {X,CX,CCX} with an analogue of the Hadamard gate and an optional phase gate.