Add LinearCombination block encoding (#1133)

* Add `LinearCombination` `BlockEncoding` * Update comment * Fix circular import * Add support for negative coeffs * Update math in docstring * Address feedback --------- Co-authored-by: Matthew Harrigan <[email protected]>
quantumlib · Jul 16, 2024 · cebe529 · cebe529
1 parent 6dec788
commit cebe529
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diff --git a/dev_tools/autogenerate-bloqs-notebooks-v2.py b/dev_tools/autogenerate-bloqs-notebooks-v2.py
@@ -60,6 +60,7 @@
 import qualtran.bloqs.block_encoding.chebyshev_polynomial
 import qualtran.bloqs.block_encoding.lcu_block_encoding
 import qualtran.bloqs.block_encoding.lcu_select_and_prepare
+import qualtran.bloqs.block_encoding.linear_combination
 import qualtran.bloqs.block_encoding.phase
 import qualtran.bloqs.bookkeeping
 import qualtran.bloqs.chemistry.df.double_factorization
@@ -592,6 +593,7 @@
             qualtran.bloqs.block_encoding.unitary._UNITARY_DOC,
             qualtran.bloqs.block_encoding.tensor_product._TENSOR_PRODUCT_DOC,
             qualtran.bloqs.block_encoding.product._PRODUCT_DOC,
+            qualtran.bloqs.block_encoding.linear_combination._LINEAR_COMBINATION_DOC,
             qualtran.bloqs.block_encoding.phase._PHASE_DOC,
         ],
         directory=f'{SOURCE_DIR}/bloqs/block_encoding/',

diff --git a/qualtran/bloqs/block_encoding/__init__.py b/qualtran/bloqs/block_encoding/__init__.py
@@ -21,6 +21,7 @@
     LCUBlockEncoding,
     LCUBlockEncodingZeroState,
 )
+from qualtran.bloqs.block_encoding.linear_combination import LinearCombination
 from qualtran.bloqs.block_encoding.phase import Phase
 from qualtran.bloqs.block_encoding.product import Product
 from qualtran.bloqs.block_encoding.tensor_product import TensorProduct

diff --git a/qualtran/bloqs/block_encoding/block_encoding.ipynb b/qualtran/bloqs/block_encoding/block_encoding.ipynb
@@ -1175,6 +1175,135 @@
     "show_call_graph(phase_block_encoding_g)\n",
     "show_counts_sigma(phase_block_encoding_sigma)"
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5f9d88fa",
+   "metadata": {
+    "cq.autogen": "LinearCombination.bloq_doc.md"
+   },
+   "source": [
+    "## `LinearCombination`\n",
+    "Linear combination of a sequence of block encodings.\n",
+    "\n",
+    "Builds the block encoding $B[\\lambda_1 U_1 + \\lambda_2 U_2 + \\cdots + \\lambda_n U_n]$ given\n",
+    "block encodings $B[U_1], \\ldots, B[U_n]$ and coefficients $\\lambda_i \\in \\mathbb{R}$.\n",
+    "\n",
+    "When each $B[U_i]$ is a $(\\alpha_i, a_i, \\epsilon_i)$-block encoding of $U_i$, we have that\n",
+    "$B[\\lambda_1 U_1 + \\cdots + \\lambda_n U_n]$ is a $(\\alpha, a, \\epsilon)$-block encoding\n",
+    "of $\\lambda_1 U_1 + \\cdots + \\lambda_n U_n$ where the normalization constant\n",
+    "$\\alpha = \\sum_i \\lvert\\lambda_i\\rvert\\alpha_i$, number of ancillas\n",
+    "$a = \\lceil \\log_2 n \\rceil + \\max_i a_i$, and precision\n",
+    "$\\epsilon = (\\sum_i \\lvert\\lambda_i\\rvert)\\max_i \\epsilon_i$.\n",
+    "\n",
+    "Under the hood, this bloq uses LCU Prepare and Select oracles to build the block encoding.\n",
+    "These oracles will be automatically instantiated if not specified by the user.\n",
+    "\n",
+    "#### Parameters\n",
+    " - `block_encodings`: A sequence of block encodings.\n",
+    " - `lambd`: Corresponding coefficients.\n",
+    " - `lambd_bits`: Number of bits needed to represent coefficients precisely.\n",
+    " - `prepare`: If specified, oracle preparing $\\sum_i \\sqrt{|\\lambda_i|} |i\\rangle$ (state should be normalized and can have junk).\n",
+    " - `select`: If specified, oracle taking $|i\\rangle|\\psi\\rangle \\mapsto \\text{sgn}(\\lambda_i) |i\\rangle U_i|\\psi\\rangle$. \n",
+    "\n",
+    "#### Registers\n",
+    " - `system`: The system register.\n",
+    " - `ancilla`: The ancilla register (present only if bitsize > 0).\n",
+    " - `resource`: The resource register (present only if bitsize > 0). \n",
+    "\n",
+    "#### References\n",
+    " - [Quantum algorithms: A survey of applications and end-to-end complexities]( https://arxiv.org/abs/2310.03011). Dalzell et al. (2023). Ch. 10.2.\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "6e60f66d",
+   "metadata": {
+    "cq.autogen": "LinearCombination.bloq_doc.py"
+   },
+   "outputs": [],
+   "source": [
+    "from qualtran.bloqs.block_encoding import LinearCombination"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7c3b1b39",
+   "metadata": {
+    "cq.autogen": "LinearCombination.example_instances.md"
+   },
+   "source": [
+    "### Example Instances"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "7048adb0",
+   "metadata": {
+    "cq.autogen": "LinearCombination.linear_combination_block_encoding"
+   },
+   "outputs": [],
+   "source": [
+    "from qualtran.bloqs.basic_gates import Hadamard, TGate, XGate, ZGate\n",
+    "from qualtran.bloqs.block_encoding.unitary import Unitary\n",
+    "\n",
+    "linear_combination_block_encoding = LinearCombination(\n",
+    "    (Unitary(TGate()), Unitary(Hadamard()), Unitary(XGate()), Unitary(ZGate())),\n",
+    "    lambd=(0.25, -0.25, 0.25, -0.25),\n",
+    "    lambd_bits=1,\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a3c0e843",
+   "metadata": {
+    "cq.autogen": "LinearCombination.graphical_signature.md"
+   },
+   "source": [
+    "#### Graphical Signature"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "403264e7",
+   "metadata": {
+    "cq.autogen": "LinearCombination.graphical_signature.py"
+   },
+   "outputs": [],
+   "source": [
+    "from qualtran.drawing import show_bloqs\n",
+    "show_bloqs([linear_combination_block_encoding],\n",
+    "           ['`linear_combination_block_encoding`'])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a506b326",
+   "metadata": {
+    "cq.autogen": "LinearCombination.call_graph.md"
+   },
+   "source": [
+    "### Call Graph"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9df2892b",
+   "metadata": {
+    "cq.autogen": "LinearCombination.call_graph.py"
+   },
+   "outputs": [],
+   "source": [
+    "from qualtran.resource_counting.generalizers import ignore_split_join\n",
+    "linear_combination_block_encoding_g, linear_combination_block_encoding_sigma = linear_combination_block_encoding.call_graph(max_depth=1, generalizer=ignore_split_join)\n",
+    "show_call_graph(linear_combination_block_encoding_g)\n",
+    "show_counts_sigma(linear_combination_block_encoding_sigma)"
+   ]
   }
  ],
  "metadata": {