Quantum homomorphic encryption for circuits of low $T$-gate complexity
Bibliographic record
Abstract
Fully homomorphic encryption is an encryption method with the property that any computation on the plaintext can be performed by a party having access to the ciphertext only. Here, we formally define and give schemes for quantum homomorphic encryption, which is the encryption of quantum information such that quantum computations can be performed given the ciphertext only. Our schemes allows for arbitrary Clifford group gates, but become inefficient for circuits with large complexity, measured in terms of the non-Clifford portion of the circuit (we use the "$π/8$" non-Clifford group gate, which is also known as the $T$-gate). More specifically, two schemes are proposed: the first scheme has a decryption procedure whose complexity scales with the square of the number of $T$-gates (compared with a trivial scheme in which the complexity scales with the total number of gates); the second scheme uses a quantum evaluation key of length given by a polynomial of degree exponential in the circuit's $T$-gate depth, yielding a homomorphic scheme for quantum circuits with constant $T$-depth. Both schemes build on a classical fully homomorphic encryption scheme. A further contribution of ours is to formally define the security of encryption schemes for quantum messages: we define quantum indistinguishability under chosen plaintext attacks in both the public and private-key settings. In this context, we show the equivalence of several definitions. Our schemes are the first of their kind that are secure under modern cryptographic definitions, and can be seen as a quantum analogue of classical results establishing homomorphic encryption for circuits with a limited number of multiplication gates. Historically, such results appeared as precursors to the breakthrough result establishing classical fully homomorphic encryption.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.004 | 0.003 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".